1. Introduction
Hyperkähler manifolds play a central role in algebraic geometry and complex geometry. However, very few examples of them are known and it is challenging to construct explicit examples of projective hyperkähler manifolds.
One of the most studied examples comes from K3 surfaces: the Hilbert schemes of points on K3 surfaces (see [Reference BeauvilleBea83]), or more generally, moduli spaces of stable sheaves on them (see [Reference MukaiMuk87]). More recently, instead of considering the moduli space of stable sheaves on K3 surfaces, people consider categories that behave homologically as actual K3 surfaces, known as noncommutative K3 surfaces or K3 categories, and construct projective hyperkähler varieties from them as Bridgeland moduli spaces of stable objects, provided suitable stability conditions are introduced.
An important source of K3 categories is provided by the Kuznetsov component of a cubic fourfold or a Gushel–Mukai fourfold/sixfold
$X$
, which is a quadric section of the projective cone over the Grassmannian
$\mathrm{Gr}(2,5)$
. The Kuznetsov component, denoted by
$\mathcal{K}u(X)$
, is the right orthogonal complement of a collection of rigid bundles on
$X$
. In [Reference Bayer, Lahoz, Macrì and StellariBLMS23, BLM+21, Reference Perry, Pertusi and ZhaoPPZ22], a family of Bridgeland stability conditions on
$\mathcal{K}u(X)$
is constructed and it is shown that the moduli spaces of semistable objects in
$\mathcal{K}u(X)$
are projective hyperkähler manifolds under natural assumptions.
In this paper, we mainly focus on Gushel–Mukai (GM) varieties
$X$
of dimension
$n \geq 3$
(see Section 4.1 for a review). There is a semi-orthogonal decomposition
\begin{equation*}\mathrm{D}^b(X)=\langle \mathcal{K}u(X),{\mathcal{O}}_X,\mathcal{U}^{\vee }_X,\ldots ,{\mathcal{O}}_X((n-3)H),\mathcal{U}_X^{\vee }((n-3)H)\rangle ,\end{equation*}
where
$\mathcal{U}_X$
is the pull-back of the tautological subbundle on
$\textrm {Gr}(2, 5)$
.
-
– If
$n=3$
or
$5$
, then
$\mathcal{K}u(X)$
is an Enriques category, i.e. its Serre functor is of the form
$T_X \circ [2]$
for a non-trivial involution
$T_X$
of
$\mathcal{K}u(X)$
. The numerical Grothendieck group of
$\mathcal{K}u(X)$
is of rank 2, generated by classes
$\lambda _1$
and
$\lambda _2$
, and there is a unique Serre-invariant stability conditionFootnote
1
on
$\mathcal{K}u(X)$
. -
– If
$n=4$
or
$6$
, then
$\mathcal{K}u(X)$
is a K3 category, i.e. its Serre functor is
$[2]$
. The numerical Grothendieck group of
$\mathcal{K}u(X)$
contains a canonical rank two sublattice, generated by classes
$\Lambda _1$
and
$\Lambda _2$
. All stability conditions on
$\mathcal{K}u(X)$
are Serre invariant, and there is a set of stability conditions
$\textrm {Stab}^{\circ }(\mathcal{K}u(X))$
constructed in [Reference Perry, Pertusi and ZhaoPPZ22, Theorem 4.12]. Moreover, when
$X$
is very general, there is a unique stability condition on
$\mathcal{K}u(X)$
(cf. Proposition 4.13).
1.1 Preservation of stability under push-forward and pull-back
Let
$j\colon Y \hookrightarrow X$
be a smooth hyperplane section of a GM variety
$X$
. The restriction of the derived pull-back to
$\mathcal{K}u(Y)$
yields the functor
\begin{equation*} j^* \colon \mathcal{K}u(X) \to \mathcal{K}u(Y). \end{equation*}
However, the image of the push-forward does not always lie in the Kuznetsov component, so we need to project it into
$\mathcal{K}u(X)$
. This gives us the functor
\begin{equation*} \mathrm{pr}_X \circ j_* \colon \mathcal{K}u(Y) \to \mathcal{K}u(X). \end{equation*}
See Section 4.1 for more details. The primary result of our paper proves the preservation of stability under both these functors.
Theorem 1.1 (Theorem 4.8). Let
$X$
be a GM variety of dimension
$n\geq 4$
and
$j \colon Y\hookrightarrow X$
be a smooth hyperplane section. Let
$\sigma _Y$
and
$\sigma _X$
be Serre-invariant stability conditions on
$\mathcal{K}u(Y)$
and
$\mathcal{K}u(X)$
, respectively. We additionally assume that whichever of
$X$
and
$Y$
has even dimensionality is considered to be very general.
-
(i) An object
$E\in \mathcal{K}u(Y)$
is
$\sigma _Y$
-semistable if and only if
$\mathrm{pr}_X(j_*E)\in \mathcal{K}u(X)$
is
$\sigma _X$
-semistable.
-
(ii) An object
$F\in \mathcal{K}u(X)$
is
$\sigma _X$
-semistable if and only if
$j^*F\in \mathcal{K}u(Y)$
is
$\sigma _Y$
-semistable.
We refer to Theorem 4.8 for a more complete version, describing when
$\mathrm{pr}_X(j_*E)$
or
$j^*F$
are stable. The above theorem can be viewed as a noncommutative analog of the stability of push-forward/pull-back objects for the embedding of a curve into a K3 surface.
Via a deformation argument, we can generalize Theorem 1.1(i) to general GM fourfolds.
Theorem 1.2 (Theorem 4.16). Let
$X$
be a general GM fourfold, and
$j \colon Y\hookrightarrow X$
be a smooth hyperplane section. Given
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
and a Serre-invariant stability condition
$\sigma _Y$
on
$\mathcal{K}u(Y)$
. If
$E\in \mathcal{K}u(Y)$
is a
$\sigma _Y$
-semistable object, then
$\mathrm{pr}_X(j_*E)$
is
$\sigma _X$
-semistable.
For a generalization of Theorem 1.1(ii), see Theorem 4.17. Once a construction of stability conditions of families of GM fivefolds and sixfolds is known, we can generalize both parts of Theorem 1.1 to general GM varieties. See Remark 4.18. We also conjecture that Theorem 1.1 holds generically for every GM fourfold and cubic fourfold (cf. Conjecture A.1).
Using Theorems 1.1 and 1.2, we can construct families of Lagrangian subvarieties as follows. We denote by
$M^X_{\sigma _X}(a,b)$
(respectively
$M^Y_{\sigma _Y}(a,b)$
) the moduli space that parameterizes S-equivalence classes of
$\sigma _X$
-semistable (respectively
$\sigma _Y$
-semistable) objects of class
$a\Lambda _1 +b\Lambda _2$
(respectively
$a\lambda _1 +b\lambda _2$
) in
$\mathcal{K}u(X)$
(respectively
$\mathcal{K}u(Y)$
). Then, by [Reference Perry, Pertusi and ZhaoPPZ22], for any pair of coprime integers
$a,b$
and a generic
$\sigma _X$
, the space
$M^X_{\sigma _X}(a,b)$
is a projective hyperkähler manifold. In this case, Theorems 1.1 and 1.2 induce a morphism
\begin{equation*}M^Y_{\sigma _Y}(a,b)\to M^X_{\sigma _X}(a,b),\quad E\mapsto \mathrm{pr}_X(j_*E)\end{equation*}
which is finite and unramified when
$Y\subset X$
is a general hyperplane section, and we show that its image is a Lagrangian subvariety of the hyperkähler manifold
$M^X_{\sigma _X}(a,b)$
(cf. Theorem 5.3). Therefore, as we vary the hyperplane section
$Y\subset X$
, we obtain a family of Lagrangian subvarieties of
$M^X_{\sigma _X}(a,b)$
, which can be arranged into a Lagrangian family as detailed below.
Theorem 1.3 (Theorem 5.8). Let
$a,b$
be a pair of coprime integers and
$X$
be a general GM fourfold. Then for any
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
generic
Footnote
2
with respect to
$a\Lambda _1+b\Lambda _2$
, there is a Lagrangian family (cf. Definition 5.6) of
$M^X_{\sigma _X}(a,b)$
over an open dense subset of
$|{\mathcal{O}}_X(H)|$
.
For GM sixfolds, there are similar constructions in Corollary 5.9. One may wonder if the Lagrangian families constructed above are Lagrangian covering families (cf. Definition 5.6), provided
$\dim M^X_{\sigma _X}(a,b)$
is small. We can verify this except in the case when
$\dim M^X_{\sigma _X}(a,b)=12$
, as discussed in Section 5.4. In a sequel [Reference Feyzbakhsh, Guo, Liu and ZhangFGLZ25], we demonstrate that all known constructions of Lagrangian covering families of hyperkähler manifolds coming from GM fourfolds or cubic fourfolds can be recovered by the categorical method described in Theorem 5.8.
In Section 6, we discuss two other applications of Theorems 1.1 and 1.2. We first confirm a conjecture proposed in [Reference Perry, Pertusi and ZhaoPPZ22, Section 5.4.3].
Corollary 1.4 (Corollary 6.1). Let
$X$
be a general GM fourfold. For any
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
generic with respect to
$\Lambda _1+2\Lambda _2$
, the functor
$\mathrm{pr}_X$
induces a rational map
\begin{equation*}X\dashrightarrow M^X_{\sigma _X}(1,2)\end{equation*}
which sends the structure sheaf at a general point to its projection to the Kuznetsov component. Moreover, the map is generically an embedding.
As a second application, we prove the projectivity of the moduli spaces of semistable objects in
$\mathcal{K}u(Y)$
of any class for a general GM threefold
$Y$
, which improves [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.3(2)] and confirms the expectation in [Reference Perry, Pertusi and ZhaoPPZ23, Remark 1.4] for general GM threefolds (which can be either ordinary or special).
Corollary 1.5 (Corollary 6.4). Let
$Y$
be a general GM threefold. For any pair of integers
$a,b$
and any Serre-invariant stability condition
$\sigma _Y$
on
$\mathcal{K}u(Y)$
, the moduli space of
$\sigma _Y$
-semistable objects of class
$a\lambda _1 +b\lambda _2$
is a projective scheme.
1.2 Lagrangian constant cycle subvarieties
As an application of studying a GM fourfold/sixfold
$X$
via its hyperplane sections, we can construct a Lagrangian constant cycle subvariety for each Bridgeland moduli space of stable objects in
$\mathcal{K}u(X)$
. Recall that there is an involution
$T_X$
defined on
$\mathcal{K}u(X)$
(cf. Lemma 4.3). And there is a family of stability conditions
$\textrm {Stab}^{\circ }(\mathcal{K}u(X))$
on
$\mathcal{K}u(X)$
constructed in [Reference Perry, Pertusi and ZhaoPPZ22, Theorem 4.12]. According to [Reference Bayer and PerryBP23, Remark 5.8] and [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.6], the functor
$T_X$
induces an anti-symplectic involution on
$M_{\sigma _X}^X(a,b)$
and the fixed locus
$\mathrm{Fix}(T_X)$
is a smooth Lagrangian subvariety of
$M_{\sigma _X}^X(a,b)$
.
Corollary 1.6 (Theorem 6.7). Let
$X$
be a GM fourfold or sixfold and
$a,b$
be a pair of coprime integers. For any stability condition
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
which is generic with respect to
$a\Lambda _1+b\Lambda _2$
, the fixed locus
$\mathrm{Fix}(T_X)$
is a Lagrangian constant cycle subvariety of
$M_{\sigma _X}^X(a,b)$
, i.e. all points of
$\mathrm{Fix}(T_X)$
are rationally equivalent in
$M_{\sigma _X}^X(a,b)$
.
The existence of Lagrangian constant cycle subvarieties for any projective hyperkähler manifold is conjectured by Voisin in [Reference VoisinVoi16, Conjecture 0.4]. Therefore, Corollary 1.6 provides an infinite series of locally complete
$20$
-dimensional families of hyperkähler varieties satisfying this conjecture. By [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.8(2)], this result can be regarded as a noncommutative version of [Reference BeckmannBec20, Proposition 4.2].
1.3 Related work
The idea of constructing Lagrangian subvarieties of hyperkähler varieties arising from cubic or GM fourfolds using their hyperplane sections has been realized in several classical results. For instance, see the Fano variety of lines of a cubic fourfold by [Reference VoisinVoi92] and the Lehn–Lehn–Sorger–van Straten (LLSvS) eightfold associated with a general cubic fourfold due to [Reference Shinder and SoldatenkovSS17, Proposition 6.9]. On the other hand, for a GM fourfold
$X$
, according to [Reference Iliev and ManivelIM11, Propositions 5.2 and 5.6], one can construct concrete Lagrangians of the double Eisenbud–Popescu–Walter (EPW) sextic either from hyperplane sections of
$X$
or fivefolds that contain
$X$
as a hyperplane section.
In the recent paper [Reference Perry, Pertusi and ZhaoPPZ23], the authors studied the fixed locus of the anti-symplectic involution of hyperkähler varieties constructed from the Kuznetsov components of GM fourfolds and showed that they are non-empty, and thus Lagrangian. The difference is that their method relies on the relation between the Kuznetsov component of a GM threefold and a special GM fourfold branched along it, so the Lagrangian subvariety can not vary in the family of hyperplane sections.
As for the hyperkähler varieties admitting Lagrangian covering families, the works of [Reference VoisinVoi16, Reference VoisinVoi22, Reference BaiBai23] exploit their geometry and establish the Lefschetz standard conjecture of degree two and other well-behaved cohomological properties.
The notion of Lagrangian constant cycle subvarieties was introduced in [Reference HuybrechtsHuy14] for the case of K3 surfaces. As a specific case of [Reference VoisinVoi16, Conjecture 0.4], it is predicted that there exists a Lagrangian constant cycle subvariety in any projective hyperkähler manifold. In [Reference LinLin20], the author constructs Lagrangian constant cycle subvarieties for hyperkähler varieties admitting a Lagrangian fibration. The existence of such subvarieties has also been verified for some cases, such as [Reference HuybrechtsHuy14, Reference VoisinVoi16, Reference BeckmannBec20, Reference ZhangZha24, Reference Li, Yu and ZhangLYZ23].
1.4 Plan of the paper
In Section 2, we provide the necessary definitions and properties of semi-orthogonal decompositions. Then we introduce the Kuznetsov components of several Fano manifolds and recall the construction of stability conditions on them. In Section 3, we prove Theorem 3.1, which shows that the stability of objects is preserved under push-forward and pull-back between triangulated categories satisfying additional assumptions. Then in Section 4, we apply Theorem 3.1 to non-Hodge-special GM varieties of dimension
$n\geq 4$
and prove Theorem 4.8. Then we obtain Theorem 4.16 via a deformation argument. In Section 5, we use Theorems 4.8 and 4.16 to construct Lagrangian families of those Bridgeland moduli spaces on the Kuznetsov components of non-Hodge-special GM fourfolds in Theorem 5.8. Next, in Section 5.4, we discuss when the family in Theorem 5.8 is a Lagrangian covering family.
In Section 6, we discuss other applications and prove Corollary 6.1, Corollary 6.4, and Theorem 6.7. Finally, in Appendix A, we state Conjecture A.1 and explain a general recipe to construct Lagrangian subvarieties of hyperkähler varieties as Bridgeland moduli spaces on GM fourfolds and cubic fourfolds.
1.5 Notation and conventions
-
– All triangulated categories are assumed to be
$\mathbb{C}$
-linear of finite type, i.e.
$\sum _{i\in \mathbb{Z}} \dim _{\mathbb{C}} \textrm {Ext}^i(E,F)$
is finite for any two objects
$E,F$
. -
– We use
$\hom$
and
$\textrm {ext}^{i}$
to represent the dimension of the vector spaces
$\textrm {Hom}$
and
$\textrm {Ext}^{i}$
. We denote
$\textrm {RHom}(-,-)=\bigoplus _{i\in \mathbb{Z}} \textrm {Hom}(-,-[i])[-i]$
and
$\chi (-,-)=\sum _{i\in \mathbb{Z}} (-1)^i \textrm {ext}^i(-,-)$
. -
– For a triangulated category
$\mathcal{T}$
, its Grothendieck group and numerical Grothendieck group are denoted by
$\mathrm{K}(\mathcal{T})$
and
$\mathrm{K}_{\mathrm{num}}(\mathcal{T}):=\mathrm{K}(\mathcal{T})/\ker (\chi )$
, respectively.
2. Preliminaries
In this section, we recall some basic definitions and properties of semi-orthogonal decompositions and the notion of Bridgeland stability conditions. We mainly follow [Reference BridgelandBri07] and [Reference Bayer, Lahoz, Macrì and StellariBLMS23].
2.1 Semi-orthogonal decompositions
Let
$\mathcal{T}$
be a triangulated category and
$\mathcal{D}\subset \mathcal{T}$
a full triangulated subcategory. We define the right orthogonal complement of
$\mathcal{D}$
in
$\mathcal{T}$
as the full triangulated subcategory
\begin{equation*} \mathcal{D}^\bot = \{ X \in \mathcal{T} \mid \textrm {Hom}(Y, X) =0 \text{ for all } Y \in \mathcal{D} \}. \end{equation*}
The left orthogonal complement is defined similarly as
\begin{equation*} {}^\bot \mathcal{D} = \{ X \in \mathcal{T} \mid \textrm {Hom}(X, Y) =0 \text{ for all } Y \in \mathcal{D} \}. \end{equation*}
We say a full triangulated subcategory
$\mathcal{D} \subset \mathcal{T}$
is admissible if the inclusion functor
$i \colon \mathcal{D} \hookrightarrow \mathcal{T}$
has left adjoint
$i^*$
and right adjoint
$i^!$
. Let
$( \mathcal{D}_1, \dots , \mathcal{D}_m )$
be a collection of admissible full subcategories of
$\mathcal{T}$
. We say that
$\mathcal{T} = \langle \mathcal{D}_1, \dots , \mathcal{D}_m \rangle$
is a semi-orthogonal decomposition of
$\mathcal{T}$
if
$\mathcal{D}_j \subset \mathcal{D}_i^\bot$
for all
$i \gt j$
, and the subcategories
$(\mathcal{D}_1, \dots , \mathcal{D}_m )$
generate
$\mathcal{T}$
, i.e. the category resulting from taking all shifts and cones in the categories
$(\mathcal{D}_1, \dots , \mathcal{D}_m )$
is equivalent to
$\mathcal{T}$
.
Let
$i\colon \mathcal{D} \hookrightarrow \mathcal{T}$
be an admissible full subcategory. Then the left mutation functor
${\boldsymbol {\mathrm{L}}}_{\mathcal{D}}$
through
$\mathcal{D}$
is defined as the functor lying in the canonical functorial exact triangle
\begin{equation*} i i^! \rightarrow \textrm {id} \rightarrow {\boldsymbol {\mathrm{L}}}_{\mathcal{D}} \end{equation*}
and the right mutation functor
$\boldsymbol {\mathrm{R}}_{\mathcal{D}}$
through
$\mathcal{D}$
is defined similarly by the triangle
\begin{equation*} \boldsymbol {\mathrm{R}}_{\mathcal{D}} \rightarrow \textrm {id} \rightarrow i i^*. \end{equation*}
Therefore,
${\boldsymbol {\mathrm{L}}}_{\mathcal{D}}$
is exactly the left adjoint functor of
$\mathcal{D}^\bot \hookrightarrow \mathcal{T}$
and, similarly,
$\boldsymbol {\mathrm{R}}_{\mathcal{D}}$
is the right adjoint functor of
${}^\bot \mathcal{D}\hookrightarrow \mathcal{T}$
. When
$E \in \mathcal{T}$
is an exceptional object, and
$F \in \mathcal{T}$
is any object, the left mutation
${\boldsymbol {\mathrm{L}}}_E F:={\boldsymbol {\mathrm{L}}}_{\langle E\rangle } F$
fits into the triangle
\begin{equation*} E \otimes \textrm {RHom}(E, F) \rightarrow F \rightarrow {\boldsymbol {\mathrm{L}}}_E F, \end{equation*}
and the right mutation
$\boldsymbol {\mathrm{R}}_E F:=\boldsymbol {\mathrm{R}}_{\langle E\rangle } F$
fits into the triangle
\begin{equation*} \boldsymbol {\mathrm{R}}_E F \rightarrow F \rightarrow E \otimes \textrm {RHom}(F, E)^\vee. \end{equation*}
Given a semi-orthogonal decomposition
$\mathcal{T} = \langle \mathcal{D}_1 , \mathcal{D}_2 \rangle$
,
$\mathcal{T} \simeq \langle S_{\mathcal{T}}(\mathcal{D}_2), \mathcal{D}_1 \rangle \simeq \langle \mathcal{D}_2, S^{-1}_{\mathcal{T}}(\mathcal{D}_1) \rangle$
are also semi-orthogonal decompositions of
$\mathcal{T}$
, where
$S_{\mathcal{T}}$
is the Serre functor of
$\mathcal{T}$
. Moreover, [Reference KuznetsovKuz19, Lemma 2.6] shows that
\begin{equation*} S_{\mathcal{D}_2} = \boldsymbol {\mathrm{R}}_{\mathcal{D}_1} \circ S_{\mathcal{T}} \, \, \, \text{ and } \, \, \, S_{\mathcal{D}_1}^{-1} = {\boldsymbol {\mathrm{L}}}_{\mathcal{D}_2} \circ S_{\mathcal{T}}^{-1} . \end{equation*}
2.2 Stability conditions
Let
$\mathcal{T}$
be a triangulated category and
$\mathrm{K}(\mathcal{T})$
be its Grothendieck group. Fix a surjective morphism to a finite rank lattice
$v \colon \mathrm{K}(\mathcal{T}) \rightarrow \Lambda$
.
Definition 2.1. A stability condition on
$\mathcal{T}$
is a pair
$\sigma = (\mathcal{A}, Z)$
, where
$\mathcal{A}$
is the heart of a bounded t-structure on
$\mathcal{T}$
and
$Z \colon \Lambda \rightarrow \mathbb{C}$
is a group homomorphism such that:
-
(i) the composition
$Z \circ v : \mathrm{K}(\mathcal{A}) \cong \mathrm{K}(\mathcal{T}) \rightarrow \mathbb{C}$
is a stability function on
$\mathcal{A}$
, i.e. for any
$E \in \mathcal{A}$
we have
$\textrm {Im} Z(v(E)) \geq 0$
and if
$\textrm {Im} Z(v(E)) = 0$
,
$\textrm {Re} Z(v(E)) \lt 0$
. From now on, we write
$Z(E)$
rather than
$Z(v(E))$
.
For any object
$E \in \mathcal{A}$
, we define the slope function
$\mu _{\sigma }(-)$
as
\begin{equation*} \mu _\sigma (E) := \begin{cases} - \frac {\textrm {Re} Z(E)}{\textrm {Im} Z(E)} & \text{for }\textrm {Im} Z(E) \gt 0 ,\\ + \infty, & \text{else}. \end{cases} \end{equation*}
An object
$0 \neq E \in \mathcal{A}$
is said to be
$\sigma$
-(semi)stable if, for any proper subobject
$F \subset E$
, we have
$\mu _\sigma (F) (\leq ) \mu _\sigma (E)$
.
-
(ii) Any object
$E \in \mathcal{A}$
has a Harder–Narasimhan filtration in terms of
$\sigma$
-semistability defined above. -
(iii) There exists a quadratic form
$Q$
on
$\Lambda \otimes \mathbb{R}$
such that
$Q|_{\ker Z}$
is negative definite and
$Q(E) \geq 0$
for all
$\sigma$
-semistable objects
$E \in \mathcal{A}$
. This is known as the support property.
The phase of a
$\sigma$
-semistable object
$E\in \mathcal{A}$
is defined as
\begin{equation*}\phi _{\sigma }(E):=\frac {1}{\pi }\mathrm{arg}(Z(E))\in (0,1].\end{equation*}
For
$n\in \mathbb{Z}$
, we set
$\phi _{\sigma }(E[n]):=\phi _{\sigma }(E)+n$
. Given a non-zero object
$E\in \mathcal{T}$
, we will denote by
$\phi ^+_{\sigma }(E)$
(respectively
$\phi ^-_{\sigma }(E)$
) the biggest (respectively smallest) phase of a Harder–Narasimhan factor of
$E$
with respect to
$\sigma$
.
A slicing
$\mathcal{P}_{\sigma }$
of
$\mathcal{T}$
with respect to the stability condition
$\sigma$
consists of full additive subcategories
$\mathcal{P}_{\sigma }(\phi ) \subset \mathcal{T}$
for each
$\phi \in \mathbb{R}$
such that the subcategory
$\mathcal{P}_{\sigma }(\phi )$
contains the zero object and all
$\sigma$
-semistable objects whose phase is
$\phi$
. For any interval
$I\subset \mathbb{R}$
, we denote by
$\mathcal{P}_{\sigma }(I)$
the category given by the extension closure of
$\{\mathcal{P}_{\sigma }(\phi )\}_{\phi \in I}$
. We will use both the notation
$\sigma = (\mathcal{A},Z)$
and
$\sigma = (\mathcal{P}_{\sigma },Z)$
for a stability condition
$\sigma$
with heart
$\mathcal{A} = \mathcal{P}_{\sigma }((0,1])$
.
We finish this section with the following definition.
Definition 2.2. Let
$\mathcal{T}$
be a triangulated category and
$\Phi$
be an auto-equivalence of
$\mathcal{T}$
. We say a stability condition
$\sigma$
on
$\mathcal{T}$
is
$\Phi$
-invariantFootnote
3
if
\begin{equation*}\Phi \cdot \sigma =\sigma \cdot \widetilde {g}\end{equation*}
for an element
$\widetilde {g}\in \widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
. We say
$\sigma$
is Serre invariant if it is
$S_{\mathcal{T}}$
-invariant, where
$S_{\mathcal{T}}$
is the Serre functor of
$\mathcal{T}$
.
3. Stability of push-forward and pull-back objects
In this section, we are going to prove Theorem 3.1, which will be applied to the Kuznetsov components of GM varieties in Section 4. We consider two triangulated categories
$\mathcal{K}_1$
and
$\mathcal{K}_2$
with the following properties.
-
(C1) We have
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}_1) = \mathbb{Z} \lambda _1 \oplus \mathbb{Z} \lambda _2$
whose Euler pairing is(1)
\begin{equation} \left [ \begin{array}{c@{\quad}c} -1 & 0 \\ 0 & -1\\ \end{array} \right ]. \end{equation}
-
(C2) We have
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}_2) = \mathbb{Z} \Lambda _1 \oplus \mathbb{Z} \Lambda _2$
whose Euler pairing is(2)
\begin{equation} \left [ \begin{array}{c@{\quad}c} -2 & 0 \\ 0 & -2\\ \end{array} \right ]. \end{equation}
-
(C3) There exist exact auto-equivalences
$T_i$
of
$\mathcal{K}_i$
for
$i=1, 2$
such that
$T_i^2= \mathrm{id}_{\mathcal{K}_i}$
and
$T_i$
acts trivially on
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}_i)$
. -
(C4) We have the Serre functors
$S_{\mathcal{K}_2} = [2]$
and
$S_{\mathcal{K}_1} = T_1\circ [2]$
. -
(C5) We have exact functors
with adjoint pairs
\begin{equation*}\Phi ^*\colon \mathcal{K}_2\to \mathcal{K}_1,\quad \Phi _*\colon \mathcal{K}_1\to \mathcal{K}_2\end{equation*}
$\Phi _* \circ T_1 \dashv \Phi ^* \dashv \Phi _*$
so that
$\Phi _*(\lambda _i) = \Lambda _i$
and
$\Phi ^*(\Lambda _i) = 2\lambda _i$
for
$i=1, 2$
. Moreover, for any object
$F \in \mathcal{K}_2$
, we have
$\Phi ^*T_2(F) \cong T_1(\Phi ^*F)$
.
-
(C6) There are stability conditions
$\sigma _i =(Z_i, \mathcal{A}_i)$
on
$\mathcal{K}_i$
for
$i=1, 2$
such that:-
(a)
$\sigma _i$
is
$T_i$
-invariant; -
(b)
$Z_1(\lambda _1) = Z_2(\Lambda _1) =-1$
and
$Z_1(\lambda _2) = Z_2(\Lambda _2) =\mathfrak{i}$
; -
(c) there are infinitely many
$\sigma _1$
-stable objects of class
$\lambda _i$
for each
$i=1, 2$
; -
(d) for any
$\sigma _1$
-stable object
$E \in \mathcal{K}_1$
of class
$\lambda _i$
for
$i=1, 2$
, we know
$\Phi _*E$
is
$\sigma _2$
-stable with
$\phi _{\sigma _1}(E)=\phi _{\sigma _2}(\Phi _*E)$
.
-
-
(C7) Any object
$E \in \mathcal{K}_1$
lies in an exact triangle(3)
\begin{equation} T_1(E) \to \Phi ^*\Phi _*E \to E. \end{equation}
-
(C8) Any object
$E \in \mathcal{K}_2$
lies in an exact triangle(4)
\begin{equation} E \to \Phi _*\Phi ^*E \to T_2(E). \end{equation}
Moreover, for simplicity, we assume that
$\mathcal{K}_i$
for each
$i=1,2$
is a full triangulated subcategory of the derived category of coherent sheaves on a smooth variety.Footnote
4
We denote the phase function and the slope function of
$\sigma _i$
by
$\phi _i(-)$
and
$\mu _i(-)$
, respectively.
The main theorem that we prove in this section is as follows.
Theorem 3.1.
Consider triangulated categories
$\mathcal{K}_1$
and
$\mathcal{K}_2$
satisfying the above conditions (C1) to (C8).
-
(i) An object
$E \in \mathcal{K}_1$
is
$\sigma _1$
-semistable of phase
$\phi$
if and only if
$\Phi _*E$
is
$\sigma _2$
-semistable of phase
$\phi$
. Moreover,
$\Phi _*(E)$
is
$\sigma _2$
-stable if and only if
$E$
is
$\sigma _1$
-stable and for any
$\sigma _2$
-stable object
$F$
we have
$\Phi ^*F\neq E$
. -
(ii) An object
$F \in \mathcal{K}_2$
is
$\sigma _2$
-semistable of phase
$\phi$
if and only if
$\Phi ^*F$
is
$\sigma _1$
-semistable of phase
$\phi$
. Moreover,
$\Phi ^*(F)$
is
$\sigma _1$
-stable if and only if
$F$
is
$\sigma _2$
-stable and for any
$\sigma _1$
-stable object
$E$
we have
$\Phi _*E\neq F$
.
3.1 Collection of lemmas
Before proving the theorem, we investigate some of the further properties of categories
$\mathcal{K}_1$
and
$\mathcal{K}_2$
.
Lemma 3.2.
Let
$\sigma$
be a stability condition on a triangulated category
$\mathcal{T}$
. Assume that
$T$
is an exact auto-equivalence of
$\mathcal{T}$
with
$T^n=\mathrm{id}_{\mathcal{T}}$
and
$\sigma$
is
$T$
-invariant. Then for any
$\sigma$
-(semi)stable object
$E\in \mathcal{T}$
, we have
$T(E)$
is
$\sigma$
-(semi)stable with
$\phi _{\sigma }(E)=\phi _{\sigma }(T(E))$
.
Proof.
As
$\sigma$
is
$T$
-invariant,
$E$
is
$\sigma$
-(semi)stable if and only if
$T(E)$
is
$\sigma$
-(semi)stable. It remains to show
$\phi _{\sigma }(E)=\phi _{\sigma }(T(E))$
. Assume that
$T\cdot \sigma =\sigma \cdot \widetilde {g}$
for
$\widetilde {g}=(M, g)\in \widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
, where
$M\in \mathrm{GL}^+(2, \mathbb{R})$
and
$g\colon \mathbb{R}\to \mathbb{R}$
is an increasing function such that
$g(x+1)=g(x)+1$
. As
$T^n=\mathrm{id}_{\mathcal{T}}$
, we have
\begin{equation*}\phi _{\sigma }(E)=\phi _{\sigma }(T^n(E))=g(\phi _{\sigma }(T^{n-1}(E)))=\cdots =g^n(\phi _{\sigma }(E)).\end{equation*}
As
$g$
is an increasing function, we obtain
$\phi _{\sigma }(E)=\phi _{\sigma }(T(E))=g(\phi _{\sigma }(E))$
.
Therefore, by Lemma 3.2, (C7) and (C8), we have:
-
– for any
$\sigma _1$
-semistable object
$E\in \mathcal{K}_1$
,
$\Phi ^*\Phi _*E$
is also
$\sigma _1$
-semistable with
\begin{equation*}\phi _1(E)=\phi _1(\Phi ^*\Phi _*E);\end{equation*}
-
– for any
$\sigma _2$
-semistable object
$F\in \mathcal{K}_2$
,
$\Phi _*\Phi ^*F$
is also
$\sigma _2$
-semistable with
\begin{equation*}\phi _2(F)=\phi _2(\Phi _*\Phi ^*F).\end{equation*}
The next lemma simplifies the statement that we need to prove in Theorem 3.1.
Lemma 3.3.
Consider triangulated categories
$\mathcal{K}_1$
and
$\mathcal{K}_2$
satisfying the above conditions (C1) to (C8). Assume that:
-
(i) if
$E \in \mathcal{K}_1$
is
$\sigma _1$
-semistable, then
$\Phi _*E$
is
$\sigma _2$
-semistable with
$\phi _{1}(E)=\phi _2(\Phi _*E)$
; and
-
(ii) if
$F \in \mathcal{K}_2$
is
$\sigma _2$
-semistable, then
$\Phi ^*F$
is
$\sigma _1$
-semistable with
$\phi _{2}(F)=\phi _1(\Phi ^*F)$
.
Then Theorem 3.1 holds.
Proof.
Given an object
$E \in \mathcal{K}_1$
such that
$\Phi _*E$
is
$\sigma _2$
-semistable. If
$A\to E\to B$
is an exact triangle such that
$\phi ^-_1(A)\gt \phi ^+_1(B)$
, then we have
$\phi ^-_2(\Phi _*A)\gt \phi ^+_2(\Phi _*B)$
by (i), which contradicts the semistability of
$\Phi _*E$
as we have an exact triangle
$\Phi _*A\to \Phi _*E\to \Phi _*B$
. Therefore, the semistability of
$\Phi _*E$
implies the semistability of
$E$
and they have the same phase by (i). Similarly, the semistability of
$\Phi ^*F$
implies the semistability of
$F$
for any object
$F\in \mathcal{K}_2$
and they have the same phase by (ii). This proves the semistability part of Theorem 3.1.
Now we assume that
$E \in \mathcal{K}_1$
is
$\sigma _1$
-semistable such that
$\Phi _*E$
is
$\sigma _2$
-stable. Then, by (i), it is clear that
$E$
is
$\sigma _1$
-stable. Indeed, if
$E$
is strictly
$\sigma _1$
-semistable, then we can find a
$\sigma _1$
-stable object
$E'$
with
$\textrm {Hom}(E', E)\neq 0$
and
$\phi _1(E)=\phi _1(E')$
. Applying
$\textrm {Hom}(-,E)$
to the triangle (3) associated with
$E'$
and using
$\phi _1(E')=\phi _1(T_1(E'))$
, we get
\begin{equation*}\textrm {Hom}(\Phi ^*\Phi _*(E'),E)=\textrm {Hom}(\Phi _*(E'),\Phi _*(E))\neq 0,\end{equation*}
which contradicts the stability of
$\Phi _*E$
as we have
$\phi _2(\Phi _*(E'))=\phi _1(E')=\phi _1(E)=\phi _2(\Phi _*(E))$
by the assumption (i). Now if there is a
$\sigma _1$
-stable object
$F$
such that
$\Phi ^*F\cong E$
, then, by (4) and Lemma 3.2, we know that
$\Phi _*\Phi ^*(F)\cong \Phi _*(E)$
is strictly
$\sigma _2$
-semistable, a contradiction.
Next, assume that
$E \in \mathcal{K}_1$
is
$\sigma _1$
-semistable such that
$E$
is
$\sigma _1$
-stable and for any
$\sigma _2$
-stable object
$F$
we have
$\Phi ^*F\neq E$
, we need to prove the
$\sigma _2$
-stability of
$\Phi _*E$
. Up to shift, we can assume that
$\Phi _*E\in \mathcal{A}_2$
. If
$\Phi _*E$
is strictly
$\sigma _2$
-semistable, let
$\{F_k\}_{k\in I}$
be the set of Jordan–Hölder factors of
$\Phi _*E$
, where
$I$
is a finite index set. By (3) and Lemma 3.2,
$\Phi ^*\Phi _*E$
is strictly
$\sigma _1$
-semistable and the Jordan–Hölder factors are
$E$
and
$T_1(E)$
. As
$\Phi ^*F_k$
is
$\sigma _1$
-semistable with
$\phi _1(\Phi ^*F_k)=\phi _1(E)=\phi _2(\Phi _*E)$
for each
$k\in I$
by (ii), there exists
$k\in I$
such that
$\Phi ^*F_k\cong E$
by the stability of
$E$
and the uniqueness of Jordan–Hölder factors, which makes a contradiction. This proves Theorem 3.1(i). The proof of Theorem 3.1(ii) is similar.
Our approach to Theorem 3.1 is by induction on
$\textrm {ext}^1$
. Therefore, the first step is to control
$\textrm {ext}^1$
in several situations.
Lemma 3.4 (Weak Mukai Lemma). Let
$A\to E\to B$
be an exact triangle in a triangulated category
$\mathcal{T}$
. If
$\textrm {Hom}(A,B)=\textrm {Hom}(B,A[2])=0$
, then
\begin{equation*}\textrm {ext}^1(A,A)+\textrm {ext}^1(B,B)\leq \textrm {ext}^1(E,E).\end{equation*}
Proof. The result follows from the same argument as in [Reference Bayer and BridgelandBB17, Lemma 2.5].
Lemma 3.5.
Consider triangulated categories
$\mathcal{K}_1$
and
$\mathcal{K}_2$
and stability conditions
$\sigma _1$
and
$\sigma _2$
satisfying the above conditions.
-
(i) Any non-zero object
$E\in \mathcal{K}_i$
satisfies
$\textrm {ext}^1(E,E)\geq 2\,i$
for each
$i=1, 2$
. -
(ii) The heart
$\mathcal{A}_i$
has homological dimension
$2$
, i.e.
$\textrm {Hom}(A,B[k])=0$
for any
$k\gt 2$
and any two objects
$A,B\in \mathcal{A}_i$
.
Proof.
As
$\sigma _i$
is Serre invariant for each
$i=1, 2$
, part (ii) follows from [Reference Feyzbakhsh and PertusiFP23, Proposition 3.4]. Now by (ii), (C1), and (C2), every
$\sigma _i$
-stable object
$E\in \mathcal{K}_i$
satisfies
$\textrm {ext}^1(E,E)\geq 2i$
. Then part (i) follows from applying [Reference Feyzbakhsh and PertusiFP23, Proposition 3.4(b)].
Lemma 3.6.
For each
$i=1, 2$
, we have the following:
-
(i) Any object
$E \in \mathcal{K}_i$
satisfies
In particular, if
\begin{equation*}\sum _k \textrm {ext}^1(\mathcal{H}^k_{\mathcal{A}_i}(E), \mathcal{H}^k_{\mathcal{A}_i}(E))\leq \textrm {ext}^1(E,E).\end{equation*}
$E$
is not in
$\mathcal{A}_i[m]$
for any
$m \in \mathbb{Z}$
, then
$\textrm {ext}^1(\mathcal{H}^k_{\mathcal{A}_i}(E), \mathcal{H}^k_{\mathcal{A}_i}(E))\leq \textrm {ext}^1(E,E)-2\,i$
for each
$k$
.
-
(ii) Let
$\{E_j\}_{j\in I}$
be the set of the Harder–Narasimhan factors of
$E$
. Then
In particular, if
\begin{equation*}\sum _{j\in I} \textrm {ext}^1(E_j, E_j)\leq \textrm {ext}^1(E,E).\end{equation*}
$E$
is not
$\sigma _i$
-semistable, then
$\textrm {ext}^1(E_j, E_j)\leq \textrm {ext}^1(E,E)-2\,i$
for each
$j\in I$
.
-
(iii) If
$E$
is strictly
$\sigma _i$
-semistable, then each of the Jordan–Hölder factors
$A$
of
$E$
satisfies
\begin{equation*}\textrm {ext}^1(A, A)\lt \textrm {ext}^1(E,E).\end{equation*}
Proof.
For part (i), we can assume that
$E$
has at least two non-zero cohomology objects with respect to
$\mathcal{A}_i$
. Let
$k_0$
be the minimum value such that
$\mathcal{H}^{k_0}_{\mathcal{A}_i}(E) \neq 0$
. This gives rise to an exact triangle
$\mathcal{H}_{\mathcal{A}_i}^{k_0}(E)[-k_0] \to E \to E'$
, where
$E' \in \mathcal{P}_{\sigma _i}(-\infty , -k_0]$
. Consequently, according to (C6), we can deduce that
$T_1(E') \in \mathcal{P}_{\sigma _i}(-\infty , -k_0]$
. As
$S_{\mathcal{K}_1} = T_1\circ [2]$
, the conditions of Lemma 3.4 are satisfied for this exact triangle, yielding
\begin{equation*} \textrm {ext}^1(E', E') + \textrm {ext}^1(\mathcal{H}^{k_0}_{\mathcal{A}_i}(E), \mathcal{H}^{k_0}_{\mathcal{A}_i}(E)) \leq \textrm {ext}^1(E, E). \end{equation*}
Thus, the first claim in part (i) follows by induction on the number of non-zero cohomologies, and the second claim in part (i) follows by combining it with Lemma 3.5. Similarly, part (ii) can be proved via induction on the length of the Harder–Narasimhan filtration.
Finally, we prove (iii). If all Jordan–Hölder factors of
$E$
have the same class
$v$
, then for any Jordan–Hölder factor
$A$
of
$E$
, we have
$\chi (A,A)\leq 2-\textrm {ext}^1(A,A)$
and
$[E]=m[A]$
for an integer
$m\geq 2$
. Hence,
$\textrm {ext}^1(E,E)=\hom (E,E)+\textrm {ext}^2(E,E)-\chi (E,E)\geq 1-m^2\chi (A,A)$
. Then
\begin{equation*}\textrm {ext}^1(A,A)\leq 2-\chi (A,A)\lt 1-m^2\chi (A,A),\end{equation*}
as
$m^2\geq 4$
and
$\chi (A,A)\leq -1$
. If
$E$
has at least two non-isomorphic Jordan–Hölder factors, then we can find an exact triangle
$E_1\to E\to E_2$
such that all Jordan–Hölder factors of
$E_1$
are isomorphic with
$\phi _{\sigma _i}(E_1)=\phi _{\sigma _i}(E_2)$
and
$\textrm {Hom}(E_1, E_2)=0$
.
When
$i=2$
, we have
$\textrm {Hom}(E_2, E_1[2])=0$
, which gives
$\textrm {ext}^1(E_1, E_1)+\textrm {ext}^1(E_2, E_2)\leq \textrm {ext}^1(E,E)$
by Lemma 3.4. By the first case, we know that (iii) holds for
$E_1$
and, hence, we only need to prove (iii) for
$E_2$
. But as the number of Jordan–Hölder factors of
$E_2$
is less than
$E$
and
$\textrm {ext}^1(E_2, E_2)\lt \textrm {ext}^1(E, E)$
by Lemma 3.4 and Lemma 3.5, the statement for
$E_2$
follows from the induction on the number of Jordan–Hölder factors.
Now we assume that
$i=1$
. If
$\textrm {Hom}(E_2, E_1[2])=\textrm {Hom}(E_1, T_1(E_2))=0$
, then the argument is the same as above. If
$\textrm {Hom}(E_1, T_1(E_2))\neq 0$
, letting
$A$
be the Jordan–Hölder factor of
$E_1$
, then
$E_2$
has
$T_1(A)$
as a subobject and a Jordan–Hölder factor. Then we have an exact triangle
$E_3\to E_2\to E_4$
such that all Jordan–Hölder factors of
$E_3$
are isomorphic to
$T_1(A)$
with
$\phi _{1}(E_3)=\phi _{1}(E_4)$
and
$\textrm {Hom}(E_3, E_4)=0$
. Let
$E_5:=\textrm {cone}(E\to E_4)[-1]$
. Then
$E_5$
is an extension of
$E_1$
and
$E_3$
and, hence, every Jordan–Hölder factor of
$E_5$
is isomorphic to
$A$
or
$T_1(A)$
. Moreover,
$\textrm {Hom}(E_5, E_4)=0$
and
$\textrm {Hom}(E_4, E_5[2])=\textrm {Hom}(E_5,T_1(E_4))=0$
since
$A$
and
$T_1(A)$
are not Jordan–Hölder factors of
$E_4$
by construction. Then, by Lemma 3.4, we get
$\textrm {ext}^1(E_5, E_5)+\textrm {ext}^1(E_4, E_4)\leq \textrm {ext}^1(E, E)$
and the remaining argument is similar to the
$i=2$
case.
Using exact triangles (3) and (4), we are able to bound
$\textrm {ext}^1$
after acting by
$\Phi _*$
and
$\Phi ^*$
.
Lemma 3.7.
Let
$E \in \mathcal{K}_1$
be a
$\sigma _1$
-stable object with
$\textrm {Ext}^1(E, E) = \mathbb{C}^n$
. If
$T_1(E) \neq E$
, then
\begin{equation*} \textrm {RHom}(\Phi _*E, \Phi _*E) = \mathbb{C} \oplus \mathbb{C}^{2n}[-1] \oplus \mathbb{C}[-2]. \end{equation*}
Otherwise, either:
-
(i)
$\textrm {RHom}(\Phi _*E, \Phi _*E) = \mathbb{C} \oplus \mathbb{C}^{2n-2}[-1] \oplus \mathbb{C}[-2]$
; or
-
(ii)
$ \textrm {RHom}(\Phi _*E, \Phi _*E) = \mathbb{C}^2 \oplus \mathbb{C}^{2n}[-1] \oplus \mathbb{C}^2[-2]$
.
Moreover, case (ii) happens if and only if
$T_1(E)=E$
and the triangle (3) is splitting.
Proof.
We know
$\textrm {Hom}(E, E[2]) = \textrm {Hom}(E, T_1(E))^{\vee }$
, which is
$\mathbb{C}$
if
$T_1(E) \cong E$
and zero otherwise. Hence,
$\chi (\Phi _*E, \Phi _*E) = 2\chi (E, E)$
is equal to
$2(n-1)$
if
$T_1(E) \neq E$
and
$2(n-2)$
otherwise. Thus, the claim follows from applying
$\textrm {Hom}(-, E)$
to the exact triangle (3) as
$\textrm {RHom}(\Phi _*E, \Phi _*E) = \textrm {RHom}(\Phi ^*\Phi _*E, E)$
. Note that
$\textrm {Hom}(\Phi _*E, \Phi _*E[2]) = \textrm {Hom}(\Phi _*E, \Phi _*E)^{\vee }$
as
$\mathcal{K}_2$
is
$2$
-Calabi–Yau.
A similar argument and (C8) imply the following.
Lemma 3.8.
Let
$E \in \mathcal{K}_2$
be a
$\sigma _2$
-stable object with
$\textrm {Ext}^1(E, E) = \mathbb{C}^{2n}$
. Then either:
-
(i)
$\textrm {RHom}(\Phi ^*E, \Phi ^*E) = \mathbb{C} \oplus \mathbb{C}^{4n-3+\delta }[-1] \oplus \mathbb{C}^{\delta }[-2]$
; or
-
(ii)
$\textrm {RHom}(\Phi ^*E, \Phi ^*E) = \mathbb{C}^2 \oplus \mathbb{C}^{4n-1+\delta }[-1] \oplus \mathbb{C}^{1+\delta }[-2]$
for
$\delta = 0$
or
$1$
. Moreover, case (ii) happens if and only if
$E\cong T_2(E)$
and the triangle (4) is splitting.
In the following, for
$i=1, 2$
, let
$A_i$
denote any
$\sigma _1$
-stable object of class
$\lambda _i$
in the heart
$\mathcal{A}_1$
. By (C6)(d),
$\Phi _*A_i \in \mathcal{A}_2$
is
$\sigma _2$
-stable with
$\phi _{\sigma _2}(\Phi _*A_i) = \phi _{\sigma _1}(A_i)$
.
Lemma 3.9.
Let
$E \in \mathcal{A}_1$
(respectively
$F \in \mathcal{A}_2$
) be a semistable object with respect to
$\sigma _1$
(respectively
$\sigma _2$
) of phase
$\phi$
. Then
\begin{equation*} a\lt \phi ^+_2(\Phi _*E) \ , \ \phi ^-_2(\Phi _*E)\lt b \quad \text{and} \quad a\lt \phi ^+_1(\Phi ^*F) \ , \ \phi ^-_1(\Phi ^*F)\lt b, \end{equation*}
where:
-
(i)
$a = -\frac {1}{2}$
and
$b=1$
if
$0\lt \phi \lt \frac {1}{2}$
; -
(ii)
$a = -\frac {1}{2}$
and
$b= \frac {3}{2}$
if
$\phi = \frac {1}{2}$
; -
(iii)
$a =0$
and
$b= \frac {3}{2}$
if
$\frac {1}{2}\lt \phi \lt 1$
; and
-
(iv)
$a =0$
and
$b= 2$
if
$\phi =1$
.
Proof.
Firstly, we consider a
$\sigma _1$
-semistable object
$E \in \mathcal{A}_1$
. If
$0\lt \phi _1(E) = \phi \lt \frac {1}{2}$
, we have:
-
(i)
$\chi (\lambda _1, E)=\chi (E, \lambda _1)=\textrm {Re}[Z_1(E)] \gt 0$
; and -
(ii)
$\chi (\lambda _2, E)=\chi (E, \lambda _2)=-\textrm {Im}[Z_1(E)] \lt 0$
.
As
$\phi _{1}(A_1)=1\geq \phi _1(E)$
, by the finiteness of Jordan–Hölder factors, there are only at most finitely many
$\sigma _1$
-stable objects
$A_1\in \mathcal{A}_1$
such that
$\textrm {Hom}(E, A_1[2])=\textrm {Hom}(T_1(A_1), E)\neq 0$
. Then, by Lemma 3.5(ii), (i) gives
$\textrm {Hom}(E, A_1) \neq 0$
for every but finitely many
$\sigma _1$
-stable object
$A_1\in \mathcal{A}_1$
with
$[A_1]=\lambda _1$
. Thus applying
$\textrm {Hom}(-, A_1)$
to (3) implies
$\textrm {Hom}(\Phi ^*\Phi _*E, A_1) = \textrm {Hom}(\Phi _*E, \Phi _*A_1) \neq 0$
. By (C6)(c), (d) and the finiteness of Jordan–Hölder factors, we obtain
$\phi ^-_2(\Phi _*E) \lt 1$
. At the same time, (ii) implies that
$\textrm {Hom}(A_2, E[1]) \neq 0.$
Since
$\textrm {Hom}(T_1(A_2), E) = 0$
by
$\phi \lt \frac {1}{2}$
, applying
$\textrm {Hom}(-, E)$
to (3), we get
$\textrm {Hom}(\Phi ^*\Phi _*A_2, E[1]) = \textrm {Hom}(\Phi _*A_2, \Phi _*E[1]) \neq 0$
, which gives
$-\frac {1}{2} \lt \phi ^+_2(\Phi _*E)$
.
If
$\frac {1}{2}\lt \phi _1(E) =\phi \lt 1$
, we have:
-
(i)
$\chi (\lambda _1, E)=\chi (E, \lambda _1)=\textrm {Re}[Z_1(E)] \lt 0$
; and -
(ii)
$\chi (\lambda _2, E)=\chi (E, \lambda _2)=-\textrm {Im}[Z_1(E)] \lt 0$
.
Then a similar argument as above gives
$0\lt \phi _2^+(\Phi _*E)$
and
$\phi _2^-(\Phi _*E) \lt \frac {3}{2}$
. If
$\phi _1(E) = \frac {1}{2}$
, then we have
$\chi (\lambda _2, E) =\chi (E, \lambda _2) \lt 0$
. Thus
$\textrm {Hom}(E, A_2[1]) \neq 0$
for every
$A_2$
. But from the finiteness of Jordan–Hölder factors,
$\textrm {Hom}(T_1(E), A_2) = 0$
for every but finitely many
$A_2$
, so
$\textrm {Hom}(\Phi ^*\Phi _*E, A_2[1]) \neq 0$
for infinitely many
$A_2$
, which gives
$\phi _2^-(\Phi _*E) \lt \frac {3}{2}$
. Also we have
$\textrm {Hom}(A_2, E[1]) \neq 0$
and
$\textrm {Hom}(T_1(A_2), E)=0$
for infinitely many
$A_2$
, which implies
$-\frac {1}{2} \lt \phi _2^+(\Phi _*E)$
. If
$\phi _1(E) =1$
, then
$\chi (\lambda _1, E) = \chi (E, \lambda _1) \lt 0$
. Then a similar argument as above gives
$\phi _2^-(\Phi _*E) \lt 2$
and
$0\lt \phi _2^+(\Phi _*E)$
.
The second part for
$\phi _1^{\pm }(\Phi ^*F)$
where
$F \in \mathcal{A}_2$
is
$\sigma _2$
-semistable follows via the same argument as above. We only need to use
$\Phi _*A_i$
for
$i=1, 2$
and the exact triangle (4) instead.
3.2 Induction argument
Now we are prepared to prove the main theorem of this section.
Proof of Theorem 3.1
. By Lemma 3.3, we only need to show the following two claims for any
$n\gt 0$
.
$\mathbf{A}_n$
: If
$E \in \mathcal{K}_1$
is
$\sigma _1$
-stable with
$\textrm {ext}^1(E, E) \leq n$
, then
$\Phi _*E$
is
$\sigma _2$
-semistable with the same phase
$\phi _{2}(\Phi _*E) = \phi _{1}(E)$
.
$\mathbf{B}_n$
: If
$F \in \mathcal{K}_2$
is
$\sigma _2$
-stable with
$\textrm {ext}^1(F, F) \leq n$
, then
$\Phi ^*F$
is
$\sigma _1$
-semistable with the same phase
$\phi _{1}(\Phi ^*F) = \phi _{2}(F)$
.
We use a double induction argument to prove the above two statements. According to (C6)(d), the assertions
$\textbf {A}_n$
are automatically satisfied when
$n = 3$
, as any
$\sigma _1$
-stable object
$A$
with
$\textrm {ext}^1(A,A)\leq 3$
is of class
$\lambda _i$
. Then the extension to arbitrary
$n$
follows from Lemmas 3.11 and 3.14.
It remains to prove Lemmas 3.11 and 3.14, which is the goal of the rest of this section. The proof of each lemma is divided into two steps. For example, in the first step of Lemma 3.11, we prove
$\Phi _*(E)\in \mathcal{A}_2$
for any
$\sigma _1$
-stable object
$E\in \mathcal{A}_1$
. Then, in the second step, the
$\sigma _2$
-semistability of
$\Phi _*(E)$
will be deduced from a slope-comparison argument.
Remark 3.10. Using Lemma 3.6(ii) and (iii), we have the following equivalent forms of
$\mathbf{A}_n$
and
$\mathbf{B}_n$
by reducing to the case when
$E$
and
$F$
are stable.
$\mathbf{A}_n$
: If
$E \in \mathcal{K}_1$
is a non-zero object with
$\textrm {ext}^1(E, E) \leq n$
, then
$\Phi _*$
preserves the Harder–Narasimhan filtration of
$E$
.
$\mathbf{B}_n$
: If
$F \in \mathcal{K}_2$
is a non-zero object with
$\textrm {ext}^1(F, F) \leq n$
, then
$\Phi ^*$
preserves the Harder–Narasimhan filtration of
$F$
.
Here, preserving the Harder–Narasimhan filtration through a functor entails that the Harder–Narasimhan filtration of the image corresponds to the image of the Harder–Narasimhan filtration of the original object, and the components retain the same phases.
Lemma 3.11.
For any
$n \gt 1$
, the validity of
$\mathbf{B}_{n-1}$
implies
$\mathbf{A}_n$
.
Proof.
Take a
$\sigma _1$
-stable object
$E$
with
$\textrm {ext}^1(E, E) \leq n$
. Up to shift, we can assume that
$E\in \mathcal{A}_1$
. Firstly, we claim that we can furthermore assume that
$0 \lt \phi _1(E) \leq \frac {1}{2}$
. If
$\frac {1}{2}\lt \phi _1(E) \leq 1$
, then we can find
$\widetilde {g}\in \widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
such that
$\phi _{\sigma _1\cdot \widetilde {g}}(A_1)=\phi _{\sigma _2\cdot \widetilde {g}}(\Phi _*A_1)=\frac {1}{2}$
and
$\phi _{\sigma _1\cdot \widetilde {g}}(A_2[1])=\phi _{\sigma _2\cdot \widetilde {g}}(\Phi _*A_2[1])=1$
and, hence,
$0 \lt \phi _{\sigma _1\cdot \widetilde {g}}(E) \leq \frac {1}{2}$
. Therefore, if we replace
$\sigma _i$
by
$\sigma _i\cdot \widetilde {g}$
in (C6),
$\lambda _2$
(respectively
$\Lambda _2$
) by
$-\lambda _2$
(respectively
$-\Lambda _2$
) in (C1) (respectively (C2)), all conditions (C1) to (C8) will not change. Thus we only need to deal with the case
$0 \lt \phi _1(E) \leq \frac {1}{2}$
.
Write
$[E]=a\lambda _1+b\lambda _2$
for integers
$a,b\in \mathbb{Z}$
. By our assumption on
$\phi _1(E)$
, we have
$a\leq 0$
and
$b\gt 0$
. We know
$[\Phi _*E]=a\Lambda _1+b\Lambda _2$
and
\begin{equation} Z_1(E)=Z_2(\Phi _*E)=-a+b\mathfrak{i}=\chi (\lambda _1, E)-\chi (\lambda _2, E)\mathfrak{i}. \end{equation}
Step 1. The first step is to show that
$\Phi _*E \in \mathcal{A}_2$
. Suppose
$\Phi _*E$
has non-zero cohomology objects
$E^{x_i} := \mathcal{H}^{x_i}_{\mathcal{A}_2}(\Phi _*E)[-x_i]$
for
$0\leq i \leq m$
, where
\begin{equation} x_0\lt x_1\lt \cdots \lt x_m. \end{equation}
If
$x_0=x_m$
, then
$\Phi _*E \in \mathcal{A}[-x_0]$
. Then (5) shows that
$x_0 \in 2\mathbb{Z}$
, and thus, by Lemma 3.9, it must be zero as required. Assume, for a contradiction, that
$x_0 \neq x_m$
. We claim that there exists
$l \in \{0, m\}$
such that
\begin{equation} \textrm {ext}^1(E^{x_l}, E^{x_l})\leq \tfrac {1}{2}\textrm {ext}^1(\Phi _*E, \Phi _*E) ,\end{equation}
and
$\Phi ^*$
preserves the Harder–Narasimhan filtration of
$E^{x_l}$
. From Lemma 3.6(i), we know that (7) holds for at least one
$l\in \{0,m\}$
. If the inequality in (7) is strict for
$l=0$
or
$m$
, then the claim follows from
$\mathbf{B}_{n-1}$
. Thus, by Lemma 3.6(i), the only remaining case is when we have equality in (7) for both
$l=0$
and
$m$
. In this case,
$E^{x_0}$
and
$E^{x_m}$
cannot both be
$\sigma _2$
-stable at the same time by Lemma 3.13. Hence, at least one
$E^{x_l}$
must be either strictly
$\sigma _2$
-semistable or not
$\sigma _2$
-semistable. Then, by Lemma 3.6(ii) and (iii), any Jordan–Hölder factor
$E^{x_l '}$
of any Harder–Narasimhan factor of
$E^{x_l}$
satisfies
$\textrm {ext}^1(E^{x_l '},E^{x_l '})\lt \textrm {ext}^1(E^{x_l}, E^{x_l})$
and the claim follows from applying
$\mathbf{B}_{n-1}$
to each
$E^{x_l '}$
. Therefore, we divide the proof into the following two situations, and we will rule out them case by case.
Case I.
$\Phi ^*$
preserves the Harder–Narasimhan filtration of
$E^{x_0}$
and (7) holds for
$l=0$
.
In this case, we get
$\Phi ^*(E^{x_0}[x_0]) \in \mathcal{A}_1$
. Let
$E^{x_0}_{\max }$
denote the factor with the highest phase in the Harder–Narasimhan filtration of
$E^{x_{0}}$
. It is also the factor with the maximum phase in the Harder–Narasimhan filtration of
$\Phi _*E$
. Thus
$\textrm {Hom}(\Phi ^*E^{x_0}_{\max }, E) =\textrm {Hom}(E^{x_0}_{\max }, \Phi _*E) \neq 0$
, so we have
\begin{equation} -x_0\lt \phi ^-_{\sigma _1}(\Phi ^*E^{x_0}) \leq \phi ^+_{\sigma _1}(\Phi ^*E^{x_0})=\phi _{\sigma _1}(\Phi ^*E^{x_0}_{\mathrm{max}}) \leq \phi _{\sigma _1}(E) \leq \tfrac {1}{2}, \end{equation}
which implies
$-x_0\lt 1$
. Hence, combined with Lemma 3.9, we obtain
$ x_0\in \{0,1\}$
. Thus
$\Phi _*E$
lies in an exact triangle
$E_{\gt -1} \to \Phi _*E \to E_{\leq -1}$
where
$E_{\gt -1} \in \mathcal{P}_{\sigma _2}(-1, 1]$
and
$E_{\leq -1} \in \mathcal{P}_{\sigma _2}(-\infty , -1]$
. We claim
\begin{equation} \textrm {Im}[Z_2(E_{\leq -1})] = -\chi (\Lambda _2, E_{\leq -1}) \gt 0. \end{equation}
If
$x_0 =1$
, then the claim is trivial as
$\textrm {Im}[Z_2(E)] \gt 0$
and
$E_{\gt -1}=E^{x_0}\in \mathcal{A}_2[-1]$
, so we may assume that
$x_0 = 0$
. In this case, we define
$E':=\textrm {cone}(E^{x_0} \to \Phi _*E)$
. As
$E'$
is an extension of
$E_{\leq -1}$
and
$E^1$
, we only need to show
\begin{equation} \textrm {Im}[Z_2(E')] \gt 0. \end{equation}
As
$x_0=0$
, we know that
$\textrm {Re}[Z_2(E^{x_0})]\geq 0$
and
\begin{equation} \mu _2(E^{x_0})\leq \mu _2(\Phi _*E)=\mu _1(E)\leq 0. \end{equation}
So if
$\textrm {Re}[Z_2(E^{x_0})] = 0$
, then
$\textrm {Re}[Z_2(\Phi _*E)] = 0$
as well. Then we have
\begin{eqnarray} 2(\textrm {Im}[Z_2(E^{x_0})])^2 & = & -\chi (E^{x_0}, E^{x_0}) \leq -2+ \textrm {ext}^1(E^{x_0}, E^{x_0}) \lt -2 + \tfrac {1}{2}\textrm {ext}^1(\Phi _*E, \Phi _*E) \nonumber\\[6pt]&\leq & -2 + \tfrac {1}{2} (4-\chi (\Phi _*E, \Phi _*E)) = (\textrm {Im}[Z_2(E)])^2 , \end{eqnarray}
and so
$0 \lt \textrm {Im}[Z_2(E^{x_0})] \lt \textrm {Im}[Z_2(E)]$
, which proves the claim (10). Hence, we may assume that
\begin{equation*}\textrm {Re}[Z_2(E^{x_0})] \gt 0.\end{equation*}
Using (11), to prove (10), it suffices to show
\begin{equation} \textrm {Re}[Z_2(E')] \geq 0 ,\end{equation}
as
$Z_2(E') +Z_2(E^{x_0}) = Z_2(\Phi _*E)$
.
To prove (13), we investigate homomorphisms to
$A_1$
. By (8)
\begin{equation} \textrm {Hom}(\Phi ^*E^{x_0}, A_1[k])=\textrm {Hom}(\Phi ^*\Phi _*E, A_1[k])=0,\,k\notin \{0,1\}. \end{equation}
Hence, applying
$\textrm {Hom}(-,\Phi _*A_1)$
to the exact triangle
$E^{x_0} \to \Phi _*E \to E'$
gives
\begin{equation*}\textrm {Hom}(\Phi ^*E', A_1[k])=\textrm {Hom}(E', \Phi _*A_1[k])=0,\end{equation*}
for any
$k\notin \{0,1,2\}$
. Thus, if
$\textrm {Re}[Z_2(E')]=\chi (\Lambda _1, E')\lt 0$
, we have
$\textrm {Hom}(\Phi _*A_1, E'[1])\neq 0$
, which is not possibleFootnote
5
as
$E' \in \mathcal{P}_{\sigma _2}(-\infty , 0]$
. Hence,
$\textrm {Re}[Z_2(E')] = \chi (\Lambda _1, E') \geq 0$
as claimed in (13). This ends the proof of (9).
Finally, we investigate morphisms to
$A_2$
. We have
\begin{equation*}\textrm {Hom}(E_{\gt -1}, \Phi _*A_2[k_1])= 0 =\textrm {Hom}(\Phi ^*\Phi _*E, A_2[k_2]),\end{equation*}
for
$k_1 \notin \{-1, 0,1\}$
and
$k_2 \notin \{0, 1\}$
for infinitely many
$A_2$
. Note that the vanishing for
$k_1=2$
follows from the phase ordering (8). Thus applying
$\textrm {Hom}(\Phi _*A_2, -)$
to
$E_{\gt -1}\to \Phi _*E\to E_{\leq -1}$
, the claim (9) gives
$\textrm {Hom}(\Phi _*A_2, E_{\leq -1}[1]) \neq 0$
, which is not possible as
$E_{\leq -1} \in \mathcal{P}_{\sigma _2}(-\infty , -1]$
.
Case II.
$\Phi ^*$
preserves the Harder–Narasimhan filtration of
$E^{x_m}$
and (7) holds for
$l=m$
.
By Lemma 3.9, either:
-
(i)
$\phi ^-_{\sigma _2}(\Phi _*E) \lt 1$
; or -
(ii)
$\phi ^-_{\sigma _2}(\Phi _*E) \lt \frac {3}{2}$
and
$[E]$
is a multiple of
$\lambda _2$
.
Let
$E^{x_m}_{\max }$
denote the factor with the smallest phase in the Harder–Narasimhan filtration of
$E^{x_{m}}$
. It is also the factor with the minimal phase in the Harder–Narasimhan filtration of
$\Phi _*E$
. From (C5), we know that
$\textrm {Hom}(T_1(E), \Phi ^*E^{x_m}_{min})=\textrm {Hom}(\Phi _*E, E^{x_m}_{min})\neq 0$
, so
\begin{equation} 0\lt \phi _{\sigma _1}(E)\leq \phi _{\sigma _1}(\Phi ^*E^{x_m}_{\mathrm{min}})=\phi ^-_{\sigma _1}(\Phi ^*E^{x_m})\leq -x_m+1 ,\end{equation}
which gives
$x_m\leq 0$
. Thus
$x_m =0$
in case (i) and
$x_m \in \{0, -1\}$
in case (ii). We define
\begin{equation*}E':=\textrm {cone}(\Phi _*E\to E^{x_m})[-1].\end{equation*}
If
$x_m=0$
, we have
$E' \in \mathcal{P}_{\sigma _2}(1, +\infty )$
, so
$\textrm {Hom}^{\leq 0}(E', \Phi _*A_i) = 0$
for
$i=1, 2$
. Since
\begin{equation*}\textrm {Hom}(\Phi ^*E^{x_m}, A_i[k_1])= 0 = \textrm {Hom}(\Phi ^*\Phi _*E, A_i[k_2]),\end{equation*}
for any
$k_1\notin \{0,1, 2\}$
and
$k_2 \notin \{0, 1\}$
, applying
$\textrm {Hom}(-, \Phi _*A_i)$
to
$E'\to \Phi _*E\to E^{x_m}$
, we get
\begin{equation*}\textrm {Hom}(E', \Phi _*A_i[k]) = 0,\end{equation*}
for
$i=1, 2$
and
$k\neq 1$
. Thus
$\textrm {Re}[Z_2(E')] \leq 0$
and
$\textrm {Im}[Z_2(E')] \geq 0$
. Then
$\textrm {Re}[Z_2(E^{x_m})] \geq 0$
as
$\textrm {Re}[Z_2(\Phi _*E)] \geq 0$
. So if
$[E]$
is not a multiple of
$[\lambda _2]$
, then the slope
$\mu _2(E')$
is bigger than
$\mu _2(\Phi _*E)$
, which is not possible by (15). Thus the only possibility is when both
$[\Phi _*E]$
and
$[E^{x_m}]$
are multiples of
$\Lambda _2$
. But then the same argument as in (12) gives
$\textrm {Im}[Z_2(E^{x_m})] \lt \textrm {Im}[Z_2(\Phi _*E)]$
which implies
$\textrm {Im}[Z_2(E')] \gt 0$
. As
$E'$
lies in the triangle
$E'_{\gt 2} \to E' \to \mathcal{H}^{-1}_{\mathcal{A}_2}(E')[1]$
such that
$E'_{\gt 2}\in \mathcal{P}_{\sigma _2}(2,+\infty )$
, we see that
$-\chi (E'_{\gt 2}, \Lambda _2)=2\textrm {Im}[Z_2(E'_{\gt 2})] \gt 0$
. Since we have already seen that
$\textrm {Hom}(E', \Phi _*A_2[k]) = 0$
for
$k\neq 1$
and
$\textrm {Hom}(\mathcal{H}^{-1}_{\mathcal{A}_2}(E')[1], \Phi _*A_2[k])=0$
for
$k\notin \{1,2,3\}$
, applying
$\textrm {Hom}(-,\Phi _*A_2)$
to
$E'_{\gt 2}\to E'\to \mathcal{H}^{-1}_{\mathcal{A}_2}(E')[1]$
and using
$\chi (E'_{\gt 2}, \Lambda _2)\lt 0$
implies
$\textrm {ext}^1(E'_{\gt 2}, \Phi _*A_2) \neq 0$
, a contradiction.
If
$x_m=-1$
, there is an exact sequence in
$\mathcal{A}_1$
,
\begin{equation*}0\to \Phi ^*E^{x_m}[-1]\to \Phi ^*E'\to \Phi ^*\Phi _*E\to 0.\end{equation*}
Hence,
$\chi (\Phi ^*E', \lambda _2)\lt 0$
and so
$\textrm {Hom}(\Phi ^*E', A_2[1])=\textrm {Hom}(E', \Phi _*A_2[1])\neq 0$
, contradicting
$E' \in \mathcal{P}_{\sigma _2}(2, +\infty )$
.
Step 2. Now we know that
$\Phi _*E \in \mathcal{A}_2$
and it remains to show that
$\Phi _*E$
is
$\sigma _2$
-semistable. Assume not, and let
$E^{\max }$
(respectively
$E^{\min }$
) be the factor with the maximum (respectively minimum) phase in the Harder–Narasimhan filtration of
$\Phi _*E$
. Then we have
$E^{\max }, E^{\min }\in \mathcal{A}_2$
and
\begin{equation} \mu _2(E^{\max })\gt \mu _2(\Phi _*E)\gt \mu _2(E^{\min }). \end{equation}
Using Lemma 3.4, one can show via the same argument as in Step 1 that there is an
$A \in \{E^{\min }, E^{\max }\}$
such that
$\Phi ^*A$
is
$\sigma _1$
-semistable of the same phase
$\phi _{\sigma _1}(\Phi ^*A) = \phi _{\sigma _2}(A)$
. However, both
\begin{equation*}\textrm {Hom}(E^{\max }, \Phi _*E) = \textrm {Hom}(\Phi ^*E^{\max }, E) \neq 0\end{equation*}
and
\begin{equation*}\textrm {Hom}(\Phi _*E, E^{\min }) = \textrm {Hom}(E, T_1(\Phi ^*E^{\min })) \neq 0\end{equation*}
contradict (16).
To roll out the maximal dimensional scenario in Lemma 3.15, we utilize the following standard spectral sequence, see e.g. [Reference PirozhkovPir23, Lemma 2.27].
Lemma 3.12.
Let
$X$
be a smooth algebraic variety, and let
$A \to B\to C$
be an exact triangle in
$\mathrm{D}^b(X)$
. Then there exists a spectral sequence which degenerates at
$E_3$
and converges to
$\textrm {Ext}^*_X(C, C)$
, with
$E_1$
-page
\begin{equation*}E^{p,q}_1= \left \{ \begin{array}{l@{\quad}l} \mathrm{Ext}_X^q(B,A) &\text{for } p=-1, \\[4pt]\mathrm{Ext}_X^q(A,A)\oplus \mathrm{Ext}_X^q(B,B) &\text{for }p=0, \\[4pt]\mathrm{Ext}_X^q(A, B) &\text{for }p=1, \\[4pt] 0, &\text{otherwise} \end{array} \right .\end{equation*}
with differentials
$d_r^{p,q}\colon E^{p,q}_1\to E^{p+r, q-r+1}_1$
. Moreover,
$d_1^{p,q}$
is given by composition with the morphism
$A\to B$
.
Lemma 3.13.
Let
$E\in \mathcal{K}_1$
be a
$\sigma _1$
-stable object with
$\textrm {ext}^1(E,E)=n$
. If
$\Phi _*E$
is not
$\sigma _2$
-semistable with the Harder–Narasimhan filtration
\begin{equation} E_1\to \Phi _*E\to E_2 \end{equation}
such that
$E_1$
and
$E_2$
are
$\sigma _2$
-stable, then there exists
$i\in \{1,2\}$
such that
$\textrm {ext}^1(E_i, E_i)\lt n$
.
Proof.
As
$\phi _{\sigma _2}(E_1)\gt \phi _{\sigma _2}(E_2)$
, we see that
\begin{equation} \textrm {Hom}(E_1, E_2[i])=\textrm {Hom}(E_2, E_1[2-i])=0 ,\end{equation}
for
$i\leq 0$
. By Lemmas 3.4 and 3.7, we have
\begin{equation*}\textrm {ext}^1(E_1, E_1)+\textrm {ext}^1(E_2, E_2)\leq \textrm {ext}^1(\Phi _*E, \Phi _*E)\leq 2n.\end{equation*}
Thus, it is enough to show that the case
$\textrm {ext}^1(E_1, E_1)=\textrm {ext}^1(E_2, E_2)=n$
cannot happen. Suppose this happens. Applying Lemma 3.12 to the exact triangle
$E_2[-1]\to E_1\to \Phi _*E$
, we get a spectral sequence with the first page
\begin{equation*} E^{p,q}_1= \begin{array}{c@{\quad}c|c@{\quad}c} 0 & \textrm {Ext}^3(E_1,E_2) & 0 & 0\\ 0 & \textrm {Ext}^2(E_1,E_2) & 0 & 0\\ 0 & \textrm {Ext}^1(E_1,E_2) & \textrm {Ext}^2(E_1,E_1)\oplus \textrm {Ext}^2(E_2,E_2) & 0\\ 0 & 0 & \textrm {Ext}^1(E_1,E_1)\oplus \textrm {Ext}^1(E_2,E_2) & 0 \\ 0 & 0 & \textrm {Hom}(E_1,E_1)\oplus \textrm {Hom}(E_2,E_2) & \textrm {Ext}^1(E_2,E_1) \\ \hline 0 & 0 & 0 & \textrm {Hom}(E_2,E_1) \end{array}\end{equation*}
which degenerates at
$E_3$
and converges to
$\textrm {Ext}^*(\Phi _*E, \Phi _*E)$
. Note that the differential
\begin{equation*}d_1^{0,0}\colon \textrm {Hom}(E_1, E_1)\oplus \textrm {Hom}(E_2, E_2)=\mathbb{C}^{2}\to \textrm {Ext}^1(E_2, E_1)\end{equation*}
is given by compositions, hence has a non-zero kernel. Moreover, as
\begin{equation*}E^{0,1}_1=E^{0,1}_{\infty }=\textrm {Ext}^1(\Phi _*E, \Phi _*E)=\mathbb{C}^{2n} = \textrm {Ext}^1(E_1,E_1)\oplus \textrm {Ext}^1(E_2,E_2) ,\end{equation*}
we know that
$d_1^{0,0}$
is surjective. By Lemma 3.7, we have the following two cases.
Case I.
$\textrm {RHom}(\Phi _*E, \Phi _*E)=\mathbb{C}\oplus \mathbb{C}^{2n}[-1]\oplus \mathbb{C}[-2]$
.
In this case, we have
\begin{align} 2-2n = \chi (\Phi _*E, \Phi _*E) = & \chi (E_1, E_1) +\chi (E_2, E_2) +2 \chi (E_1, E_2) \notag\\ = & (2-n) + (2 -n) + 2\chi (E_1, E_2) \end{align}
and so
$\chi (E_1, E_2) =-1$
, which leads to a contradiction since the Euler pairing in
$\mathcal{K}_2$
is an even number.
Case II.
$\textrm {RHom}(\Phi _*E, \Phi _*E)=\mathbb{C}^2\oplus \mathbb{C}^{2n}[-1]\oplus \mathbb{C}^2[-2]$
.
In this case,
$T_1(E) \cong E$
and
$\Phi ^*\Phi _*E \cong E \oplus E$
by Lemma 3.7. Since
$d_1^{0, 0}$
is surjective and has a non-zero kernel, we must have either
$\textrm {Ext}^1(E_2, E_1) = 0$
or
$\mathbb{C}$
. We first show that the Harder–Narasimhan sequence (17) does not split so the former case cannot happen. Otherwise,
$\Phi _*E \cong E_1 \oplus E_2$
and so
$\Phi ^*\Phi _*E \cong \Phi ^*E_1 \oplus \Phi ^*E_2$
. Since
$E$
is
$\sigma _1$
-stable, we get
$\Phi ^*E_1 \cong \Phi ^*E_2 \cong E$
and so
$\Phi _*\Phi ^*E_1 \cong \Phi _*\Phi ^*E_2$
, which is not possible as
$\phi _2(\Phi _*\Phi ^*E_1) = \phi _2(E_1) \gt \phi _2(E_2) = \phi _2(\Phi _*\Phi ^*E_2)$
. Thus, we may assume that
\begin{equation} \textrm {Ext}^1(E_2, E_1) = \mathbb{C} = \textrm {Hom}(E_2, E_1), \end{equation}
as
$\chi (E_2, E_1) = 0$
via the same argument as in (19). Applying
$\Phi ^*$
to (17) gives the exact triangle
\begin{equation*} \Phi ^*E_1 \xrightarrow {h} E\oplus E \xrightarrow {h'} \Phi ^*E_2. \end{equation*}
The next step is to show that
$h$
and
$h'$
are non-zero by showing that
\begin{equation} \textrm {Hom}(E, \Phi ^*E_2) = \mathbb{C} = \textrm {Hom}(E, \Phi ^*E_1). \end{equation}
Since
$\textrm {Hom}(E_1[k], E_2) = 0$
for
$k\geq 0$
, taking
$\textrm {Hom}(-, E_2)$
from the sequence (17) gives
\begin{equation*} \mathbb{C} = \textrm {Hom}(\Phi _*E, E_2) = \textrm {Hom}(E, \Phi ^*E_2)\end{equation*}
as the left-hand claim in (21). Similarly, by taking
$\textrm {Hom}(-, E_1)$
and using assumption (20), we obtain the long exact sequence
\begin{equation*} \cdots \to \textrm {Hom}(E_1[1], E_1) = 0 \to \textrm {Hom}(E_2, E_1) \cong \mathbb{C} \to \textrm {Hom}(\Phi _*E, E_1) \to \textrm {Hom}(E_1, E_1) \cong \mathbb{C} \to \cdots. \end{equation*}
Since the sequence (17) does not split, we get
$\textrm {Hom}(\Phi _*E, E_1) = \mathbb{C}$
. Then the right-hand claim in (21) follows by adjunction in (C5). Similarly, we have
$\textrm {Hom}(\Phi ^*E_2, E)=\textrm {Hom}(E_2, \Phi _*E)=\mathbb{C}$
as (17) does not split.
Therefore, there is a map
$s\colon E\oplus E\to E$
such that
$s\circ h$
is non-zero. Since
$E$
is
$\sigma _1$
-stable, we know that
$\textrm {cone}(s)[-1]\cong E$
, and denote the natural map
$\textrm {cone}(s)[-1]\cong E\to E\oplus E$
by
$s'$
. Define
$F := \textrm {cone}(s \circ h)[-1]$
, then we have the following commutative diagram.
At first, we show that
$h'\circ s'\neq 0$
. As
$s\circ h\neq 0$
, we know that the induced map
\begin{equation*}\textrm {Hom}(E, E)=\mathbb{C}\to \textrm {Hom}(\Phi ^*E_1, E)\end{equation*}
is non-zero, and hence it is injective. Then, from
$\textrm {Hom}(\Phi ^*E_1, E[-1])=\textrm {Hom}(E_1, \Phi _*E[-1])=0$
, we get
$\textrm {Hom}(F, E[-1])=0$
. Therefore, if we apply
$\textrm {Hom}(-, E)$
to the exact triangle in the first row, we see that the induced map
$\textrm {Hom}(\Phi ^*E_2, E)=\mathbb{C}\to \textrm {Hom}(E, E)$
is injective as well. This implies that
$h'\circ s'$
is non-zero.
Now, taking
$\textrm {Hom}(E, -)$
from the first row and applying (21) shows that
$\textrm {Hom}(E, F) = 0$
because
$\textrm {Hom}(E, \Phi ^*E_2[-1]) = \textrm {Hom}(\Phi _*E, E_2[-1]) = 0$
as
$\phi _2^{-}(\Phi _*E_2) = \phi _2(E_2)$
. Thus, applying
$\textrm {Hom}(E, -)$
to the first column gives the isomorphism
$\mathbb{C} = \textrm {Hom}(E, \Phi ^*E_1) \to \mathbb{C} = \textrm {Hom}(E, E)$
. This shows that the first column is splitting, so by (21), we have
\begin{equation*} \textrm {Hom}(E_1, \Phi _*\Phi ^*E_2) = \textrm {Hom}(\Phi ^*E_1, \Phi ^*E_2) \cong \textrm {Hom}(E \oplus F , \Phi ^*E_2) \neq 0, \end{equation*}
which is not possible as
$\phi _2(E_1) \gt \phi _2(j_*j^*E_2)=\phi _2(E_2)$
and both are
$\sigma _2$
-semistable.
Lemma 3.14.
For any
$n\gt 1$
, the validity of
$\mathbf{A}_{n-1}$
implies that
$\mathbf{B}_n$
holds.
Proof.
The proof closely resembles that of Lemma 3.11. Therefore, we will only provide an outline of the key steps. For any object
$F \in \mathcal{K}_2$
, we know
$\textrm {ext}^1(F, F) = -\chi (F, F) + 2\hom (F, F) \in 2\mathbb{Z}$
. Thus we only need to prove the claim when
$n= 2k$
for
$k\in \mathbb{Z}$
. Therefore, we take a
$\sigma _2$
-stable object
$F$
with
$\textrm {ext}^1(F, F) =2k$
. As in the proof of Lemma 3.11, we may assume that
$F \in \mathcal{A}_2$
and
$0\lt \phi _2(F) \leq \frac {1}{2}$
.
Step 1. The first step is to show that
$\Phi ^*F \in \mathcal{A}_1$
. If this is not true, we have non-zero cohomology objects
$F^{x_i} := \mathcal{H}^{x_i}_{\mathcal{A}_1}(\Phi ^*F)[-x_i]$
for
$0 \leq i \leq m$
, where
$x_0 \lt x_1 \lt \cdots \lt x_m$
. If
$x_0 = x_m$
, then since
$Z_1(\Phi ^*F) = 2Z_2(F)$
, we have
$x_0 \in 2 \mathbb{Z}$
and so
$x_0 =0$
by Lemma 3.9. Thus, we may assume that
$x_0 \lt x_m$
. Then
$\textbf {A}_{2k-1}$
and the next Lemma 3.15 show that there exists
$l \in \{0, m\}$
such that
$\textrm {ext}^1(F^{x_l}, F^{x_l}) \leq 2k$
and
$\Phi _*$
preserves the Harder–Narasimhan filtration of both
$E^{x_{l}}$
and
$T_1(E^{x_{l}})$
as in the first step of Lemma 3.11.
Case I. Firstly, we assume that
$l= 0$
. Let
$F^{x_0}_{\max }$
denote the factor with the highest phase in the Harder–Narasimhan filtration of
$F^{x_{0}}$
. Since
$\textrm {Hom}(\Phi _*T_1(F^{x_0}_{\max }), F) =\textrm {Hom}(F^{x_0}_{\max }, \Phi ^*F) \neq 0$
, we have
\begin{equation} -x_0\lt \phi ^{-}_{\sigma _1}(F^{x_0}) \leq \phi ^{+}_{\sigma _1}(F^{x_0}) = \phi ^+_{\sigma _2}(\Phi _*T_1(F^{x_0}_{\max })) \leq \phi _{\sigma _2}(F) \leq \tfrac {1}{2}, \end{equation}
which implies
$-x_0\lt 1$
. Hence, combined with Lemma 3.9, we obtain
$ x_0\in \{0,1\}$
. Thus
$\Phi ^*F$
lies in an exact triangle
$F_{\gt -1} \to \Phi ^*F \to F_{\leq -1}$
where
$F_{\gt -1} \in \mathcal{P}_{\sigma _1}(-1, \frac {1}{2}]$
and
$F_{\leq -1} \in \mathcal{P}_{\sigma _1}(-\infty , -1]$
. We claim
\begin{equation} \textrm {Im}[Z_1(F_{\leq -1})] = -\chi (\lambda _2, F_{\leq -1}) \gt 0. \end{equation}
If
$x_0 =1$
, then the claim is trivial as
$\textrm {Im}[Z_1(\Phi ^*F)]\gt 0$
, so we may assume that
$x_0 = 0$
. We define
\begin{equation*}F':=\textrm {cone}(F^{x_0} \to \Phi ^*F).\end{equation*}
Because
$F'$
is an extension of
$F_{\leq -1}$
and
$F^1=\mathcal{H}^1_{\mathcal{A}_1}(\Phi ^*F)[-1]$
, we only need to show
$\textrm {Im}[Z_1(F')] \gt 0$
. Since the slope of
$Z_1(F^{x_0})$
is smaller than or equal to the slope of
$Z_1(\Phi ^*F)$
by (22), we can assume that
$\textrm {Re}[Z_1(F^{x_0})]\gt 0$
via the same argument as (11) and (12). Then, as (13), it suffices to prove
\begin{equation} \textrm {Re}[Z_1(F')] \geq 0. \end{equation}
If (24) does not hold, then
$\chi (\lambda _1, F')\lt 0$
. On the other hand, applying
$\textrm {Hom}(A_1, -)$
to the exact triangle
$F^{x_0}\to \Phi ^*F \to F'$
yields
$\textrm {Hom}(A_1, F'[k]) = 0$
for
$k \notin \{0, 1, 2\}$
for infinitely many
$A_1$
. Consequently,
$\textrm {Hom}(A_1, F'[1]) \neq 0$
for infinitely many
$A_1$
, which is not possible since
$F' \in \mathcal{P}_{\sigma _1}(-\infty , 0]$
. Therefore, (24) holds, and so (23) holds.
Next, we investigate morphisms from
$A_2$
by taking
$\textrm {Hom}(A_2, -)$
from the exact triangle
\begin{equation*}F_{\gt -1}\to \Phi ^*F \to F_{\leq -1}.\end{equation*}
We have
\begin{equation*}\textrm {Hom}(A_2, F_{\gt -1}[1+k])= 0 =\textrm {Hom}(A_2, \Phi ^*F[k]) = \textrm {Hom}(\Phi _*T_1(A_2), F[k])\end{equation*}
for
$k \notin \{0,1, 2\}$
and infinitely many
$A_2$
, where the first vanishing follows from
$F_{\gt -1} \in \mathcal{P}_{\sigma _1}(-1, \frac {1}{2}]$
. Thus
$\textrm {Hom}(A_2, F_{\leq -1}[k])=0$
for
$k\notin \{0,1,2\}$
. Then (23) gives
$\textrm {Hom}(A_2, F_{\leq -1}[1]) \neq 0$
, which contradicts
$F_{\leq -1} \in \mathcal{P}_{\sigma _1}(-\infty , -1]$
.
Case II. Now assume that
$l=m$
. Let
$F^{x_m}_{\min }$
be the object with the minimum phase in the Harder–Narasimhan filtration of
$F^{x_m}$
. We have
$\textrm {Hom}(F, \Phi _*F^{x_m}_{\min })=\textrm {Hom}(\Phi ^*F, F^{x_m}_{\min })\neq 0$
, so
\begin{equation} 0\lt \phi _{\sigma _2}(F)\leq \phi _{\sigma _2}(\Phi _*F^{x_m}_{\mathrm{min}})=\phi ^-_{\sigma _1}(F^{x_m})\leq \phi _{\sigma _1}^+(F^{x_m}) \leq -x_m+1, \end{equation}
which gives
$x_m\leq 0$
. Hence,
$x_m \in \{0, -1\}$
by Lemma 3.9. We define
$F':=\textrm {cone}(\Phi ^*F\to F^{x_m})[-1]$
.
If
$x_m=0$
, we have
$F' \in \mathcal{P}_{\sigma _1}(1, +\infty )$
, so
$\textrm {Hom}^{\leq 0}(F', A_i) = 0$
for
$i=1, 2$
. Since, for each
$i=1,2$
,
\begin{equation*}\textrm {Hom}(F^{x_m}, A_i[k_1])= 0 = \textrm {Hom}(\Phi ^*F, A_i[k_2])=\textrm {Hom}(F, \Phi _*A_i[k_2]),\end{equation*}
for any
$k_1\notin \{0,1, 2\}$
and
$k_2 \notin \{0, 1\}$
and infinitely many
$A_i$
, applying
$\textrm {Hom}(-, A_i)$
to
$F'\to \Phi ^*F\to F^{x_m}$
, we get
$\textrm {Hom}(F', A_i[k]) = 0$
for
$i=1, 2$
and
$k\neq 1$
. Thus
$\textrm {Re}[Z_1(F')] \leq 0$
and
$\textrm {Im}[Z_1(F')] \geq 0$
. Then
$\textrm {Re}[Z_1(F^{x_m})] \geq 0$
as
$\textrm {Re}[Z_1(\Phi ^*F)] \geq 0$
. So if
$[F]$
is not a multiple of
$[\Lambda _2]$
, then the slope
$\mu _1(F')$
is bigger than
$\mu _1(\Phi ^*F)$
, which is not possible by (25). Thus the only possibility is when both
$[\Phi ^*F]$
and
$[F^{x_m}]$
are multiples of
$\lambda _2$
. But then the same argument as in (12) gives
$\textrm {Im}[Z_1(F^{x_m})] \lt \textrm {Im}[Z_1(\Phi ^*F)]$
, which implies that
$\textrm {Im}[Z_1(F')] \gt 0$
. As
$F'$
lies in the triangle
$F'_{\gt 2} \to F' \to \mathcal{H}^{-1}_{\mathcal{A}_1}(F')[1]$
such that
$F'_{\gt 2}\in \mathcal{P}_{\sigma _1}(2,+\infty )$
, we see that
$-\chi (F'_{\gt 2}, \lambda _2)=\textrm {Im}[Z_1(F'_{\gt 2})] \gt 0$
. Since we have already seen that
$\textrm {Hom}(F', A_2[k]) = 0$
for
$k\neq 1$
and
$\textrm {Hom}(\mathcal{H}^{-1}_{\mathcal{A}_1}(F')[1], A_2[k])=0$
for
$k\notin \{1,2,3\}$
, applying
$\textrm {Hom}(-, A_2)$
to
$F'_{\gt 2}\to F'\to \mathcal{H}^{-1}_{\mathcal{A}_1}(F')[1]$
and using
$\chi (F'_{\gt 2}, \lambda _2)\lt 0$
implies that
$\textrm {ext}^1(F'_{\gt 2}, A_2) \neq 0$
, a contradiction.
If
$x_m = -1$
, there is an exact sequence in
$\mathcal{A}_2$
,
\begin{equation*}0\to \Phi _*F^{x_m}[-1]\to \Phi _*F'\to \Phi _*\Phi ^*F\to 0.\end{equation*}
Hence,
$\chi (\Phi _*F', \Lambda _2)\lt 0$
and so
\begin{equation*}\textrm {Hom}(\Phi _*A_2, \Phi _*F'[1])=\textrm {Hom}(\Phi ^*\Phi _*A_2, F'[1]) = \textrm {Hom}(F',T_1(\Phi ^*\Phi _*A_2)[1])\neq 0,\end{equation*}
contradicting
$F' \in \mathcal{P}_{\sigma _1}(2, +\infty )$
as
$T_1(\Phi ^*\Phi _*A_2)$
is
$\sigma _1$
-semistable of phase
$\phi _1(A_2)=\frac {1}{2}$
.
Step 2. Now we know
$\Phi ^*F \in \mathcal{A}_1$
and we only need to show that it is
$\sigma _1$
-semistable. Assume not, and let
$F^{\max }$
(respectively
$F^{\min }$
) be the factor with the maximum (respectively minimum) phase in the Harder–Narasimhan filtration of
$\Phi ^*F$
, so
$F^{\max }, F^{\min }\in \mathcal{A}_1$
and
\begin{equation} \mu _1(F^{\max })\gt \mu _1(\Phi ^*F)\gt \mu _1(F^{\min }). \end{equation}
Using Lemma 3.4, one can show via the same argument as in Step 1 of the proof of Lemma 3.11 that there exists
$A \in \{F^{\min }, F^{\max }\}$
such that
$\Phi _*A$
and
$\Phi _*T_1(A)$
are
$\sigma _2$
-semistable of the same phase
\begin{equation*}\phi _{\sigma _2}(\Phi _*A)=\phi _{\sigma _2}(\Phi _*T_1(A)) = \phi _{\sigma _1}(A).\end{equation*}
However,
$0 \neq \textrm {Hom}(F^{\max }, \Phi ^*F) = \textrm {Hom}(\Phi _*T_1(F^{\max }), F)$
and
$0 \neq \textrm {Hom}(\Phi ^*F, F^{\min }) = \textrm {Hom}(F, \Phi _*F^{\min })$
, which are not possible by (26).
Lemma 3.15.
Let
$F\in \mathcal{K}_2$
be a
$\sigma _2$
-stable object with
$\textrm {ext}^1(F,F)=2k$
. If
$\Phi ^*F\in \mathcal{K}_1$
is not
$\sigma _1$
-semistable with the Harder–Narasimhan filtration
\begin{equation} F_1\to \Phi ^*F\to F_2 \end{equation}
such that
$F_1$
and
$F_2$
are
$\sigma _1$
-stable, then there exists
$i\in \{1,2\}$
such that
$\textrm {ext}^1(F_i, F_i)\lt 2k$
.
Proof.
The proof is similar to the proof of Lemma 3.13, so we only outline the main steps. By Lemmas 3.4 and 3.7, we only need to consider the case that
$\textrm {ext}^1(F_1, F_1) = \textrm {ext}^1(F_2, F_2) = 2k$
and
$\textrm {ext}^1(\Phi ^*F, \Phi ^*F) = 4k$
. Then Lemma 3.8 implies that
$\textrm {Hom}(\Phi ^*F, \Phi ^*F) = \mathbb{C}^2$
,
$T_2(F) = F$
and
$\Phi _*\Phi ^*F \cong F \oplus F$
. By (C5), we have
$T_1(\Phi ^*F)\cong \Phi ^*(T_2(F))\cong \Phi ^*F$
.
As in Lemma 3.13, we get a spectral sequence with the first page
\begin{equation*} E^{p,q}_1= \begin{array}{c@{\quad}c|c@{\quad}c} 0 & 0 & 0 & 0\\[2pt] 0 & \textrm {Ext}^2(F_1,F_2) & 0 & 0\\[2pt] 0 & \textrm {Ext}^1(F_1,F_2) & \textrm {Ext}^2(F_1,F_1)\oplus \textrm {Ext}^2(F_2,F_2) & 0\\[2pt] 0 & 0 & \mathbb{C}^{2n} & 0 \\[2pt] 0 & 0 & \mathbb{C}^2 & \textrm {Ext}^1(F_2,F_1) \\[2pt] \hline 0 & 0 & 0 & \textrm {Hom}(F_2,F_1) \end{array}\end{equation*}
which degenerates at
$E_3$
and converges to
$\textrm {Ext}^*(\Phi ^*F, \Phi ^*F)$
. Moreover,
$d^{0,0}_1$
has a non-zero kernel and is surjective. We know that (27) does not split, otherwise the
$\sigma _2$
-stability of
$F$
forces
$\Phi _*F_1 \cong \Phi _*F_2 \cong F$
, and so
$\Phi ^*\Phi _*F_1 \cong \Phi ^*\Phi _*F_2$
, which is not possible by the phase ordering of
$F_1$
and
$F_2$
. Therefore,
\begin{equation} \textrm {Ext}^1(F_2, F_1) = \mathbb{C} = \textrm {Hom}(F_2, F_1). \end{equation}
Taking
$\Phi _*$
from (27) gives the exact triangle
$\Phi _*F_1 \xrightarrow {h} F \oplus F \xrightarrow {h'} \Phi _*F_2$
so that both
$h$
and
$h'$
are non-zero because
\begin{equation} \textrm {Hom}(F, \Phi _*F_2) = \mathbb{C} = \textrm {Hom}(F, \Phi _*F_1). \end{equation}
Therefore, we can find an exact triangle
$F \xrightarrow {s'} F \oplus F \xrightarrow {s} F$
so that both
$s \circ h$
and
$h' \circ s'$
are non-zero. Defining
$G := \textrm {cone}(s \circ h)[-1]$
, we then have the following commutative diagram.
Taking
$\textrm {Hom}(F, -)$
from the first row and applying (29) shows that
$\textrm {Hom}(F, G) = 0$
. Thus taking
$\textrm {Hom}(F, -)$
from the first column shows that the first column is splitting. Then we have
$\Phi _*F_1 \cong G \oplus F$
, which implies
$\textrm {Hom}(\Phi _*F_1, \Phi _*F_2) \neq 0$
and contradicts
$\phi _1(F_1)=\phi _1(\Phi ^*\Phi _*F_1) \gt \phi _1(F_2)$
.
4. Gushel–Mukai varieties
This section aims to utilize Theorem 3.1 from the preceding section to establish Theorem 1.1 or, more precisely, Theorem 4.8. At the end of this section, we will use a deformation argument to generalize a part of Theorem 4.8 from very general cases to general cases (cf. Theorem 4.16). We begin by investigating some properties of the Kuznetsov components of GM varieties and then explore the required conditions mentioned in the earlier section.
4.1 Gushel–Mukai varieties
Recall that a GM variety
$X$
of dimension
$n$
is a smooth intersection
\begin{equation*}X=\mathrm{Cone}(\textrm {Gr}(2,5))\cap Q,\end{equation*}
where
$\mathrm{Cone}(\textrm {Gr}(2,5))\subset \mathbb{P}^{10}$
is the projective cone over the Plücker embedded Grassmannian
$\textrm {Gr}(2,5)\subset \mathbb{P}^9$
, and
$Q\subset \mathbb{P}^{n+4}$
is a quadric hypersurface. Then
$n\leq 6$
and we have a natural morphism
$\gamma _X\colon X\to \textrm {Gr}(2,5)$
. We say
$X$
is ordinary if
$\gamma _X$
is a closed special if
$\gamma _X$
is a double covering onto its image.
Definition 4.1. Let
$X$
be a GM variety of dimension
$n=4$
or
$6$
. We say
$X$
is Hodge-special if
\begin{equation*}\mathrm{H}^{\frac {n}{2}, \frac {n}{2}}(X)\cap \mathrm{H}_{\mathrm{van}}^n(X, \mathbb{Q}) \neq 0,\end{equation*}
where
$\mathrm{H}_{\mathrm{van}}^n(X, \mathbb{Q}):=\mathrm{H}_{\mathrm{van}}^n(X, \mathbb{Z})\otimes \mathbb{Q}$
and
$\mathrm{H}_{\mathrm{van}}^n(X, \mathbb{Z})$
is defined as the orthogonal complement of
\begin{equation*}\gamma _X^*\mathrm{H}^n(\textrm {Gr}(2,5), \mathbb{Z})\subset \mathrm{H}^n(X, \mathbb{Z})\end{equation*}
with respect to the intersection form.
By [Reference Debarre, Iliev and ManivelDIM15, Corollary 4.6],
$X$
is non-Hodge-special when
$X$
is very general among all ordinary GM varieties of the same dimension or very general among all special GM varieties of the same dimension.
The semi-orthogonal decomposition of
$\mathrm{D}^b(X)$
for a GM variety
$X$
of dimension
$n\geq 3$
is given by
\begin{equation*}\mathrm{D}^b(X)=\langle \mathcal{K}u(X),{\mathcal{O}}_X,\mathcal{U}^{\vee }_X,\ldots ,{\mathcal{O}}_X((n-3)H),\mathcal{U}_X^{\vee }((n-3)H)\rangle ,\end{equation*}
where
$\mathcal{U}_X$
is the pull-back of the tautological subbundle via
$\gamma _X$
. We refer to
$\mathcal{K}u(X)$
as the Kuznetsov component of
$X$
. We define the projection functors
\begin{equation*}\mathrm{pr}_X:=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}{\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X}\cdots {\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X((n-4)H)}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X((n-4)H)}\colon \mathrm{D}^b(X)\to \mathcal{K}u(X)\end{equation*}
when
$n\geq 4$
and
$\mathrm{pr}_X:={\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X}$
when
$n=3$
.
When
$n=3$
, according to the proof of [Reference KuznetsovKuz09, Proposition 3.9],
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
is a rank two lattice generated by
$\lambda _1$
and
$\lambda _2$
, where
\begin{equation} \mathrm{ch}(\lambda _1)=-1+\tfrac {1}{5}H^2,\quad \mathrm{ch}(\lambda _2)=2-H+\tfrac {1}{12}H^3, \end{equation}
with the Euler pairing
\begin{equation} \left [ \begin{array}{c@{\quad}c} -1 & 0 \\[2pt] 0 & -1\\[2pt] \end{array} \right ].\end{equation}
When
$n=4$
, there is a rank two sublattice in
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
generated by
$\Lambda _1$
and
$\Lambda _2$
, where
\begin{equation} \mathrm{ch}(\Lambda _1)=-2+(H^2-\gamma ^*_X \sigma _2)-\tfrac {1}{20}H^4, \quad \mathrm{ch}(\Lambda _2)=4-2H+\tfrac {1}{6}H^3, \end{equation}
whose Euler pairing is
\begin{equation} \left [ \begin{array}{c@{\quad}c} -2 & 0 \\[2pt] 0 & -2\\[2pt] \end{array} \right ], \end{equation}
where
$\gamma ^*_X \sigma _2$
is the pull-back of the Schubert cycle
$\sigma _2\in \mathrm{H}^4(\textrm {Gr}(2, 5), \mathbb{Z})$
. When
$X$
is non-Hodge-special, by [Reference Kuznetsov and PerryKP18, Proposition 2.25], we have
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))=\mathbb{Z}\Lambda _1\oplus \mathbb{Z}\Lambda _2$
.
When
$n=5$
, by [Reference Kuznetsov and PerryKP23, Corollary 6.5], we can find a smooth GM threefold
$X'$
with an equivalence
$\mathcal{K}u(X')\simeq \mathcal{K}u(X)$
. Hence,
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
is also a rank two lattice with the Euler pairing (31).
When
$n=6$
, by [Reference Kuznetsov and PerryKP23, Corollary 6.5] again, we can find a smooth GM fourfold
$X'$
with an equivalence
$\mathcal{K}u(X')\simeq \mathcal{K}u(X)$
. Hence,
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
contains a rank two lattice whose Euler pairing is the same as (33), which is the whole numerical Grothendieck group when
$X$
is non-Hodge-special by [Reference Debarre, Iliev and ManivelDIM15, Corollary 4.6] and [Reference Kuznetsov and PerryKP18, Proposition 2.25].
Now let
$Y$
(respectively
$X$
) be a GM variety of dimension
$n-1$
(respectively
$n$
) where
$n\geq 4$
, and
$j \colon Y \hookrightarrow X$
represents a hyperplane section. For any object
$E\in \mathcal{K}u(Y)$
, we have
\begin{equation} \mathrm{pr}_X(j_*E)=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}(j_*E). \end{equation}
This can be easily deduced from the adjunction of
$j^*$
and
$j_*$
. For example, when
$X$
is a GM fourfold, we have
\begin{equation*}\textrm {RHom}_X(\mathcal{U}^{\vee }_X,j_*E)=\textrm {RHom}_Y(\mathcal{U}^{\vee }_Y,E)=0,\quad \textrm {RHom}_X({\mathcal{O}}_X,j_*E)=\textrm {RHom}_Y({\mathcal{O}}_Y,E)=0,\end{equation*}
since
$E\in \mathcal{K}u(Y)$
.
Lemma 4.2.
For any object
$F\in \mathcal{K}u(X)$
, we have
$j^*F\in \mathcal{K}u(Y)$
.
Proof.
By Serre duality and adjunction, the functor
$j^!=j_*\circ -\otimes {\mathcal{O}}_Y(H)[-1]$
is the left adjoint of
$j^*$
. Thus, in the GM fourfold case, we need to show that
\begin{equation*}\textrm {RHom}_X(j_*(\mathcal{U}^{\vee }_Y(H))[-1], F)=0,\quad \textrm {RHom}_X(j_*({\mathcal{O}}_Y(H))[-1], F)=0.\end{equation*}
Then the result follows from the fact that
$F\in \mathcal{K}u(X)$
and applying
$\textrm {Hom}_X(-, F)$
to the exact triangles
${\mathcal{O}}_X\to {\mathcal{O}}_X(H)\to j_*{\mathcal{O}}_Y(H)$
and
$\mathcal{U}^{\vee }_X\to \mathcal{U}^{\vee }_X(H)\to j_*\mathcal{U}^{\vee }_Y(H)$
. The other cases are similar.
Therefore, we have well-defined functors
\begin{equation*}\mathrm{pr}_X\circ j_* \colon \mathcal{K}u(Y) \to \mathcal{K}u(X) \quad \text{and} \quad j^*\colon \mathcal{K}u(X)\to \mathcal{K}u(Y).\end{equation*}
Lemma 4.3.
Let
$X$
be a GM variety of dimension
$n \geq 3$
. We define
\begin{equation*}T_X:=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X} \boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}\circ (-\otimes {\mathcal{O}}_X(-H))[1]\colon \mathcal{K}u(X)\to \mathcal{K}u(X).\end{equation*}
-
(i) If
$n$
is odd, then
$T_X=S_{\mathcal{K}u(X)}[-2]$
is an involution on
$\mathcal{K}u(X)$
that acts trivially on
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
. -
(ii) If
$n$
is even and
$X$
is non-Hodge-special, then
$T_X$
is an involution on
$\mathcal{K}u(X)$
that acts trivially on
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
. Moreover,
$\mathcal{K}u(X)$
has the Serre functor
$S_{\mathcal{K}u(X)}=[2]$
.
Proof.
Note that
$T_X$
is the inverse of
${\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X}\circ (-\otimes {\mathcal{O}}_X(H))[-1]|_{\mathcal{K}u(X)}$
, so both of them are involutions on
$\mathcal{K}u(X)$
by [Reference Bayer and PerryBP23, Theorem 4.15]. Thus, (ii) follows from [Reference Bayer and PerryBP23, Proposition 5.7]. And (i) can be deduced from a direct computation using the Euler pairing, as
\begin{equation*}\chi (v, T_X(w))=\chi (v, S_{\mathcal{K}u(X)}(w))=\chi (w, v)=\chi (v, w),\end{equation*}
for any two classes
$v,w\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
.
Lemma 4.4.
-
(i) If
$n=4,6$
, we have adjoint pairs
and isomorphisms of functors
\begin{equation*}\mathrm{pr}_X\circ j_*\circ T_Y\dashv j^* \dashv \mathrm{pr}_X\circ j_*\end{equation*}
$j^*\circ T_X\cong T_Y\circ j^*$
and
$\mathrm{pr}_X\circ j_* \circ T_Y\cong T_X\circ \mathrm{pr}_X\circ j_*$
.
-
(ii) If
$n=5$
, we have adjoint pairs
and isomorphisms of functors
\begin{equation*}j^* \dashv \mathrm{pr}_X\circ j_* \dashv j^*\circ T_X\end{equation*}
$j^*\circ T_X\cong T_Y\circ j^*$
and
$\mathrm{pr}_X\circ j_* \circ T_Y\cong T_X\circ \mathrm{pr}_X\circ j_*$
.
Proof.
Firstly, we have the adjunction
$j^* \dashv \mathrm{pr}_X\circ j_*$
because, for any pair of objects
$F\in \mathcal{K}u(X)$
and
$E\in \mathcal{K}u(Y)$
, we have
\begin{equation*}\textrm {Hom}_Y(j^*F, E)=\textrm {Hom}_X(F, j_*E)=\textrm {Hom}_X(F, \mathrm{pr}_X(j_*E)),\end{equation*}
where the last equality follows from adjunctions of right mutations as
$\mathrm{pr}_X(j_*E)=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}(j_*E)$
by (34). Then the existence of adjoint pairs in (i) and (ii) follows from the description of Serre functors on
$\mathcal{K}u(Y)$
and
$\mathcal{K}u(X)$
.
As decompositions of
$\mathrm{D}^b(Y)$
and
$\mathrm{D}^b(X)$
fit into the situation of [Reference Kuznetsov and PerryKP21, Corollary 4.19(1)], the isomorphism
$j^*\circ T_X\cong T_Y\circ j^*$
in (i) follows from [Reference Kuznetsov and PerryKP21, Lemma 4.12]. Then, for any pair of objects
$E,F\in \mathcal{K}u(X)$
, we have functorial isomorphisms
\begin{align*}\textrm {Hom}_X(E, \mathrm{pr}_X(j_*T_Y(F)))\cong \textrm {Hom}_Y(j^*E, T_Y(F))&\cong \textrm {Hom}_Y(T_Y(j^*E), F),\\\cong \textrm {Hom}_Y(j^*(T_Y(E)), F)\cong \textrm {Hom}_X(T_X(E), \mathrm{pr}_X(j_*F))&\cong \textrm {Hom}_X(E, T_X(\mathrm{pr}_X(j_*F))),\end{align*}
where the first and the fourth isomorphism follow from the adjunction of functors, the second and the last one follow from Lemma 4.3 that
$T_X$
and
$T_Y$
are involutions, and the third one is given by
$j^*\circ T_X\cong T_Y\circ j^*$
. Therefore, the last statement of (ii) follows from the Yoneda Lemma. The last statement of (i) is similar.
The following will be useful later in our computations.
Proposition 4.5.
For any object
$E\in \mathrm{D}^b(Y)$
, we have
\begin{equation*}\mathrm{pr}_X(j_*E)\cong \mathrm{pr}_X(j_*\mathrm{pr}_Y(E)).\end{equation*}
Proof.
As the cone of the natural map
$E\to \mathrm{pr}_Y(E)$
is contained in
${}^\bot \mathcal{K}u(Y)$
, we only need to show that
$\mathrm{pr}_X(j_*E)\cong 0$
for any
$E\in \langle {\mathcal{O}}_Y, \mathcal{U}^{\vee }_Y,\ldots ,{\mathcal{O}}_X((n-3)H),\mathcal{U}_X^{\vee }((n-3)H)\rangle$
.
First of all, we assume that
$X$
is a GM fourfold. Recall that
$\mathrm{pr}_X=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}{\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}_X^{\vee }}$
. Then from the exact sequence
$0\to \mathcal{U}_X\to \mathcal{U}_X^{\vee }\to j_*\mathcal{U}_Y^{\vee }\to 0$
we see that
$\mathrm{pr}_X(j_*\mathcal{U}^{\vee }_Y)=\mathrm{pr}_X(\mathcal{U}_X[1])$
. Since we have
$\textrm {RHom}_X(\mathcal{U}^{\vee }_X, \mathcal{U}_X)=0$
by [Reference Guo, Liu and ZhangGLZ24, Lemma 5.4(2)], then
${\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}_X^{\vee }}(\mathcal{U}_X)=\mathcal{U}_X$
and, hence, we deduce that
$\mathrm{pr}_X(\mathcal{U}_X)=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}(\mathcal{U}_X)=0$
. Similarly, we have
$\mathrm{pr}_X(j_*{\mathcal{O}}_Y)=0$
and the result follows. The arguments for other cases are similar.
4.1.1 Stability conditions on the Kuznetsov components
In [Reference Bayer, Lahoz, Macrì and StellariBLMS23], the authors provide a way to construct stability conditions on a semi-orthogonal component from weak stability conditions on a larger category. For GM varieties, we have the following.
Theorem 4.6 [Reference Perry, Pertusi and ZhaoPPZ22]. Let
$X$
be a GM variety of dimension
$n \geq 3$
. Then there exists a family of stability conditions on
$\mathcal{K}u(X)$
.
When
$X$
is a GM fourfold, we denote by
$\textrm {Stab}^{\circ }(\mathcal{K}u(X))$
the family of stability conditions on
$\mathcal{K}u(X)$
constructed in [Reference Perry, Pertusi and ZhaoPPZ22, Theorem 4.12].
When
$Y$
is a GM threefold, it is proved in [Reference Pertusi and RobinettPR23] that stability conditions on
$\mathcal{K}u(Y)$
constructed in [Reference Bayer, Lahoz, Macrì and StellariBLMS23] are Serre invariant. Furthermore, they all belong to the same
$\widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
-orbit.
Theorem 4.7 [Reference Jacovskis, Lin, Liu and ZhangJLLZ24, Reference Feyzbakhsh and PertusiFP23]. Let
$Y$
be a GM threefold, then all Serre-invariant stability conditions on
$\mathcal{K}u(Y)$
are contained in the same
$\widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
-orbit.
4.2 Stability of pull-back and push-forward
One of the main goals of this subsection is to prove the following theorem.
Theorem 4.8.
Let
$X$
be a GM variety of dimension
$n\geq 4$
and
$j \colon Y\hookrightarrow X$
be a smooth hyperplane section. Furthermore, we assume that the one between
$X$
and
$Y$
with even dimension is non-Hodge-special. Let
$\sigma _Y$
and
$\sigma _X$
be Serre-invariant stability conditions on
$\mathcal{K}u(Y)$
and
$\mathcal{K}u(X)$
, respectively.
-
(i) An object
$E\in \mathcal{K}u(Y)$
is
$\sigma _Y$
-semistable if and only if
$\mathrm{pr}_X(j_*E)\in \mathcal{K}u(X)$
is
$\sigma _X$
-semistable. Moreover,
$\mathrm{pr}_X(j_*E)$
is
$\sigma _X$
-stable if and only if
$E$
is
$\sigma _Y$
-stable and there is no
$\sigma _X$
-stable object
$F \in \mathcal{K}u(X)$
such that
$j^*F \cong E$
. -
(ii) An object
$F\in \mathcal{K}u(X)$
is
$\sigma _X$
-semistable if and only if
$j^*F\in \mathcal{K}u(Y)$
is
$\sigma _Y$
-semistable. Moreover,
$j^*F$
is
$\sigma _Y$
-stable if and only if
$F$
is
$\sigma _X$
-stable and there is no
$\sigma _Y$
-stable object
$E$
such that
$\mathrm{pr}_X(j_*E) \cong F$
.
Remark 4.9. In Section 4.3, we will explain how to generalize this theorem from very general cases to general cases.
To prove Theorem 4.8, we only need to verify conditions (C1) to (C8) in the earlier section for the functors
$j^*$
and
$\mathrm{pr} \circ j_*$
. To accomplish this, we need to further analyze these two adjoint functors.
Lemma 4.10.
Let
$X$
be a GM variety of dimension
$n$
and
$j\colon Y\hookrightarrow X$
be a smooth hyperplane section.
-
(i) If
$n=4,6$
, we have exact triangles
(35)and
\begin{equation} T_Y\to j^*\circ (\mathrm{pr}_X\circ j_*) \to \mathrm{id}_{\mathcal{K}u(Y)} \end{equation}
(36)
\begin{equation} \mathrm{id}_{\mathcal{K}u(X)}\to (\mathrm{pr}_X\circ j_*)\circ j^*\to T_X. \end{equation}
-
(ii) If
$n=5$
, we have exact triangles
(37)and
\begin{equation} T_X\to (\mathrm{pr}_X\circ j_*)\circ (j^*\circ T_X) \to \mathrm{id}_{\mathcal{K}u(X)} \end{equation}
(38)
\begin{equation} \mathrm{id}_{\mathcal{K}u(Y)}\to (j^*\circ T_X)\circ (\mathrm{pr}_X\circ j_*) \to T_Y. \end{equation}
Proof.
By [Reference Kuznetsov and PerryKP21, Corollary 4.19], the functor
$j^*\colon \mathcal{K}u(X)\to \mathcal{K}u(Y)$
is a spherical functor in the sense of [Reference Kuznetsov and PerryKP21, Definition 2.1]. Then applying Lemma 4.4 and [Reference Kuznetsov and PerryKP21, Corollary 2.3], the functor
$\mathrm{pr}_X\circ j_*$
is spherical as well. Thus according to Lemma 4.4 and [Reference Kuznetsov and PerryKP21, Equation (2.5)], we have an exact triangle
\begin{equation*}j^*\circ \mathrm{pr}_X\circ j_* \to \mathrm{id}_{\mathcal{K}u(Y)} \to \mathbf{T}_{j^*, \mathrm{pr}_X\circ j_*}\end{equation*}
where the functor
$\mathbf{T}_{j^*, \mathrm{pr}_X\circ j_*}$
is defined in [Reference Kuznetsov and PerryKP21, Definition 2.1].
When
$n=4,6$
, to show the existence of (35), we only need to determine
$\mathbf{T}_{j^*, \mathrm{pr}_X\circ j_*}$
. From [Reference Kuznetsov and PerryKP21, Corollary 4.19(1)], we see that
$S_{\mathcal{K}u(Y)}\cong \mathbf{T}_{j^*, \mathrm{pr}_X\circ j_*}\circ [1]$
, which gives (35).
For (36), since
$j^*E\in \mathcal{K}u(Y)$
for any object
$E\in \mathcal{K}u(X)$
, by (34) we have a functorial isomorphism
$\mathrm{pr}_X(j_*j^*E)=\boldsymbol {\mathrm{R}}_{\mathcal{U}_X} \boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}(j_*j^*E)$
. Hence, (36) follows from composing
$\boldsymbol {\mathrm{R}}_{\mathcal{U}_X} \boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}$
with the standard exact triangle
$\mathrm{id}_{\mathrm{D}^b(X)}\to j_*\circ j^*\to (-\otimes {\mathcal{O}}_X(-H))[1]$
.
Now assume that
$n=5$
. As in the case of odd dimension, we have exact triangles
\begin{equation} T_Y\to j^*\circ (\mathrm{pr}_X\circ j_*) \to \mathrm{id}_{\mathcal{K}u(Y)} \end{equation}
and
\begin{equation} \mathrm{id}_{\mathcal{K}u(X)}\to (\mathrm{pr}_X\circ j_*)\circ j^*\to T_X. \end{equation}
Therefore, (38) follows from composing
$T_Y$
with (39) and using the isomorphism
\begin{equation*}T_Y\circ j^*\circ (\mathrm{pr}_X\circ j_*)\cong j^*\circ T_X\circ (\mathrm{pr}_X\circ j_*)\end{equation*}
given in Lemma 4.4(ii). Similarly, (37) follows from composing
$T_X$
with (40) and using isomorphisms
\begin{equation*}T_X\circ (\mathrm{pr}_X\circ j_*)\circ j^*\cong (\mathrm{pr}_X\circ j_*)\circ T_Y \circ j^*\cong (\mathrm{pr}_X\circ j_*)\circ j^*\circ T_X\end{equation*}
given in Lemma 4.4(ii).
Now we investigate how our functors act on the numerical Grothendieck groups.
Lemma 4.11.
Let
$X$
be a GM variety of dimension
$n$
and
$j\colon Y\hookrightarrow X$
be a smooth hyperplane section.
-
(i) If
$n=4$
, then
$\mathrm{pr}_X(j_*\lambda _i)=\Lambda _i$
and
$j^*\Lambda _i=2\lambda _i$
for each
$i=1,2$
. -
(ii) If
$n=5$
, we define
$\lambda _i$
to be the unique numerical class satisfying
$j^*\lambda _i=\Lambda _i$
for each
$i=1,2$
. Then
$\mathrm{pr}_X(j_*\Lambda _i)=2\lambda _i$
for each
$i=1,2$
and
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))=\mathbb{Z}\lambda _1\oplus \mathbb{Z} \lambda _2$
with the Euler pairing
(31)
. -
(iii) If
$n=6$
, we define
$\Lambda _i:=\mathrm{pr}_X(j_*\lambda _i)$
. Then the restriction of the Euler pairing to the sublattice
$\mathbb{Z}\Lambda _1\oplus \mathbb{Z}\Lambda _2\subset \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
is given by
(33)
.
Proof.
When
$n=4$
, as each class has been described in (30) and (32), it is straightforward to verify that
$\mathrm{pr}_X(j_*\lambda _i)=\Lambda _i$
and
$j^*\Lambda _i=2\lambda _i$
for each
$i=1,2$
.
When
$n=5$
, by Lemma 4.3, Lemma 4.4, and (37), we have
\begin{equation*}\chi (j^*v, j^*w)=\chi (v, \mathrm{pr}_X(j_*j^*w))=\chi (v, \mathrm{pr}_X(j_*j^*T_X(w)))=2\chi (v,w),\end{equation*}
for
$i=1,2$
and any classes
$v,w\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
. Thus,
$\chi (\lambda _i, \lambda _j)=-\delta _{ij}$
. Hence,
$\lambda _1$
and
$\lambda _2$
generate
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
from the description of the Euler pairing (31). Therefore, if we take any class
$v\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
, then we can write
$v=a_1\lambda _1+a_2\lambda _2$
, which gives
$j^*v=a_1\Lambda _1+a_2\Lambda _2$
and
\begin{equation*}\chi (v, \mathrm{pr}_X(j_*\Lambda _i))=\chi (j^*v, \Lambda _i)=-2a_i=\chi (v,2\lambda _i).\end{equation*}
This shows that
$\mathrm{pr}_X(j_*\Lambda _i)=2\lambda _i$
.
When
$n=6$
, by Lemma 4.3, Lemma 4.4, and (35), we have
\begin{equation*}\chi (\mathrm{pr}_X(j_*v), \mathrm{pr}_X(j_*w))=\chi (j^*\mathrm{pr}_X(j_*v), w)=2\chi (v,w),\end{equation*}
for each
$i=1,2$
and classes
$v,w\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(Y))$
. Hence,
$\Lambda _1$
and
$\Lambda _2$
generate a rank two lattice, and the description of the Euler pairing follows from (ii).
The final step in proving Theorem 4.8 is to explore further stability conditions on
$\mathcal{K}u(X)$
and
$\mathcal{K}u(Y)$
. We begin by examining the stability of specific objects in these categories.
Lemma 4.12.
Let
$X$
be a GM variety of dimension
$n \geq 3$
.
-
(i) If
$n=3,5$
, any object
$E \in \mathcal{K}u(X)$
with
$\textrm {ext}^1_X(E, E) \leq 3$
is stable with respect to every Serre-invariant stability condition on
$\mathcal{K}u(X)$
. -
(ii) If
$n=4, 6$
and
$X$
is non-Hodge-special, any object
$E\in \mathcal{K}u(X)$
with
$\mathrm{ext}_X^1(E,E)\lt 8$
is stable with respect to every stability condition on
$\mathcal{K}u(X)$
.
Proof.
Part (i) follows from [Reference Feyzbakhsh and PertusiFP23, Proposition 3.4(c)]. For part (ii), one can easily check that Lemma 3.5(i) holds for
$\mathcal{K}u(X)$
. Then, by Lemma 3.4,
$E$
is
$\sigma$
-semistable for any stability condition
$\sigma$
on
$\mathcal{K}u(X)$
. If
$E$
is strictly
$\sigma$
-semistable and has at least two non-isomorphic Jordan–Hölder factors, then, by looking at the Jordan–Hölder filtration of
$E$
, we can also find an exact triangle
$A\to E\to B$
such that
$A$
is semistable with all Jordan–Hölder factors being a Jordan–Hölder factor
$A'$
of
$E$
and
$B$
does not have
$A'$
as a factor (see also [[Reference Fan, Liu and MaFLM23], Lemma 4.1]). This gives
\begin{equation*}\textrm {Hom}_X(A, B)=\textrm {Hom}_X(B, A[2])=0.\end{equation*}
However, this also leads to a contradiction as above by
$\textrm {ext}^1_X(E, E)\lt 8$
, Lemma 3.4, and Lemma 3.5(i). If
$E$
is strictly
$\sigma$
-semistable such that all Jordan–Hölder factors are isomorphic, then
$[E]\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
is not primitive. As
\begin{equation*}\chi (E,E)\geq 2-\textrm {ext}^1_X(E,E)\gt -6,\end{equation*}
from (33) we know that
$\chi (E,E)=-2$
or
$-4$
. But in each case,
$[E]$
is a primitive class, which makes a contradiction.
Proposition 4.13.
Let
$X$
be a non-Hodge-special GM fourfold or sixfold. Then all stability conditions on
$\mathcal{K}u(X)$
are in the same
$\widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
-orbit.
Proof.
By [Reference Kuznetsov and PerryKP23, Corollary 6.5], we only need to prove the statement when
$\dim X=4$
. Since
$X$
is non-Hodge-special,
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
is generated by
$\Lambda _1$
and
$\Lambda _2$
. Let
$j\colon Y\hookrightarrow X$
be a smooth GM threefold and
$F_1, F^{\prime}_1$
and
$F_2$
be objects in
$\mathcal{K}u(Y)$
defined in [Reference Jacovskis, Lin, Liu and ZhangJLLZ24, Lemma A.8]. Denote by
$\sigma _Y$
a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
. Then, by [Reference Jacovskis, Lin, Liu and ZhangJLLZ24, Lemma A.8], we have
\begin{equation*}\phi _{\sigma _Y}(F_2)-1\lt \phi _{\sigma _Y}(F_1)=\phi _{\sigma _Y}(F^{\prime}_1)\lt \phi _{\sigma _Y}(F_2),\end{equation*}
with
$\textrm {Hom}_Y(F_2, F_1[1])\neq 0$
and
$\textrm {Hom}_Y(F^{\prime}_1, F_2)\neq 0$
. Moreover, we have
\begin{equation*}\textrm {RHom}_Y(F_1, F_1)=\textrm {RHom}_Y(F^{\prime}_1, F^{\prime}_1)=\textrm {RHom}_Y(F_2, F_2)=\mathbb{C}\oplus \mathbb{C}^3[-1],\end{equation*}
as in the proof of [Reference Jacovskis, Lin, Liu and ZhangJLLZ24, Lemma A.7].
The same argument as in Lemma 3.7 implies that, for any stable object
$E\in \mathcal{K}u(Y)$
with respect to a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
with
$\textrm {Ext}^1_Y(E, E)=\mathbb{C}^n$
, we have
\begin{equation*}\textrm {ext}^1_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E))\leq 2n.\end{equation*}
Thus, by Lemma 4.12,
$\mathrm{pr}_X(j_*F_1), \mathrm{pr}_X(j_*F^{\prime}_1)$
and
$\mathrm{pr}_X(j_*F_2)$
are stable with respect to any stability condition on
$\mathcal{K}u(X)$
.
Since
$\phi _{\sigma _Y}(F^{\prime}_1)=\phi _{\sigma _Y}(T_Y(F^{\prime}_1))\gt \phi _{\sigma _Y}(F_2)-1$
, we have
$\textrm {Hom}(T_Y(F^{\prime}_1), F_2[-1])=0$
. After applying
$\textrm {Hom}(-,F_2)$
to the exact triangle
\begin{equation*}T_Y(F^{\prime}_1)\to j^*\mathrm{pr}_X(j_*F^{\prime}_1)\to F^{\prime}_1\end{equation*}
in (3), we obtain an injection
$0\neq \textrm {Hom}_Y(F^{\prime}_1, F_2)\hookrightarrow \textrm {Hom}_Y(j^*\mathrm{pr}_X(j_*F^{\prime}_1), F_2)$
, which implies
\begin{equation*}\textrm {Hom}_Y(j^*\mathrm{pr}_X(j_*F^{\prime}_1), F_2)=\textrm {Hom}_X(\mathrm{pr}_X(j_*F^{\prime}_1), \mathrm{pr}_X(j_*F_2))\neq 0.\end{equation*}
Similarly, since
$\phi _{\sigma _Y}(F_2)=\phi _{\sigma _Y}(T_Y(F_2))\gt \phi _{\sigma _Y}(F_1)$
, applying
$\textrm {Hom}_Y(-,F_1)$
to the exact triangle
\begin{equation*}T_Y(F_2)\to j^*\mathrm{pr}_X(j_*F_2)\to F_2\end{equation*}
in (3), we get an injection
$0\neq \textrm {Hom}_Y(F_2, F_1[1])\hookrightarrow \textrm {Hom}_Y(j^*\mathrm{pr}_X(j_*F_2), F_1[1])$
, which implies
\begin{equation*}\textrm {Hom}_Y(j^*\mathrm{pr}_X(j_*F_2), F_1[1])=\textrm {Hom}_X(\mathrm{pr}_X(j_*F_2), \mathrm{pr}_X(j_*F_1)[1])\neq 0.\end{equation*}
By the same argument, we have
$\textrm {Hom}_Y(j^*\mathrm{pr}_X(j_*F_1), F^{\prime}_1[3])=\textrm {Hom}_X(\mathrm{pr}_X(j_*F_1), \mathrm{pr}_X(j_*F^{\prime}_1)[3])= 0$
as well. Therefore, if we define
$Q_2:=\mathrm{pr}_X(j_*F^{\prime}_1)$
,
$Q_2':=\mathrm{pr}_X(j_*F_1)$
, and
$Q_1:=\mathrm{pr}_X(j_*F_2)$
, then they satisfy the assumptions in [Reference Feyzbakhsh and PertusiFP23, Lemma 3.7]. Thus the result follows from [Reference Feyzbakhsh and PertusiFP23, Theorem 3.2].
Now we are prepared to prove the main theorem in this subsection.
Proof of Theorem 4.8 . We start by fixing the following data:
-
– categories
when
\begin{equation*}\mathcal{K}_1:=\mathcal{K}u(Y),\quad \mathcal{K}_2:=\mathcal{K}u(X)\end{equation*}
$n$
is even, andwhen
\begin{equation*}\mathcal{K}_1:=\mathcal{K}u(X),\quad \mathcal{K}_2:=\mathcal{K}u(Y)\end{equation*}
$n$
is odd;
-
– classes
$\lambda _i$
and
$\Lambda _i$
as in (30), (32), and Lemma 4.11 for each
$i=1,2$
; -
– auto-equivalences
$T_1$
(respectively
$T_2$
) are
$T_Y$
(respectively
$T_X$
) as in Lemma 4.3 when
$n$
is even and
$T_X$
(respectively
$T_Y$
) when
$n$
is odd; -
– functors
when
\begin{equation*}\Phi ^*:=j^*|_{\mathcal{K}u(X)},\quad \Phi _*:=\mathrm{pr}_X\circ j_*\end{equation*}
$n$
is even, andwhen
\begin{equation*}\Phi ^*:=\mathrm{pr}_X\circ j_*,\quad \Phi _*:=j^*\circ T_X\end{equation*}
$n$
is odd.
We fix Serre-invariant stability conditions
$\sigma _1 = (Z_1, \mathcal{A}_1)$
and
$\sigma _2 = (Z_2, \mathcal{A}_2)$
on
$\mathcal{K}_1$
and
$\mathcal{K}_2$
, respectively. By Proposition 4.13 and Theorem 4.7, up to
$\widetilde {\mathrm{GL}}^+(2,\mathbb{R})$
-action, we can assume that
\begin{equation*}Z_1(\lambda _1) = Z_2(\Lambda _1) = -1, \quad Z_1(\lambda _2) = Z_2(\Lambda _2) = \mathfrak{i}.\end{equation*}
Claim. We can furthermore assume that, for any
$\sigma _1$
-stable object
$A\in \mathcal{K}_1$
with
$[A]\in \{\pm \lambda _1, \pm \lambda _2\}$
, the object
$\Phi _*(A)$
is
$\sigma _2$
-stable with
$\phi _{\sigma _1}(A)=\phi _{\sigma _2}(\Phi _*(A))$
.
Proof of the Claim.
Since
$\textrm {ext}^1(A,A)\leq 3$
, we have
$\textrm {ext}^1(\Phi _*(A),\Phi _*(A))\leq 6$
via the same argument as in Lemma 3.7 and so
$\Phi _*(A)$
is
$\sigma _2$
-stable by Lemma 4.12. Since
$Z_1=Z_2$
, we know that
\begin{equation} \phi _{\sigma _1}(A)-\phi _{\sigma _2}(\Phi _*(A))\in 2\mathbb{Z}. \end{equation}
Let
$D_1$
and
$D_2$
be two
$\sigma _1$
-stable objects defined in [Reference Fan, Liu and MaFLM23, Lemma 5.6] when
$\mathcal{K}_1$
is the Kuznetsov component of a GM threefold. When
$\mathcal{K}_1$
is the Kuznetsov component of a GM fivefold, we denote the image of
$D_1$
and
$D_2$
under the equivalence in [Reference Kuznetsov and PerryKP23] by
$D_1$
and
$D_2$
as well. Up to shift and relabeling the subscript, we can fix the hearts
$\mathcal{A}_1$
and
$\mathcal{A}_2$
so that
$D_1, D_2\in \mathcal{A}_3$
,
$[D_i]=\lambda _i$
, and
\begin{equation*}\phi _{\sigma _1}(D_1)=\phi _{\sigma _2}(\Phi _*(D_1))=1.\end{equation*}
As in [Reference Fan, Liu and MaFLM23, Lemma 5.6], we have
$\textrm {Hom}(D_2, D_1)\neq 0$
.
Firstly, we claim that
\begin{equation*}\phi _{\sigma _1}(D_2)=\phi _{\sigma _2}(\Phi _*(D_2))=\tfrac {1}{2}.\end{equation*}
Indeed, applying
$\textrm {Hom}(-, D_1)$
to the exact triangle in (3), we get
\begin{equation*}\textrm {Hom}(\Phi ^*\Phi _*(D_2), D_1)=\textrm {Hom}(\Phi _*(D_2), \Phi _*(D_1))\neq 0.\end{equation*}
Thus
\begin{equation*}\phi _{\sigma _2}(\Phi _*(D_1))-2\lt \phi _{\sigma _2}(\Phi _*(D_2))\lt \phi _{\sigma _2}(\Phi _*(D_1)),\end{equation*}
since
\begin{equation*}\textrm {Hom}(\Phi _*(D_2), \Phi _*(D_1))=\textrm {Hom}(\Phi _*(D_1), \Phi _*(D_2)[2])\neq 0.\end{equation*}
As
$\phi _{\sigma _2}(\Phi _*(D_1))=1$
, we get
\begin{equation*}-1\lt \phi _{\sigma _2}(\Phi _*(D_2))\lt 1,\end{equation*}
which implies that
$\phi _{\sigma _2}(\mathrm{pr}_X(j_*D_2))=\phi _{\sigma _1}(D_2)=\frac {1}{2}$
by (41) and the claim follows.
Now let
$A\in \mathcal{A}_3$
be an arbitrary
$\sigma _1$
-stable object with
$[A]=\lambda _i$
for
$i\in \{1,2\}$
. From
$\chi (D_i, A)\lt 0$
and the fact that
$\mathcal{A}_1$
has homological dimension
$2$
, we get
\begin{equation*}\textrm {Hom}(D_i, A[1])=\textrm {Hom}(A, T_1(D_i)[1])\neq 0.\end{equation*}
Hence, applying
$\textrm {Hom}(-, A)$
to the exact triangle in (3), we see that
\begin{equation*}\textrm {Hom}(\Phi ^*\Phi _*(D_i), A[1])=\textrm {Hom}(\Phi _*(D_i), \Phi _*(A)[1])\neq 0,\end{equation*}
which implies that
$\phi _{\sigma _2}(\Phi _*(D_i))-1\lt \phi _{\sigma _2}(\Phi _*(A))$
. Similarly, we have
$\phi _{\sigma _2}(\Phi _*(A))\lt \phi _{\sigma _2}(\Phi _*(D_i))+1$
as well. Since
$\phi _{\sigma _1}(A)=\phi _{\sigma _1}(D_i)=\phi _{\sigma _2}(\Phi _*(D_i))$
, we get
\begin{equation*}\phi _{\sigma _1}(A)-1\lt \phi _{\sigma _2}(\Phi _*(A))\lt \phi _{\sigma _1}(A)+1,\end{equation*}
and the result follows from (41).
Finally, by Theorem 4.7 and Proposition 4.13,
$\sigma _i$
is
$T_i$
-invariant for each
$i=1, 2$
. Moreover, [Reference Jacovskis, Lin, Liu and ZhangJLLZ24, Theorem 7.12, Theorem 8.9] show that there is a two-dimensional family of stable objects of class
$\lambda _i$
for each
$i=1,2$
in
$\mathcal{K}_1$
. Therefore, Lemmas 4.3, 4.4, 4.10, and 4.11 demonstrate that the above given data satisfies all conditions (C1) to (C8) in Section 3, and thus the claim follows from Theorem 3.1.
As a corollary, we have the following.
Corollary 4.14.
Let
$X$
be a non-Hodge-special GM variety of dimension
$n=4,6$
and
$j: Y\hookrightarrow X$
be a smooth hyperplane section. If
$E\in \mathcal{K}u(Y)$
is a stable object with respect to Serre-invariant stability conditions such that
$[E]\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(Y))$
is primitive, then
$\mathrm{pr}_X(j_*E)$
is stable with respect to any stability condition on
$\mathcal{K}u(X)$
.
Proof.
By Theorem 4.8,
$\mathrm{pr}_X(j_*E)$
is semistable with respect to any stability condition on
$\mathcal{K}u(X)$
. As
$[E]$
is primitive, we know that
$[\mathrm{pr}_X(j_*E)]$
is primitive as well by Lemma 4.11(i). Hence,
$\mathrm{pr}_X(j_*E)$
is stable.
Similarly, we have the following.
Corollary 4.15.
Let
$X$
be a non-Hodge-special GM fourfold and let
$j: X\hookrightarrow Y$
realize
$X$
as a smooth hyperplane section of a GM fivefold
$Y$
. If
$E\in \mathcal{K}u(Y)$
is an object stable with respect to Serre-invariant stability conditions such that
$[E]\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(Y))$
is primitive, then
$j^*E$
is stable with respect to any stability condition on
$\mathcal{K}u(X)$
.
4.3 General cases
In this section, we aim to extend Theorem 4.8(i) to the general case, and prove the following.
Theorem 4.16.
Let
$a,b$
be a pair of integers. Then for a general GM fourfold
$X$
and its smooth hyperplane section
$j \colon Y\hookrightarrow X$
, if
$E\in \mathcal{K}u(Y)$
is a
$\sigma _Y$
-semistable object of class
$a\lambda _1+b\lambda _2$
, then
$\mathrm{pr}_X(j_*E)$
is
$\sigma _X$
-semistable, where
$\sigma _Y$
is a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
.
Note that the generality assumption in Theorem 4.16 means that the statement holds when
$X$
is in an open dense substack of the moduli stack of smooth ordinary GM fourfolds. Moreover, as we will see in the proof, such a substack depends on the choice of integers
$a,b$
.
Proof.
Let
$\mathcal{M}^{\mathrm{GM}}_4$
be the moduli stack of smooth ordinary GM fourfolds, which is a smooth irreducible Deligne–Mumford stack of finite type and separated over
$\mathbb{C}$
(cf. [Reference Kuznetsov and PerryKP18, Proposition A.2] and [Reference Debarre and KuznetsovDK20, Corollary 5.12]). For a smooth projective morphism
$\mathcal{X}\to S$
over a smooth scheme
$S$
over
$\mathbb{C}$
and a
$S$
-linear semi-orthogonal component
$\mathcal{D}\subset \mathrm{D}^b(\mathcal{X})$
, we denote by
$\mathcal{M}_{\mathrm{pug}}(\mathcal{D}/S)$
the moduli stack of universally gluable objects in
$\mathcal{D}$
over
$S$
, defined in [BLM+21, Definition 9.1]. According to [BLM+21, Lemma 21.12], when
$X$
is a GM fourfold,
$\mathcal{M}^X_{\sigma _X}(a,b)$
is an open substack of
$\mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})$
. And, by Lemma 3.7, for any smooth hyperplane section
$j\colon Y\hookrightarrow X$
we have a morphism
\begin{equation*}\gamma \colon \mathcal{M}^Y_{\sigma _Y}(a,b)\to \mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})\end{equation*}
given by
$E\mapsto \mathrm{pr}_X(j_*E)$
at the level of
$\mathbb{C}$
-point. Therefore, given a
$\mathbb{C}$
-point
$s\colon \textrm {Spec}(\mathbb{C})\to \mathcal{M}^Y_{\sigma _Y}(a,b)$
corresponding to a
$\sigma _Y$
-semistable object
$E\in \mathcal{K}u(Y)$
, to show that
$\mathrm{pr}_X(j_*E)\in \mathcal{K}u(X)$
is
$\sigma _X$
-semistable, we need to show that the composition
\begin{equation*}\textrm {Spec}(\mathbb{C})\xrightarrow {s}\mathcal{M}^Y_{\sigma _Y}(a,b)\xrightarrow {\gamma }\mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})\end{equation*}
factors through the natural open immersion
$\mathcal{M}^X_{\sigma _X}(a,b)\subset \mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})$
. By [BLM+21, Theorem 12.17(3)], this is equivalent to finding an extension
$\kappa$
of
$\mathbb{C}$
such that the composition
\begin{equation*}\textrm {Spec}(\kappa )\xrightarrow {s_{\kappa }}\mathcal{M}^Y_{\sigma _Y}(a,b)\xrightarrow {\gamma }\mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})\end{equation*}
factors through
$\mathcal{M}^X_{\sigma _X}(a,b)\subset \mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})$
. We denote by
$|\mathcal{M}|$
the associated topological space of an algebraic stack
$\mathcal{M}$
. From the definition of topological spaces of algebraic stacks (cf. [Sta25, Tag 04XG]), it is also equivalent to say
\begin{equation*} s_{\kappa }\in |\mathcal{M}^X_{\sigma _X}(a,b)|\cap \gamma (|\mathcal{M}^Y_{\sigma _Y}(a,b)|)\subset |\mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})|. \end{equation*}
Thus, to prove that
$\mathrm{pr}_X(j_*E)$
is
$\sigma _X$
-semistable for any
$\sigma _Y$
-semistable object
$E\in \mathcal{K}u(Y)$
, we only need to show the inclusion
\begin{equation*}\gamma (|\mathcal{M}^Y_{\sigma _Y}(a,b)|)\subset |\mathcal{M}^X_{\sigma _X}(a,b)|\end{equation*}
in
$|\mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(X)/\mathbb{C})|$
. Then the result can be deduced from Theorem 4.8 and Chevalley’s theorem on constructible subsets in algebraic stacks as follows.
First, we claim that there exists a smooth connected scheme
$S$
of finite type over
$\mathbb{C}$
and a family of smooth ordinary GM fourfolds
$\pi _4\colon \mathcal{X}\to S$
with a closed subscheme
$\mathcal{Y}\subset \mathcal{X}$
such that
$\pi _3\colon \mathcal{Y}\hookrightarrow \mathcal{X}\to S$
is a family of smooth GM threefolds, and for each general GM fourfold
$X$
and its smooth hyperplane section
$j\colon Y\hookrightarrow X$
there exists a point
$s\in S$
such that
$\mathcal{Y}_s\hookrightarrow \mathcal{X}_s$
is isomorphic to
$j$
.
Indeed, let
$\pi _1\colon \mathcal{X}'\to W$
be a family of smooth GM fourfolds such that
$W$
is smooth connected and of finite type over
$\mathbb{C}$
and any general smooth GM fourfold occurs as a fiber of
$\pi _1$
. Such a family exists by [Reference Kuznetsov and PerryKP18, Proposition A.2], since we can take
$W$
to be a connected component of a smooth atlas of
$\mathcal{M}_4^{\mathrm{GM}}$
which dominants the moduli stack
$\mathcal{M}_4^{\mathrm{GM}}$
of smooth ordinary GM fourfolds, and
$\pi _1$
to be the pull-back of the universal family to
$W$
. Let
$S$
be the open subscheme of the relative Hilbert scheme
$\mathrm{Hilb}_{\mathcal{X}'/W}^{{\mathcal{O}}(1)}$
of hyperplane sections of
$\mathcal{X}'$
over
$W$
, parameterizing smooth hyperplane sections. We denote by
$q\colon S\to W$
the natural morphism, which is of finite type. We define
$\mathcal{X}:=\mathcal{X}'\times _W S$
and
$\mathcal{Y}\subset \mathcal{X}$
to be the universal closed subscheme. As
$S$
is an open subset of a
$\mathbb{P}^8$
-bundle over
$W$
, we see that
$S$
is smooth connected and of finite type over
$\mathbb{C}$
. Then the other statements of the claim follow from the construction. By replacing
$W$
with its open dense subscheme and its covering, we can assume that
$\pi _4\colon \mathcal{X}\to S$
is a family of ordinary GM fourfolds satisfying two assumptions in [Reference Perry, Pertusi and ZhaoPPZ22, Proposition 5.3].
By [Reference Bayer and PerryBP23, Lemma 5.9], there is an
$S$
-linear semi-orthogonal component
$\mathcal{K}u(\mathcal{X})\subset \mathrm{D}_{\mathrm{perf}}(\mathcal{X})$
such that
$\mathcal{K}u(\mathcal{X})_s\simeq \mathcal{K}u(\mathcal{X}_s)$
for any
$s\in S$
. Therefore, using the construction in [Reference Perry, Pertusi and ZhaoPPZ22, Section 4], there is a stability condition
$\underline {\sigma }$
on
$\mathcal{K}u(\mathcal{X})$
over
$S$
such that
$\underline {\sigma }|_s\in \textrm {Stab}^{\circ }(\mathcal{K}u(\mathcal{X}_s))$
for each
$s\in S$
. Similarly, by [BLM+21, Theorem 23.1, Proposition 26.1], there exists a stability condition
$\underline {\sigma '}$
on
$\mathcal{K}u(\mathcal{Y})$
over
$S$
in the sense of [BLM+21, Definition 21.15] such that
$\underline {\sigma '}|_s$
is Serre invariant for each
$s\in S$
.
We denote by
$\mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(\mathcal{X})/S)$
the moduli stack of universally gluable objects in
$\mathcal{K}u(\mathcal{X})$
over
$S$
, defined in [BLM+21, Definition 9.1]. By Lemma 3.7, we have a morphism
\begin{equation*}\gamma '\colon \mathcal{M}^{\mathcal{Y}}_{\underline {\sigma '}}(a,b)\to \mathcal{M}_{\mathrm{pug}}(\mathcal{K}u(\mathcal{X})/S)\end{equation*}
induced by the push-forward along the embedding
$\mathcal{Y}\hookrightarrow \mathcal{X}$
and the projection functor
$\mathrm{D}_{\mathrm{perf}}(\mathcal{X})\to \mathcal{K}u(\mathcal{X})$
. We set
$\mathcal{U}:=\gamma '^{-1}(\mathcal{M}^{\mathcal{X}}_{\underline {\sigma }}(a,b))$
, which is an open substack of
$\mathcal{M}^{\mathcal{Y}}_{\underline {\sigma '}}(a,b)$
by [BLM+21, Lemma 21.12]. Let
$\mathcal{Z}$
be the complement of
$|\mathcal{U}|$
in
$|\mathcal{M}^{\mathcal{Y}}_{\underline {\sigma '}}(a,b)|$
and
$p\colon \mathcal{M}^{\mathcal{Y}}_{\underline {\sigma '}}(a,b)\to S$
be the natural morphism, which is of finite type by [BLM+21, Theorem 21.24]. Then we define
$\mathcal{V}:=|\mathcal{M}^{\mathcal{Y}}_{\underline {\sigma '}}(a,b)|\setminus p^{-1}(p(\mathcal{Z}))$
, which is a constructible subset by [Reference Laumon and Moret-BaillyLMB00, Corollaire (5.9.2), Théorème (5.9.4)]. Let
$V:=p(\mathcal{V})$
, which is a constructible subset of
$S$
. As
$p^{-1}(V)\cap \mathcal{Z}=\varnothing$
, we know that, for any
$\mathbb{C}$
-point
\begin{equation*}s\colon \textrm {Spec}(\mathbb{C})\to V\subset S \subset \mathrm{Hilb}_{\mathcal{X}'/W}^{{\mathcal{O}}(1)},\end{equation*}
we have a GM fourfold
$\mathcal{X}_s$
and a smooth hyperplane section
$j_s\colon \mathcal{Y}_s\hookrightarrow \mathcal{X}_s$
such that
$\mathrm{pr}_{\mathcal{X}_s}(j_{s*}(E))$
is
$\underline {\sigma }|_s$
-semistable for any
$\underline {\sigma '}|_s$
-semistable object
$E\in \mathcal{K}u(\mathcal{Y}_s)$
of class
$a\lambda _1+b\lambda _2$
.
We define a new constructible subset
$T:=q(S\setminus q^{-1}(q(S\setminus V)))\subset W$
. Since
$q^{-1}(T)\subset V$
, we know that, for any
$s\colon \textrm {Spec}(\mathbb{C})\to T$
, we have a GM fourfold
$\mathcal{X}'_s$
satisfying the statement of the theorem. By Theorem 4.8, any very general GM fourfold represents a point in
$T$
, and hence
$T$
is dense in
$W$
. As
$T$
is constructible and dense in
$W$
, it contains an open dense subscheme of
$W$
by [Sta25, Tag 005K] and the result follows.
Using the same idea in the proof of Theorem 4.16, but just replacing the roles of
$X,Y$
, and
$\mathrm{pr}_X\circ j_*$
by
$Y,X$
, and
$j^*$
, respectively, we have the following theorem.
Theorem 4.17.
Let
$a,b$
be a pair of integers. Then for a general GM threefold
$Y$
and a general GM fourfold
$X$
with an embedding
$j \colon Y\hookrightarrow X$
, if
$F\in \mathcal{K}u(X)$
is a
$\sigma _X$
-semistable object of class
$a\Lambda _1+b\Lambda _2$
, then
$j^*F$
is
$\sigma _Y$
-semistable, where
$\sigma _Y$
is a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
.
5. First application: Lagrangian families
In this section, we explain how to construct Lagrangian families of Bridgeland moduli spaces via Theorems 4.8 and 4.16.
5.1 Properties of moduli spaces
Firstly, we prove some basic properties of moduli spaces of semistable objects in
$\mathcal{K}u(Y)$
for a GM threefold
$Y$
.
Proposition 5.1.
Let
$Y$
be a GM threefold and
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
. Let
$a,b$
be a pair of coprime integers.
-
(i) The moduli space
$M_{\sigma _Y}^Y(a,b)$
is normal of pure dimension
$a^2+b^2+1$
. -
(ii) If
$Y$
is a general
Footnote
6
GM threefold, then
$M_{\sigma _Y}^Y(a,b)$
is smooth and projective.
Proof.
Let
$[F]\in M^Y_{\sigma _Y}(a,b)$
. As
$a,b$
are coprime, we know that
$F$
is
$\sigma _Y$
-stable. If
$F\neq S_{\mathcal{K}u(Y)}(F)[-2]$
, then
$\textrm {Ext}_Y^2(F,F)=0$
and
$[F]\in M^Y_{\sigma _Y}(a,b)$
is a smooth point. If
$F= S_{\mathcal{K}u(Y)}(F)[-2]$
, then following the notation in [Reference Arbarello and SaccàAS18, Proposition 6.1], the same argument in the proof of [Reference Arbarello and SaccàAS18, Proposition 6.1] shows that there is a quiver
$\overline {Q}$
and a dimension vector
$\mathbf{n}$
such that the moment map
\begin{equation*}\mu \colon \mathrm{Rep}(\overline {Q}, \mathbf{n})\to \mathfrak{gl}(\mathbf{n})^{\vee }\end{equation*}
corresponds to the second-order term of the Kuranishi map
$\kappa _2\colon \textrm {Ext}^1(F,F)\to \textrm {Ext}^2(F,F)$
under
$\textrm {Aut}(F)$
-equivariant isomorphisms
\begin{equation*}\mathrm{Rep}(\overline {Q}, \mathbf{n})\cong \textrm {Ext}^1(F,F), \quad \mathfrak{gl}(\mathbf{n})^{\vee }\cong \textrm {Ext}^2(F,F).\end{equation*}
And by [Reference Chen, Pertusi and ZhaoCPZ24, Proposition 3.11], there is a local analytic
$\textrm {Aut}(F)$
-equivariant isomorphism
\begin{equation*}(M^Y_{\sigma _Y}(a,b), [F])\cong (\kappa ^{-1}_2(0)\mathbin {/\mkern -5mu/}\textrm {Aut}(F), 0)\end{equation*}
of germs of analytic spaces. Therefore, we get a local
$\textrm {Aut}(F)$
-equivariant isomorphism
\begin{equation*}(M^Y_{\sigma _Y}(a,b), [F])\cong (\mu ^{-1}(0)\mathbin {/\mkern -5mu/}\textrm {Aut}(F), 0),\end{equation*}
and (i) follows from [Reference Crawley-BoeveyCB01, Corollary 1.4] and [Reference Crawley-BoeveyCB03, Theorem 1.1].
Next, we prove (ii). By [Reference Perry, Pertusi and ZhaoPPZ22, Theorem 1.3(2)], if
$Y$
is the branch divisor of a non-Hodge-special GM fourfold, then (ii) holds. Now consider a family
$\pi \colon \mathcal{Y}\to S$
of smooth GM threefolds such that
$S$
is a smooth connected scheme, of finite type over
$\mathbb{C}$
and each smooth ordinary GM threefold appears as a fiber in this family. Such
$S$
exists by [Reference Kuznetsov and PerryKP18, Proposition A.2]. By [Reference Bayer and PerryBP23, Lemma 5.9], there is a
$S$
-linear semi-orthogonal decomposition
\begin{equation*}\mathrm{D}_{\mathrm{perf}}(\mathcal{Y})=\langle \mathcal{K}u(\mathcal{Y}), \pi ^*(\mathrm{D}_{\mathrm{perf}}(S))\otimes {\mathcal{O}}_{\mathcal{Y}}, \pi ^*(\mathrm{D}_{\mathrm{perf}}(S))\otimes \mathcal{U}^{\vee }_{\mathcal{Y}}\rangle, \end{equation*}
such that
$\mathcal{K}u(\mathcal{Y})_s\simeq \mathcal{K}u(\mathcal{Y}_s)$
. From [BLM+21, Corollary 26.2], or more precisely [BLM+21, Theorem 23.1, Proposition 26.1], there is a stability condition
$\underline {\sigma }$
on
$\mathcal{K}u(\mathcal{Y})$
over
$S$
such that, for any
$s\in S$
, the restriction
$(\underline {\sigma })|_s$
is a stability condition on
$\mathcal{K}u(\mathcal{Y}_s)$
constructed in [Reference Bayer, Lahoz, Macrì and StellariBLMS23] and hence is Serre invariant.
Let
$v\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(\mathcal{Y})/S)$
be the class such that
$v_s=a\lambda _1+b\lambda _2\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(\mathcal{Y}_s))$
for any
$s\in S$
. We define
\begin{equation*}p\colon M_{\underline {\sigma }}(\mathcal{K}u(\mathcal{Y})/S, v)\to S\end{equation*}
to be the good moduli space of the moduli stack
$\mathcal{M}_{\underline {\sigma }}(\mathcal{K}u(\mathcal{Y})/S, v)$
of families of geometrically
$\underline {\sigma }$
-stable objects of class
$v$
over
$S$
in the sense of [BLM+21, Definition 21.11(1)]. According to [BLM+21, Theorem 21.24(3)], such a moduli space exists and
$p$
is proper. Then
$p^{-1}(s)\cong M^{\mathcal{Y}_s}_{\underline {\sigma }|_s}(a,b)$
for each
$s\in S$
. By [Sta25, Tag 052A], after replacing
$S$
with an open dense subset, we can assume that
$p$
is flat. Therefore, we can further shrink
$S$
to assume that
$p$
is smooth as the generic fiber of
$p$
is smooth by [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.3(2)]. This proves the smoothness part of (ii). Now the projectivity follows from [BLM+21, Theorem 21.15] and [Reference Villalobos-PazVP21, Corollary 3.4].
5.2 Lagrangian subvarieties
Let
$X$
be a GM variety of dimension
$4$
or
$6$
and
$j \colon Y \hookrightarrow X$
be a smooth hyperplane section. As before, let
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
. For any pair of integers
$a,b$
, we denote by
$M^X_{\sigma _X}(a,b)$
(respectively
$M^Y_{\sigma _Y}(a,b)$
) the moduli space of
$\sigma _X$
(respectively
$\sigma _Y$
)-semistable objects in
$\mathcal{K}u(X)$
(respectively
$\mathcal{K}u(Y)$
) of class
$a\Lambda _1 +b\Lambda _2$
(respectively
$a\lambda _1 +b\lambda _2$
). By Theorems 4.8 and 4.16, we have a morphism
$M_{\sigma _Y}^Y(a,b)\to M^X_{\sigma _X}(a,b)$
induced by
$\mathrm{pr}_X\circ j_*$
when
$X$
is general and
$\dim X=4$
, or
$X$
is non-Hodge-special and
$\dim X=6$
. The following lemma describes the local properties of this morphism.
Lemma 5.2.
Let
$E\in \mathcal{K}u(Y)$
be a
$\sigma _Y$
-semistable object.
-
(i) (Injectivity of tangent maps.) If
$E$
is
$\sigma _Y$
-stable and
$E\neq T_Y(E)$
, then the natural map
is injective.
\begin{equation*}\textrm {Ext}^1_Y(E,E)\to \textrm {Ext}^1_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E))\end{equation*}
-
(ii) (Fibers.) Let
$E'\neq E\in \mathcal{K}u(Y)$
be another
$\sigma _Y$
-semistable object with
$\phi _{\sigma _Y}(E)=\phi _{\sigma _Y}(E')$
.-
(a) If
$\mathrm{pr}_X(j_*E)\cong \mathrm{pr}_X(j_*E')$
, then
$E\oplus T_Y(E)$
is S-equivalent to
$E'\oplus T_Y(E')$
. -
(b) If
$E$
and
$E'$
are
$\sigma _Y$
-stable, then
if and only if
\begin{equation*}\textrm {Hom}_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E'))\neq 0,\end{equation*}
$E'\cong T_Y(E)$
and the triangle
(42)is splitting.
\begin{equation} T_Y(E) \to j^*\mathrm{pr}_X(j_*E) \to E \end{equation}
-
Proof.
Note that the exact triangle (42) can be obtained from applying the triangle (35) on
$E$
. Then applying
$\textrm {Hom}_Y(-, E)$
to (42), we get an exact sequence
\begin{equation*}\textrm {Hom}_Y(T_Y(E), E)\to \textrm {Ext}^1_Y(E,E)\to \textrm {Ext}^1_Y(j^*\mathrm{pr}_X(j_*E), E)=\textrm {Ext}^1_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E)).\end{equation*}
If
$E$
is
$\sigma _Y$
-stable, then
$T_Y(E)$
is
$\sigma _Y$
-stable and has the same phase as
$E$
by Lemma 3.2, so part (i) follows.
For part (ii)(b), by Lemma 4.4 we have
$\textrm {Hom}_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E'))=\textrm {Hom}_Y(j^*\mathrm{pr}_X(j_*E), E')$
, and then the result also follows from applying
$\textrm {Hom}_Y(-, E')$
to the triangle (42). For part (ii)(a), as there is an isomorphism
$\mathrm{pr}_X(j_*E)\cong \mathrm{pr}_X(j_*E')$
, we get
$j^*\mathrm{pr}_X(j_*E)\cong j^*\mathrm{pr}_X(j_*E')$
. Then the result follows from the exact triangle (42).
By [Reference Perry, Pertusi and ZhaoPPZ22], the moduli space
$M^X_{\sigma _X}(a,b)$
is a projective hyperkähler manifold when
$\gcd (a,b)=1$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
is generic with respect to
$a\Lambda _1+b\Lambda _2$
. Moreover, as
$\mathcal{K}u(X)$
is 2-Calabi–Yau, the holomorphic 2-form on
$M^X_{\sigma _X}(a,b)$
at a point
$[E]$
is given by the Yoneda pairing as in [Reference MukaiMuk87] (see also [Reference Kuznetsov and MarkushevichKM09] and [BLM+21, Lemma 32.5]):
\begin{equation*}\textrm {Ext}^1_X(E,E)\times \textrm {Ext}^1_X(E,E)\to \textrm {Ext}^2_X(E,E)\xrightarrow {\mathrm{\cong }} \mathbb{C}.\end{equation*}
Theorem 5.3.
Let
$a,b$
be a pair of integers and
$X$
be a general GM fourfold or non-Hodge-special GM sixfold with a given smooth hyperplane section
$j \colon Y \hookrightarrow X$
. Let
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
. Then the functor
$\mathrm{pr}_X\circ j_*$
induces a finite morphism
\begin{equation*}r\colon M^Y_{\sigma _Y}(a,b) \to M^X_{\sigma _X}(a,b).\end{equation*}
Moreover, if
$\gcd (a,b)=1$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
is generic with respect to
$a\Lambda _1+b\Lambda _2$
, then:
-
(i)
$r$
is generically unramified and its image is a Lagrangian subvariety in
$M^X_{\sigma _X}(a,b)$
, and
$r$
is unramified when
$Y$
is a general hyperplane section; and
-
(ii) the restriction
$r|_M$
of
$r$
to any irreducible component
$M$
of
$M^Y_{\sigma _Y}(a,b)$
is either a birational map onto its image or a finite morphism of degree
$2$
and étale on the smooth locus of
$M$
.
Proof.
By Theorem 4.8, we have a morphism
$r\colon M^Y_{\sigma _Y}(a,b) \to M^X_{\sigma _X}(a,b)$
. Moreover,
$r$
is quasi-finite by Lemma 5.2(ii)(a). As moduli spaces on both sides are proper by [BLM+21, Theorem 21.24(3)],
$r$
is finite (cf. [Sta25, Tag 0A4X]).
Now assume that
$\gcd (a,b)=1$
. As the space
$M^Y_{\sigma _Y}(a,b)$
is reduced by Proposition 5.1(i), it is generically smooth, or in other words,
$E\neq T_Y(E)$
for a general point
$[E]\in M^Y_{\sigma _Y}(a,b)$
. Then
$r$
is generically unramified by Lemma 5.2(i). Moreover, when
$Y$
is a general hyperplane section,
$M^Y_{\sigma _Y}(a,b)$
is smooth by Proposition 5.1, and hence
$r$
is unramified.
Let
$U \subset M^Y_{\sigma _Y}(a,b)$
be the open dense locus of stable objects
$E$
satisfying
$E\neq T_Y(E)$
, and let
$L$
be the image of
$r$
. First of all,
$L$
is reduced because
$M^Y_{\sigma _Y}(a,b)$
is reduced. Hence,
$L$
is generically smooth since we work over
$\mathbb{C}$
. Moreover, Lemma 5.2(ii) implies that
$r|_U$
is of relative dimension zero, and thus
$\dim \overline {U}=\dim L$
. Furthermore, by Lemma 5.2(i), the induced morphism
$r|_U$
is injective at the level of tangent maps. Therefore, shrinking
$U$
if necessary, we can assume that
$r|_U\colon U\to r(U)$
is étale. Hence, the tangent space at any point in
$r(U)$
can be naturally identified with
$\textrm {Ext}^1_Y(E,E)$
for some
$[E]\in U$
.
Since
$\dim \overline {U}=\dim L=\frac {1}{2}\dim M^X_{\sigma _X}(a,b)$
, to show that
$L$
is Lagrangian, it suffices to prove that the canonical holomorphic symplectic 2-form of
$M_{\sigma _X}^X(a,b)$
becomes zero after restricting to
$L$
. Because
$r(U)$
is dense in
$L$
, this is equivalent to proving that the restricting 2-form vanishes at every point of
$r(U)$
.
For
$[F]\in M_{\sigma _X}^X(a,b)$
,
$\mathrm{Ext}^1_X(F,F)$
is identified with the tangent space
$T_{[F]} M_{\sigma _X}^X(a,b)$
. Then the holomorphic symplectic 2-form of
$M_{\sigma _X}^X(a,b)$
is naturally given by the Yoneda pairing
\begin{equation*}\mathrm{Ext}^1_X(F,F)\times \mathrm{Ext}^1_X(F,F)\xrightarrow {\phi }\mathrm{Ext}^2_X(F,F)\cong \mathrm{Hom}_X(F,F)^{\vee }\cong \mathbb{C}.\end{equation*}
This means that, for any
$x, y\in \mathrm{Ext}^1_X(F,F)$
, the map
$\phi (x,y): F\to F[2]$
is the composition map
$x[1]\circ y$
.
Now we assume that
$[F]=r([E])$
for an object
$[E]\in U$
. We know that the functor
$\mathrm{pr}_X\circ j_*$
induces an embedding
$d_{[E]}\colon \textrm {Ext}^1_Y(E,E)\hookrightarrow \textrm {Ext}^1_X(F, F)$
by Lemma 5.2(i), mapping an element
$a \colon E\to E[1]$
to
$\mathrm{pr}_X\circ j_*(a)\colon F\to F[1]$
. For any two elements
$x, y\in d_{[E]}(\textrm {Ext}^1_Y(E, E))=T_{[F]}L$
, we can express
$x=\mathrm{pr}_X\circ j_*(a)$
and
$y=\mathrm{pr}_X\circ j_*(b)$
, where
$a,b \in \textrm {Ext}^1_Y(E,E)$
. Then the restriction of
$\phi$
on
$d_{[E]}(\textrm {Ext}^1_Y(E, E))=T_{[F]}L$
is given by
\begin{equation*}\phi (x, y)=x[1]\circ y=\mathrm{pr}_X\circ j_*(a[1]\circ b)\in \mathrm{pr}_X\circ j_*(\textrm {Ext}^2_Y(E, E)).\end{equation*}
Since
$\textrm {Ext}^2_Y(E, E)=0$
,
$\phi$
vanishes at every point of
$r(U)$
, and the desired result in part (i) follows.
Next, let
$M$
be an irreducible component of
$M^Y_{\sigma _Y}(a,b)$
and
$M_{sm}$
be its smooth locus. Recall that the natural involution
$T_X$
on
$\mathcal{K}u(X)$
induces an involution on the moduli space
$M^X_{\sigma _X}(a,b)$
, which is denoted by
$T_X\colon M^X_{\sigma _X}(a,b)\to M^X_{\sigma _X}(a,b)$
also. Let
$\mathrm{Fix}(T_X)\subset M^X_{\sigma _X}(a,b)$
be the fixed locus, which is a smooth subvariety by [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.6]. If
$r(M)$
is contained in
$\mathrm{Fix}(T_X)$
, then
$F:=r(M)$
is a connected component of
$\mathrm{Fix}(T_X)$
as
$\dim (r(M))=M=\mathrm{Fix}(T_X)$
and
$M$
is reduced. By Lemma 5.2(ii), we know that
$r|_M\colon M\to r(M)$
has degree
$2$
. Now we aim to show that
$r|_{M_{sm}}$
is étale. Note that
$r(M_{sm})\subset F$
as
$M_{sm}$
is already reduced. Hence, we have a morphism
$r|_{M_{sm}}\colon M_{sm}\to F$
. As
$M_{sm}$
and
$F$
are both smooth, from Lemma 5.2(ii), we conclude that
$r|_{M_{sm}}$
is étale.
If
$r(M)$
is not contained in
$\mathrm{Fix}(T_X)$
, then there is a smooth open dense subset
\begin{equation*}U':=(r|_M)^{-1}(r(M)\setminus \mathrm{Fix}(T_X))\cap M_{sm}\subset M,\end{equation*}
such that
$r(U')\cap \mathrm{Fix}(T_X)=\varnothing$
. Using Lemma 5.2, we know that
$r|_{U'}$
is an immersion, so
$r$
is birational onto its image.
Therefore, we can construct a family of Lagrangian subvarieties for each moduli space
$M^X_{\sigma _X}(a,b)$
using hyperplane sections
$Y\subset X$
. The next proposition demonstrates that this construction coincides with the one presented in [Reference Perry, Pertusi and ZhaoPPZ23] when
$X$
is special and
$Y$
is the branch divisor of
$X$
.
Proposition 5.4.
Let
$X$
be a special GM fourfold and
$j\colon Y\hookrightarrow X$
be its branch divisor. Let
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
and
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
. Then for any pair of integers
$a,b$
, the functor
$\mathrm{pr}_X\circ j_*$
induces a finite morphism
\begin{equation*}r\colon M^Y_{\sigma _Y}(a,b) \to M^X_{\sigma _X}(a,b)\end{equation*}
such that
$r$
is an étale double cover onto
$\mathrm{Fix}(T_X)$
.
Proof.
Recall that in [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.8], the authors show that
${\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X}(j_*E(H))$
is
$\sigma _X$
-semistable for any
$\sigma _Y$
-semistable object
$E\in \mathcal{K}u(Y)$
(cf. [Reference Perry, Pertusi and ZhaoPPZ23, Equation (8.3)]). Then it induces a morphism
$r'\colon M^Y_{\sigma _Y}(a,b) \to M^X_{\sigma _X}(a,b)$
which maps onto
$\mathrm{Fix}(T_X)$
. Moreover,
$r'([E])=[{\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X}(j_*E(H))]$
. Using Lemma 4.3, it is easy to check that
\begin{equation*}T_X({\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X}(j_*E(H)))\cong \mathrm{pr}_X(j_*E)[1],\end{equation*}
and hence
$T_X\circ r'=r$
, where
$T_X\colon M^X_{\sigma _X}(a,b)\to M^X_{\sigma _X}(a,b)$
is the induced involution on the moduli space.
Using Theorem 4.8 and Lemma 4.10, the above results can be easily generalized to GM fivefolds.
Corollary 5.5.
Let
$X$
be a non-Hodge-special GM fourfold and
$Y$
be a GM fivefold containing
$X$
as a hyperplane section with the embedding
$j \colon X\hookrightarrow Y$
. Let
$\sigma _X$
be a stability condition on
$\mathcal{K}u(X)$
and
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
. Then, for any pair of integers
$a,b$
, the functor
$j^*$
induces a finite morphism
\begin{equation*}r \colon M^Y_{\sigma _Y}(a,b) \to M^X_{\sigma _X}(a,b).\end{equation*}
Moreover, if
$\gcd (a,b)=1$
, then:
-
(i)
$r$
is generically unramified and its image is a Lagrangian subvariety in
$M^X_{\sigma _X}(a,b)$
; and
-
(ii) the restriction
$r|_M$
of
$r$
to any irreducible component
$M$
of
$M^Y_{\sigma _Y}(a,b)$
is either a birational map onto its image or a finite morphism of degree
$2$
and étale on the smooth locus of
$M$
.
5.3 Lagrangian families
Using Theorem 5.3, we obtain infinitely many Lagrangian subvarieties of
$M^X_{\sigma _X}(a,b)$
, parameterized by an open subset of
$|{\mathcal{O}}_X(H))|$
when
$\dim X$
is even. These subvarieties can form a family using stability conditions in families and relative moduli spaces. We start with a general definition of Lagrangian families, which can be viewed as a generalization of Lagrangian fibrations.
Definition 5.6. A Lagrangian family of a projective hyperkähler manifold
$M$
is a diagram
where
$p$
is flat and projective, and
$L$
and
$B$
are quasi-projective manifolds, such that, for a general point
$b\in B$
,
$q|_{p^{-1}(b)}$
is generically finite to its image, which is a Lagrangian subvariety of
$M$
. The manifold
$B$
is called the base of a Lagrangian family. A Lagrangian family is called a Lagrangian covering family if
$q$
is dominant.
Remark 5.7. There is another definition of Lagrangian covering families in [Reference VoisinVoi22, Definition 0.4], which does not assume
$p$
to be flat but requires
$B$
to be projective. It is easy to see that the existence of a Lagrangian family in the sense of Definition 5.6 implies the existence of one in the sense of [Reference VoisinVoi22, Definition 0.4] by compactifying
$p$
in Definition 5.6 to
$\overline {p}\colon \overline {L}\to \overline {B}$
, resolving singularities of
$\overline {L}$
by
$\widetilde {L}$
, and then resolving the indeterminacy locus of
$\widetilde {L}\dashrightarrow M$
.
Now let
$\mathbb{P}_0\subset |{\mathcal{O}}_X(H)|$
be the open locus parameterizing smooth divisors. Using relative moduli spaces and Theorem 4.16, we can construct a smooth Lagrangian family for each
$M^X_{\sigma _X}(a,b)$
over an open subset of
$\mathbb{P}_0$
when
$X$
is a general GM fourfold.
Theorem 5.8.
Let
$a,b$
be a pair of coprime integers. Then, for any general GM fourfold
$X$
and a stability condition
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
generic with respect to
$a\Lambda _1+b\Lambda _2$
, there exists an open subscheme
$V\subset |{\mathcal{O}}_X(H)|$
and a relative moduli space of stable objects
\begin{equation*}p\colon L(a,b)\to V\end{equation*}
with a morphism
\begin{equation*}q\colon L(a,b)\to M^X_{\sigma _X}(a,b)\end{equation*}
such that
$L(a,b)$
is a quasi-projective manifold,
$p$
is smooth and projective, and the following diagram is a Lagrangian family.
Furthermore,
$q|_{p^{-1}(s)}$
is finite and unramified for each
$s\in V$
.
Proof.
Let
$V$
be an open subset of
$\mathbb{P}_0\subset |{\mathcal{O}}_X(H)|$
such that, for every
$[Y]\in V$
, the moduli space
$M^Y_{\sigma _Y}(a,b)$
is smooth. Such a
$V$
exists by Proposition 5.1 as
$X$
is general. We denote the restriction of the universal hyperplane on
$V$
by
$\mathcal{H}_V$
. We define
${\mathcal{O}}_{\mathcal{H}_V}(1):=p_2^*{\mathcal{O}}_X(H)|_{\mathcal{H}_V}$
, where
$p_2\colon |{\mathcal{O}}_X(H)|\times X\to X$
is the projection. Then
${\mathcal{O}}_{\mathcal{H}_V}(1)|_{\mathcal{H}_s}\cong {\mathcal{O}}_{\mathcal{H}_s}(H)$
for any
$s\in V$
. In other words,
$\pi \colon \mathcal{H}_V\to V$
is a family of GM threefolds in the sense of [Reference Bayer and PerryBP23, Section 5.2]. As in the proof of Proposition 5.1, we have a relative Kuznetsov component
$\mathcal{K}u(\mathcal{H}_V)\subset \mathrm{D}_{\mathrm{perf}}(\mathcal{H}_V)$
and a stability condition
$\underline {\sigma }$
on
$\mathcal{K}u(\mathcal{H}_V)$
over
$V$
such that, for any
$s\in V$
,
$(\underline {\sigma })|_s$
is Serre invariant.
Let
$v\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(\mathcal{H}_V)/V)$
be the class such that
$v_s=a\lambda _1+b\lambda _2\in \mathrm{K}_{\mathrm{num}}(\mathcal{K}u(\mathcal{H}_s))$
for any
$s\in V$
. We define
\begin{equation*}p\colon L(a,b):=M_{\underline {\sigma }}(\mathcal{K}u(\mathcal{H}_V)/V, v)\to V\end{equation*}
to be the relative good moduli space. According to [BLM+21, Theorem 21.24(3)], such a moduli space exists and
$p$
is proper. By Theorem 4.16, the functor
$\mathrm{pr}_X\circ j_*$
induces a morphism
$q\colon L(a,b)\to M^X_{\sigma _X}(a,b)$
.
From our construction and Theorem 5.3,
$p^{-1}(s)=M^{\mathcal{H}_s}_{(\underline {\sigma })|_s}(a,b)$
is smooth projective and
$q|_{p^{-1}(s)}$
is finite and unramified with the image Lagrangian for each
$s\in V$
. Applying [Reference Perry, Pertusi and ZhaoPPZ23, Lemma 5.1] and [BLM+21, Lemma 21.12] implies the smoothness of
$p$
. Thus,
$L(a,b)$
is smooth as well. Finally, the projectivity of
$p$
follows from the smoothness, [BLM+21, Theorem 21.25] ,and [Reference Villalobos-PazVP21, Corollary 3.4], which implies that
$L(a,b)$
is a quasi-projective manifold. By definition, this yields a smooth Lagrangian family of
$M^X_{\sigma _X}(a,b)$
.
Indeed, the results above can be generalized to a non-Hodge-special GM sixfold
$X$
for an open subset
$V\subset |{\mathcal{O}}_X(H))|=\mathbb{P}^{10}$
if a family of Serre-invariant stability conditions on
$\mathcal{K}u(\mathcal{H}_V)$
over
$V$
is known. This can be done by establishing a generalization of [Reference Kuznetsov and PerryKP23, Corollary 6.5] for families of GM fivefolds. Since the proof is analogous to the GM fourfolds case, we state the corresponding results here for later use.
Corollary 5.9.
Let
$X$
be a non-Hodge-special GM sixfold and
$\sigma _X$
be a stability condition on
$\mathcal{K}u(X)$
. Assuming there exists an open subset
$V\subset |{\mathcal{O}}_X(H))|$
and a stability condition
$\underline {\sigma }$
on
$\mathcal{K}u(\mathcal{H}_V)$
over
$V$
such that
$(\underline {\sigma })|_s$
is a Serre-invariant stability condition on
$\mathcal{K}u(\mathcal{H}_s)$
for any
$s\in V$
, then the claim of Theorem
5.8
holds for the moduli space
$M^X_{\sigma _X}(a, b)$
.
A similar result can be obtained when considering a fixed GM fourfold within a family of GM fivefolds.
Corollary 5.10.
Let
$X$
be a non-Hodge-special GM fourfold and let
$\sigma _X$
be a stability condition on
$\mathcal{K}u(X)$
. Given a smooth connected quasi-projective manifold
$V$
and a family of GM fivefolds
$\mathcal{H}_V\to V$
containing
$X$
, assume that there is a stability condition
$\underline {\sigma }$
on
$\mathcal{K}u(\mathcal{H}_V)$
over
$V$
such that
$(\underline {\sigma })|_s$
is a Serre-invariant stability condition on
$\mathcal{K}u(\mathcal{H}_s)$
for any
$s\in V$
. Then the claim of Theorem
5.8
holds for the moduli space
$M^X_{\sigma _X}(a, b)$
.
5.4 Lagrangian covering families
In Theorem 5.8, we construct a Lagrangian family for each
$M^X_{\sigma _X}(a,b)$
. A natural question arises regarding whether
$q$
is dominant, indicating the presence of a Lagrangian covering family. A straightforward computation of dimension shows that if
$q$
is dominant, then
$\dim M^X_{\sigma _X}(a,b)\leq 16$
. So up to sign, we have
\begin{equation*}(a,b)=(0, 1), (1,0), (1,\pm 1), (1, \pm 2), (2, \pm 1).\end{equation*}
At this moment, we do not know an abstract method to demonstrate that the Lagrangian family in Theorem 5.8 constitutes a Lagrangian covering family. However, this can be established for the first three classes as follows, since we have a specific description of the corresponding moduli spaces.
-
–
$(a,b)=(0,1)$
or
$(1,0)$
. By [Reference Guo, Liu and ZhangGLZ24], the corresponding moduli space on a very general GM fourfold is the double EPW sextic or the double dual EPW sextic. On the other hand, the corresponding moduli space on a GM threefold is studied in [Reference Jacovskis, Lin, Liu and ZhangJLLZ24]. We will see in [Reference Feyzbakhsh, Guo, Liu and ZhangFGLZ25] that Theorem 5.8 gives a Lagrangian covering family which agrees with the classical construction in [Reference Iliev and ManivelIM11, Section 5.1]. -
–
$(a,b)=(1,\pm 1)$
. By [Reference Kapustka, Kapustka and MongardiKKM22], the corresponding moduli space on a very general GM fourfold is the double EPW cube constructed in [Reference Iliev, Kapustka, Kapustka and RanestadIKKR19]. In [Reference Feyzbakhsh, Guo, Liu and ZhangFGLZ25], we will realize it as the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on a GM fourfold. Moreover, we will apply Theorem 5.8 to construct Lagrangian covering families of double EPW cubes in [Reference Feyzbakhsh, Guo, Liu and ZhangFGLZ25]. These constructions are new in the literature. -
–
$(a,b)=(1,\pm 2)$
or
$(2, \pm 1)$
. In this case, the corresponding moduli spaces on GM fourfolds are
$12$
-dimensional hyperkähler varieties and have not been extensively studied using classical methods. According to Corollary 6.1, the moduli space for
$(a,b)=(1,2)$
on a very general GM fourfold
$X$
contains a subvariety birational to
$X$
. On the other hand, the moduli space for
$(a,b)=(1,2)$
on a GM threefold is studied in [Reference Jacovskis, Liu and ZhangJLZ22], which contains such threefold as a Brill–Noether locus. We expect that both moduli spaces can be constructed from Hilbert schemes of certain curves, similar to [Reference Bayer, Beentjes, Feyzbakhsh, Hein, Martinelli, Rezaee and SchmidtBBF+24] and [Reference Li, Pertusi and ZhaoLPZ23]. If this is the case, then it is possible to demonstrate that the family in Theorem 5.8 in this scenario indeed constitutes a covering family.
A similar discussion as above for cubic fourfolds has been included in Appendix A.
6. Further applications
In this section, we delve into further applications of studying GM fourfolds through their hyperplane sections.
6.1 Embedding Gushel–Mukai fourfolds into hyperkähler manifolds
Using Theorem 4.8, we can construct a rational embedding
$X\dashrightarrow M^X_{\sigma _X}(1,2)$
which confirms an expectation in [Reference Perry, Pertusi and ZhaoPPZ22, Section 5.4.3]. We denote by
$\mathbb{C}_x$
the skyscraper sheaf at a point
$x \in X$
.
Corollary 6.1.
Let
$X$
be a general GM fourfold or non-Hodge-special special GM fourfold. Then, for
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
generic with respect to
$\Lambda _1+2\Lambda _2$
, the functor
$\mathrm{pr}_X$
induces a rational map
\begin{align*} X & \dashrightarrow M_{\sigma _X}^{X}(1,2) \\[2pt] x & \mapsto \mathrm{pr}_X(\mathbb{C}_x) \end{align*}
which is generically an immersion.
Proof.
For any point
$x\in Y$
on a smooth GM threefold
$j\colon Y\hookrightarrow X$
, we know that
$\mathrm{pr}_Y(\mathbb{C}_x)$
is of class
$[\mathrm{pr}_Y(\mathbb{C}_x)]=\lambda _1+2\lambda _2$
and stable with respect to any Serre-invariant stability condition on
$\mathcal{K}u(Y)$
by [Reference Jacovskis, Liu and ZhangJLZ22, Theorem 5.9]. Then Theorem 4.16 and Corollary 4.14 imply that
$\mathrm{pr}_X(j_*\mathrm{pr}_Y(\mathbb{C}_x))$
is
$\sigma _X$
-stable. Moreover, by Proposition 4.5, we have
$\mathrm{pr}_X(\mathbb{C}_x)\cong \mathrm{pr}_X(j_*\mathrm{pr}_Y(\mathbb{C}_x))$
.
Let
$W\subset X$
be an open subset such that each
$x\in W$
is contained in a smooth hyperplane section
$Y$
and
\begin{equation} \mathrm{pr}_Y(\mathbb{C}_x)\neq T_Y(\mathrm{pr}_Y(\mathbb{C}_x)). \end{equation}
If
$X$
is ordinary, then (43) holds for any point
$x \in Y$
by [Reference Jacovskis, Liu and ZhangJLZ22, Lemma 6.6, Theorem 1.1]. But if both
$X$
and
$Y$
are special, then the involution
$T_Y$
on
$\mathcal{K}u(Y)$
is induced by the geometric involution of the double cover on
$Y$
. Thus, (43) holds when
$x\in Y$
is not contained in the branch divisor of
$X$
, as the double cover structures on
$X$
and
$Y$
are compatible.
Thus, we obtain a morphism
$W\to M^X_{\sigma _X}(1,2)$
, which is injective by next Lemma 6.2, and unramified by Lemma 5.2 and (43). Hence, it is an immersion.
Lemma 6.2.
Let
$X$
be a GM fourfold and
$x, x'\in X$
be two closed points. Then
$\mathrm{pr}_X(\mathbb{C}_x)\cong \mathrm{pr}_X(\mathbb{C}_{x'})$
if and only if
$x=x'$
.
Proof.
Let
$x\in X$
be a closed point. By definition, we have an exact triangle
$\mathcal{U}_X^{\vee \oplus 2}\to \mathbb{C}_x\to {\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X} \mathbb{C}_x$
, which yields the exact triangle
\begin{equation} \mathcal{Q}^{\vee \oplus 2}_X\xrightarrow {a} I_x\to {\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{\mathcal{U}^{\vee }_X} \mathbb{C}_x[-1] =: F_x[1] \end{equation}
after applying
${\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}$
. Note that
$a$
corresponds to two linearly independent maps in
$\textrm {Hom}_X(\mathcal{U}^{\vee }_X, \mathbb{C}_x)=\mathbb{C}^2$
, and thus its image is the zero locus of two linearly independent sections of
$\mathcal{Q}_X$
, i.e.
$\textrm {Gr}(0,3)\cap X$
. Therefore,
$a$
is surjective and
$F_x$
is a sheaf.
Since
$\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}\mathcal{Q}^{\vee }_X=\mathcal{Q}^{\vee }_X$
, applying
$\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}$
to (44) we obtain
\begin{equation} \mathcal{Q}^{\vee \oplus 2}_X\to \boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x\to \mathrm{pr}_X(\mathbb{C}_x)[-1]. \end{equation}
Considering
$\textrm {RHom}_X(I_x, {\mathcal{O}}_X(-H))=\mathbb{C}[-3]$
, there is an exact triangle
\begin{equation} \boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x\to \boldsymbol {\mathrm{R}}_{\mathcal{U}_X}I_x\to \boldsymbol {\mathrm{R}}_{\mathcal{U}_X}{\mathcal{O}}_X(-H)[3] =: K[2]. \end{equation}
Here,
$K$
is the cokernel of the natural map
${\mathcal{O}}_X(-H)\hookrightarrow \mathcal{U}_X^{\oplus 5}=\mathcal{U}_X\otimes H^0(\mathcal{U}^{\vee }_X)$
. Since
$\textrm {RHom}_X(I_x, \mathcal{U}_X)=\mathbb{C}^2[-3]$
, we have the exact triangle
$\mathcal{U}_X^{\oplus 2}[2] \to \boldsymbol {\mathrm{R}}_{\mathcal{U}_X}I_x \to I_x$
. Thus, taking cohomology from (46) yields a long exact sequence
\begin{equation} 0\to \mathcal{H}^{-2}(\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x)\to \mathcal{U}_X^{\oplus 2}\xrightarrow {b} K\to \mathcal{H}^{-1}(\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x)\to 0 \end{equation}
and
$\mathcal{H}^{0}(\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x)\cong I_x$
. Then, taking cohomology from (45) implies that
\begin{equation*}\mathcal{H}^{-2}(\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x) = \mathcal{H}^{-2}(\mathrm{pr}_X(\mathbb{C}_x)[-1]),\end{equation*}
and gives the long exact sequence
\begin{equation} 0 \to \mathcal{H}^{-1}(\boldsymbol {\mathrm{R}}_{\mathcal{U}_X}\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}I_x) \to \mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_x)[-1]) \to \mathcal{Q}^{\vee \oplus 2}_X\xrightarrow {a} I_x\to \mathcal{H}^{0}(\mathrm{pr}_X(\mathbb{C}_x)[-1]) \to 0. \end{equation}
Since
$a$
is surjective,
$ \mathcal{H}^{0}(\mathrm{pr}_X(\mathbb{C}_x)[-1]) = 0$
and its kernel is the sheaf
$F$
defined in (44). Consequently, from (48), we derive the long exact sequence
\begin{equation} 0\to \mathcal{H}^{-2}(\mathrm{pr}_X(\mathbb{C}_x)[-1])\to \mathcal{U}_X^{\oplus 2}\xrightarrow {b} K \to \mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_x)[-1])\to F_x\to 0. \end{equation}
Now we consider two cases depending on the morphism
$b$
.
Case I. First assume that the map
$b$
is injective, so then
$\mathcal{H}^{-2}(\mathrm{pr}_X(\mathbb{C}_x)[-1]) = 0$
and
$\mathrm{pr}_X(\mathbb{C}_x) = \mathcal{H}^{-2}(\mathrm{pr}_X(\mathbb{C}_x))[2]$
. The injective map
$b$
in (49) induces the following commutative diagram of exact triangles.
By dualizing the third row, we get
$\mathcal{E}xt_X^i(G_x, {\mathcal{O}}_X)=0$
for any
$i\gt 1$
and
$\mathcal{E}xt_X^1(G_x, {\mathcal{O}}_X)$
is supported at a point. Hence,
$G_x$
is reflexive, and it is straightforward to check that
$G_x$
is slope stable with
$\mathrm{ch}_0(G_x) = 5$
,
$\mathrm{ch}_1(G_x) = -2H$
,
$\mathrm{ch}_2(G_x).H^2 = -2$
, and
$\mathrm{ch}_3(G_x).H = \frac {8}{3}$
. Similarly, one can check that the sheaf
$F_x$
in (44) is slope stable with
$\mathrm{ch}_0(F_x) = 5$
,
$\mathrm{ch}_1(F_x) = -2H$
,
$\mathrm{ch}_2(F_x).H^2 = -2$
, and
$\mathrm{ch}_3(F_x).H = \frac {2}{3}$
. This proves that
$\textrm {Hom}_X(G_x, F_{x'}) = 0$
for any closed point
$x' \in X$
. Now assume for a contradiction that there is a closed point
$x' \neq x$
on
$X$
such that
$\mathrm{pr}_X(\mathbb{C}_x) \cong \mathrm{pr}_X(\mathbb{C}_{x'})$
, so then by (49) we have the exact triangle
$G_x \to \mathrm{pr}_X(\mathbb{C}_{x'})[-2] \to F_x$
of coherent sheaves. Thus the surjection
$\mathrm{pr}_X(\mathbb{C}_{x'})[-2] \twoheadrightarrow F_{x'}$
implies that
$\textrm {Hom}_X(F_x, F_{x'}) \neq 0$
and so
$F_x \cong F_{x'}$
which is not possible as they have different non-locally free loci.
Case II. Now assume that the map
$b$
in (49) is not injective, so then we claim either
$\ker (b)=\mathcal{U}_X$
or
$b$
is zero. Note that any map
$\mathcal{U}^{\oplus m}_X\to {\mathcal{O}}_X$
corresponding to
$4\leq m\leq 5$
linearly independent sections of
$\mathcal{U}^{\vee }_X$
is surjective, as its image is the ideal sheaf of the zero locus of these
$m$
sections, i.e.
$\textrm {Gr}(2, 5-m)\cap X$
. Therefore, dualizing the exact sequence
$0\to {\mathcal{O}}_X(-H)\to \mathcal{U}^{\oplus 5}_X\to K\to 0$
, we find that
$\mathcal{E}xt_X^i(K, {\mathcal{O}}_X)=0$
for any
$i\gt 0$
, and hence
$K$
is locally free. One can easily check that
$K$
is slope stable with
$\mathrm{ch}_0(K) = 9$
and
$\mathrm{ch}_1(K) = -4H$
. Hence if the map
$b$
in (49) is non-zero, ordering of slopes forces
$\mathrm{ch}_0(\ker b) =2$
and
$\mathrm{ch}_1(\ker b) = -H$
, and so
$\ker b \cong \mathcal{U}_X$
by the stability of
$\mathcal{U}_X$
.
As
$\mathcal{E}xt_X^3(I_x, {\mathcal{O}}_X)=\mathbb{C}_x$
, we know that
$\mathcal{E}xt_X^2(F_x, {\mathcal{O}}_X)=\mathbb{C}_x$
. We claim that
\begin{equation*}\mathcal{E}xt_X^1(G, {\mathcal{O}}_X)=\mathcal{E}xt_X^2(G, {\mathcal{O}}_X)=0,\end{equation*}
where
$G:=\mathrm{cok}(b)$
, which implies that
$\mathcal{E}xt_X^2(\mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_x)[-1]), {\mathcal{O}}_X)=\mathbb{C}_x$
. To this end, when
$b$
is zero, we have
$G=K$
, and hence the result follows from the fact that
$K$
is a bundle. When
$\ker (b)=\mathcal{U}_X$
, from the definition of
$K$
, we have an exact sequence
$0\to {\mathcal{O}}_X(-H)\to \mathcal{U}_X^{\oplus 4}\to G\to 0$
where
${\mathcal{O}}_X(-H)\to \mathcal{U}_X^{\oplus 4}$
corresponds to four linearly independent sections of
$\mathcal{U}^{\vee }_X$
. Hence, by dualizing, we know that
$\mathcal{E}xt_X^i(G, {\mathcal{O}}_X)=0$
for any
$i\gt 0$
, and the claim follows.
Now, given a point
$x' \neq x$
on
$X$
, if
$\mathrm{pr}_X(\mathbb{C}_x)\cong \mathrm{pr}_X(\mathbb{C}_{x'})$
, then we have
$\mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_x))\cong \mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_{x'}))$
, which implies that
\begin{equation*}\mathbb{C}_x=\mathcal{E}xt_X^2(\mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_x)[-1]), {\mathcal{O}}_X)\cong \mathcal{E}xt_X^2(\mathcal{H}^{-1}(\mathrm{pr}_X(\mathbb{C}_{x'})[-1]), {\mathcal{O}}_X)=\mathbb{C}_{x'},\end{equation*}
leading to a contradiction.
Remark 6.3. By Theorem 4.8, we actually have a rational map
\begin{equation*}X\dashrightarrow M_{\sigma _X}^{X}([\mathrm{pr}_X(\mathbb{C}_x)])\end{equation*}
for a very general GM variety
$X$
of dimension
$n\geq 4$
and any Serre-invariant stability condition
$\sigma _X$
on
$\mathcal{K}u(X)$
, where
$M_{\sigma _X}^{X}([\mathrm{pr}_X(\mathbb{C}_x)])$
is the moduli space of
$\sigma$
-semistable objects with class
$[\mathrm{pr}_X(\mathbb{C}_x)]$
. However, we do not have a proof for the stability of
$\mathrm{pr}_X(\mathbb{C}_x)$
for
$n=5,6$
as
$[\mathrm{pr}_X(\mathbb{C}_x)]$
is not primitive. We expect that
$\mathrm{pr}_X(\mathbb{C}_x)$
is stable for a general point
$x\in X$
, which implies that the rational map above is generically an immersion, as in Corollary 6.1.
6.2 Projectivity of Bridgeland moduli spaces of semistable objects
Using Theorem 4.8, we can also achieve projectivity of moduli spaces of semistable objects in
$\mathcal{K}u(Y)$
of any class for a general GM threefold
$Y$
, which improves [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.3(2)] and confirms the expectation in [Reference Perry, Pertusi and ZhaoPPZ23, Remark 1.4] for general (ordinary or special) GM threefolds.
Corollary 6.4.
Let
$a,b$
be a pair of integers and
$Y$
be a general GM threefold. Then, for any Serre-invariant stability condition
$\sigma _Y$
on
$\mathcal{K}u(Y)$
, the moduli space of
$\sigma _Y$
-semistable objects
$M^Y_{\sigma _Y}(a,b)$
is a projective scheme.
Proof.
Assume that
$Y$
is a general GM threefold, in the sense that
$Y$
is contained in a general ordinary GM fourfold
$X$
in the sense of Theorem 4.16 or a non-Hodge-special GM fourfold. By Theorem 5.3, we have a finite morphism
$M^Y_{\sigma _Y}(a,b) \to M^X_{\sigma _X}(a,b)$
for
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
. So it suffices to prove that
$M^X_{\sigma _X}(a,b)$
is a projective scheme.
We can assume that
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
is generic with respect to
$a\Lambda _1+b\Lambda _2$
. When
$(a,b)\neq (\pm 2,0)$
or
$(0,\pm 2)$
, we know that
$M^X_{\sigma _X}(a,b)$
is projective by the same argument as in [Reference SaccàSac23, Theorem 3.1] using the corresponding results in [Reference Perry, Pertusi and ZhaoPPZ22] instead of [BLM+21] (cf. [Reference SaccàSac23, Section 8]). When
$(a,b)= (\pm 2,0)$
or
$(0,\pm 2)$
, or, in other words, the class is of OG10-type in the sense of [Reference SaccàSac23], the projectivity of
$M^X_{\sigma _X}(a,b)$
also follows from the same argument in [Reference Li, Pertusi and ZhaoLPZ22, Section 3.7] by replacing results in [BLM+21] with [Reference Perry, Pertusi and ZhaoPPZ22].
6.3 Lagrangian constant cycle subvarieties
Finally, we can construct a Lagrangian constant cycle subvariety for each Bridgeland moduli space of stable objects. We refer to [Reference LinLin20, Section 3] for the definition and basic properties of constant cycle subvarieties.
Definition 6.5. For a projective hyperkähler manifold
$M$
, a subvariety
$Z\subset M$
is called a Lagrangian constant cycle subvariety of
$M$
if
$Z$
is Lagrangian and all points of
$Z$
are rationally equivalent in
$M$
.
It is conjectured in [Reference VoisinVoi16, Conjecture 0.4] that each projective hyperkähler manifold admits a Lagrangian constant cycle subvariety. In the following, we focus on hyperkähler manifolds arising from GM fourfolds. One of the crucial ideas is the following proposition, which follows from the same argument as in [Reference Marian and ZhaoMZ20] and [Reference Shen and YinSY20, Proposition 3.4].
Proposition 6.6 ([Reference Marian and ZhaoMZ20, Theorem], [Reference Shen and YinSY20, Proposition 3.4]). Let
$X$
be a GM fourfold or sixfold and
$M$
be a Bridgeland moduli space of stable objects in
$\mathcal{K}u(X)$
such that
$M$
is a projective hyperkähler manifold. For closed points
$[E]$
and
$[F]$
in
$M$
, if
\begin{equation*}\mathrm{ch}(E)=\mathrm{ch}(F)\in \mathrm{CH}^{*}(X)_{\mathbb{Q}},\end{equation*}
then
\begin{equation*}[E]=[F]\in \mathrm{CH}_0(M).\end{equation*}
Recall that we have an involution
$T_X$
on
$\mathcal{K}u(X)$
defined in Lemma 4.3 for each GM fourfold or sixfold
$X$
. According to [Reference Bayer and PerryBP23, Remark 5.8] and [Reference Perry, Pertusi and ZhaoPPZ23, Theorem 1.6], the functor
$T_X$
induces an anti-symplectic involution on
$M_{\sigma _X}^X(a,b)$
which we also denote by
$T_X$
, and the fixed locus is non-empty. We denote by
$\mathrm{Fix}(T_X)$
the fixed locus, so then
$\mathrm{Fix}(T_X)$
is a smooth Lagrangian subvariety of
$M_{\sigma _X}^X(a,b)$
by [Reference BeauvilleBea11, Lemma 1.1].
Theorem 6.7.
Let
$X$
be a GM fourfold or sixfold and
$a,b$
be a pair of coprime integers. Then, for any two objects
$E_1,E_2\in \mathcal{K}u(X)$
with
$\mathrm{ch}(E_1)=\mathrm{ch}(E_2)\in \mathrm{H}^*(X, \mathbb{Q})$
, we have
\begin{equation} \mathrm{ch}(E_1)+\mathrm{ch}(T_X(E_1))=\mathrm{ch}(E_2)+\mathrm{ch}(T_X(E_2))\in \mathrm{CH}^*(X)_{\mathbb{Q}}. \end{equation}
In particular, for any stability condition
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
that is generic with respect to
$a\Lambda _1+b\Lambda _2$
, the fixed locus
$\mathrm{Fix}(T_X)$
is a Lagrangian constant cycle subvariety of
$M_{\sigma _X}^X(a,b)$
.
Proof.
By Proposition 6.6, we only need to show (50). Indeed, let
$[E_1], [E_2]\in \mathrm{Fix}(T_X)$
. As
$T_X(E_i)\cong E_i$
for each
$i=1,2$
, by (50), we have
\begin{equation*}\mathrm{ch}(E_1)=\mathrm{ch}(E_2)\in \mathrm{CH}^*(X)_{\mathbb{Q}}.\end{equation*}
Thus, Proposition 6.6 implies that
$[E_1]=[E_2]\in \mathrm{CH}_0(M_{\sigma _X}^X(a,b))$
.
To prove (50), we first assume that
$X$
is a GM fourfold. By [Reference Fu and MoonenFM22, Proposition 4.2], we know that the cycle map
\begin{equation*}\mathrm{CH}^i(X)\to \mathrm{H}^{2i}(X, \mathbb{Z})\end{equation*}
is injective for each
$i\neq 3$
. Hence,
$\mathrm{ch}(E_1)=\mathrm{ch}(E_2)\in \mathrm{H}^*(X, \mathbb{Q})$
implies
\begin{equation*}\mathrm{ch}_i(E_1)=\mathrm{ch}_i(E_2)\in \mathrm{CH}^*(X)_{\mathbb{Q}}\end{equation*}
for each
$i\neq 3$
. Now for any smooth hyperplane section
$j\colon Y\hookrightarrow X$
, we have
$\mathrm{CH}^3(Y)\cong \mathrm{H}^6(Y,\mathbb{Z}) \cong \mathbb{Z}$
according to [Reference Fu and MoonenFM22, Theorem 4.1]. Thus we obtain
\begin{equation*}\mathrm{ch}(j^*E_1)=\mathrm{ch}(j^*E_2)\in \mathrm{CH}^*(Y)_{\mathbb{Q}},\end{equation*}
which implies
$\mathrm{ch}(j_*j^*E_1)=\mathrm{ch}(j_*j^*E_2)\in \mathrm{CH}^*(X)_{\mathbb{Q}}$
and
$\mathrm{ch}(\mathrm{pr}_X(j_*j^*E_1))=\mathrm{ch}(\mathrm{pr}_X(j_*j^*E_2))\in \mathrm{CH}^*(X)_{\mathbb{Q}}$
. Therefore, the equality (50) follows from the exact triangle (36).
Similarly, when
$X$
is a GM sixfold, we have
\begin{equation*}\mathrm{ch}_i(E_1)=\mathrm{ch}_i(E_2)\in \mathrm{CH}^*(X)_{\mathbb{Q}}\end{equation*}
for each
$i\neq 4$
by [Reference Fu and MoonenFM22, Theorem 4.5]. Since
$\mathrm{CH}^4(Y)\cong \mathbb{Z}$
(cf. [Reference Fu and MoonenFM22, Theorem 4.4]), we still have
$\mathrm{ch}(\mathrm{pr}_X(j_*j^*E_1))=\mathrm{ch}(\mathrm{pr}_X(j_*j^*E_2))$
in
$\mathrm{CH}^*(X)_{\mathbb{Q}}$
. Then (50) follows from (36) as well.
Remark 6.8. By the modular interpretation in [Reference Guo, Liu and ZhangGLZ24] Theorem 1.1] and [Reference Kapustka, Kapustka and MongardiKKM22, Corollary 1.3], Theorem 6.7 and [Reference Perry, Pertusi and ZhaoPPZ22, Proposition 5.16] imply that the fixed loci of natural involutions on double (dual) EPW sextics (cf. [Reference O’GradyO’G13]) and double EPW cubes (cf. [Reference Iliev, Kapustka, Kapustka and RanestadIKKR19]) associated with very general GM fourfolds are Lagrangian constant cycle subvarieties. For double EPW sextics, this is proved in [Reference ZhangZha24].
Appendix A. Cubic threefolds/fourfolds
In this section, we delve into potential generalizations of our result in Theorem 4.8 for GM varieties to the case of cubic threefolds/fourfolds. Subsequently, we will discuss its potential applications.
For a cubic fourfold
$X$
, its semi-orthogonal decomposition is given by
\begin{equation*}\mathrm{D}^b(X)=\langle \mathcal{K}u(X),{\mathcal{O}}_X,{\mathcal{O}}_X(H),{\mathcal{O}}_X(2H)\rangle =\langle {\mathcal{O}}_X(-H),\mathcal{K}u(X),{\mathcal{O}}_X,{\mathcal{O}}_X(H)\rangle ,\end{equation*}
where
$H$
is the hyperplane class of
$X$
, satisfying
$S_{\mathcal{K}u(X)}=[2]$
. We define the projection functor
\begin{equation*}\mathrm{pr}_X:=\boldsymbol {\mathrm{R}}_{{\mathcal{O}}_X(-H)}{\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}{\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X(H)}.\end{equation*}
There is a rank two lattice in the numerical Grothendieck group
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))$
generated by
$\Lambda _1$
and
$\Lambda _2$
with
\begin{equation*}\mathrm{ch}(\Lambda _1)=3-H-\tfrac {1}{2}H^2+\tfrac {1}{6}H^3+\tfrac {1}{8}H^4,\quad \mathrm{ch}(\Lambda _2)=-3+2H-\tfrac {1}{3}H^3,\end{equation*}
over which the Euler pairing is of the form
\begin{equation} \left [ \begin{array}{c@{\quad}c} -2 & 1 \\[2pt] 1 & -2\\[2pt] \end{array} \right ]. \end{equation}
When
$X$
is non-Hodge-special, we have
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(X))=\langle \Lambda _1, \Lambda _2 \rangle$
.
For a cubic threefold
$Y$
, we have a semi-orthogonal decomposition
\begin{equation*}\mathrm{D}^b(Y)=\langle \mathcal{K}u(Y), {\mathcal{O}}_Y, {\mathcal{O}}_Y(H)\rangle .\end{equation*}
In this case,
$S_{\mathcal{K}u(Y)}={\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_Y}\circ (-\otimes {\mathcal{O}}_Y(H))[1]$
and
$S^3_{\mathcal{K}u(Y)}=[5]$
. Moreover,
$\mathrm{K}_{\mathrm{num}}(\mathcal{K}u(Y))$
is a rank two lattice generated by
$\lambda _1$
and
$\lambda _2$
with
\begin{equation*}\mathrm{ch}(\lambda _1)=2-H-\tfrac {1}{6}H^2+\tfrac {1}{6}H^3 ,\quad \mathrm{ch}(\lambda _2)=-1+H-\tfrac {1}{6}H^2-\tfrac {1}{6}H^3,\end{equation*}
and the Euler pairing is
\begin{equation} \left [ \begin{array}{c@{\quad}c} -1 & 1 \\[2pt] 0 & -1\\[2pt] \end{array} \right ]. \end{equation}
We define the projection functor by
$\mathrm{pr}_Y:={\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_Y(H)}{\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_Y}$
. Let
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
. We denote by
$M^Y_{\sigma _Y}(a,b)$
the moduli space of S-equivalence classes of
$\sigma _Y$
-semistable objects in
$\mathcal{K}u(Y)$
with class
$a\lambda _1+b\lambda _2$
.
Conjecture A.1. Let
$X$
be a cubic fourfold or a GM fourfold and
$j\colon Y\hookrightarrow X$
be its smooth hyperplane section. Let
$\sigma _Y$
be a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
. If
$\mathrm{gcd}(a,b)=1$
, then there is a stability condition
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
generic with respect to
$a\Lambda _1+b\Lambda _2$
and an open smooth subscheme
$U\subset M^Y_{\sigma _Y}(a,b)$
such that
$\mathrm{pr}_X(j_*E)$
is
$\sigma _X$
-semistable for any
$[E]\in U$
.
Note that when
$\gcd (a,b)=1$
,
$M^Y_{\sigma _Y}(a,b)$
is smooth by [Reference Pertusi and YangPY22]. From now on, let
$X$
be a cubic fourfold and
$j \colon Y\hookrightarrow X$
be a smooth hyperplane section. Moreover, assume that
$\sigma _Y$
is a Serre-invariant stability condition on
$\mathcal{K}u(Y)$
and
$\sigma _X$
belongs to the family of stability conditions
$\textrm {Stab}^{\circ }(\mathcal{K}u(X))$
constructed in [BLM+21, Theorem 29.1]. According to [BLM+21, Theorem 29.2], for any pair of coprime integers
$a,b$
, and the stability condition
$\sigma _X\in \textrm {Stab}^{\circ }(\mathcal{K}u(X))$
which is generic with respect to
$a\Lambda _1+b\Lambda _2$
, the moduli space
$M^X_{\sigma _X}(a,b)$
is a projective hyperkähler manifold.
Lemma A.2.
For any object
$E\in \mathcal{K}u(Y)$
, we have an exact triangle
\begin{equation} S^{-1}_{\mathcal{K}u(Y)}(E)[2]\to j^*\mathrm{pr}_X(j_*E)\to E. \end{equation}
In particular, if
$E\in \mathcal{K}u(Y)$
is
$\sigma _Y$
-semistable for a Serre-invariant stability condition
$\sigma _Y$
on
$\mathcal{K}u(Y)$
, then
$j^*\mathrm{pr}_X(j_*E)$
is not
$\sigma _Y$
-semistable and
(53)
is the Harder–Narasimhan filtration of
$j^*\mathrm{pr}_X(j_*E)$
with
\begin{equation*}\phi _{\sigma _Y}(E)+1\gt \phi _{\sigma _Y}(S^{-1}_{\mathcal{K}u(Y)}(E)[2])\gt \phi _{\sigma _Y}(E).\end{equation*}
Proof.
As in the proof of Lemma 4.10(i), we only need to determine
$\mathbf{T}_{j^*, \mathrm{pr}_X\circ j_*}(E)$
. From [Reference Kuznetsov and PerryKP21, Proposition 4.16, Corollary 4.19(1)], we see that
$\mathbf{T}_{j^*, \mathrm{pr}_X\circ j_*}\cong S^{-1}_{\mathcal{K}u(Y)}\circ [3]$
. Hence, (53) is verified as desired. The rest follows from (53), [Reference Feyzbakhsh and PertusiFP23, Proposition 3.4(a), (d)], and Lemma 3.2.
As a result of Lemma A.2, one observes that, unlike Theorem 4.8 for GM varieties, Conjecture A.1 may not be true for
$U = M^Y_{\sigma _Y}(a,b)$
for the cubic threefold
$Y$
. Nevertheless, the same argument as in Lemma 5.2 implies the following.
Lemma A.3.
Let
$E\in \mathcal{K}u(Y)$
be a
$\sigma _Y$
-semistable object.
-
(i) (Injectivity of tangent maps.) The natural map
is injective.
\begin{equation*}\textrm {Ext}^1_Y(E,E)\to \textrm {Ext}^1_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E))\end{equation*}
-
(ii) (Fibers.) Let
$E'\neq E\in \mathcal{K}u(Y)$
be another
$\sigma _Y$
-semistable object such that
so then
\begin{equation*}\phi _{\sigma _Y}(E)=\phi _{\sigma _Y}(E')\quad \text{and}\quad \textrm {Hom}_Y(E, E')=0,\end{equation*}
\begin{equation*}\textrm {Hom}_X(\mathrm{pr}_X(j_*E), \mathrm{pr}_X(j_*E'))=0.\end{equation*}
As a result, we deduce the following immersion via the same argument as in Theorem 5.3.
Theorem A.4.
Let
$a, b$
be a pair of integers with
$\gcd (a, b) =1$
such that Conjecture A.1 holds. Then the functor
$\mathrm{pr}_X\circ j_*$
induces a rational map defined over
$U$
,
\begin{equation*}r \colon \overline {U} \dashrightarrow M^X_{\sigma _X}(a,b),\end{equation*}
such that the image
$L$
is Lagrangian in
$M^X_{\sigma _X}(a,b)$
. Furthermore,
$r|_U$
is an immersion and
$\overline {U}$
is birational onto
$L$
.
Proof.
According to Lemma A.3,
$r|_U$
is injective at the level of underlying spaces. Hence,
$r|_U$
is an immersion and
$\overline {U}$
is birational to
$L$
. The rest follows via the same argument as in Theorem 5.3.
A.1 Lagrangian covering families
If an analog of Corollary 4.14 holds for a cubic fourfold
$X$
and it yields a Lagrangian covering family, then
$\dim M^X_{\sigma _X}(a,b)\leq 10$
. Up to sign, we have
\begin{equation*}(a,b)=(0, 1), (1,0), (1,\pm 1), (2,\pm 1), (1, \pm 2).\end{equation*}
-
–
$(a,b)=(0, 1), (1,0), (1,\pm 1)$
. In this case, the functor
${\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}\circ (-\otimes {\mathcal{O}}_X(H))$
is an auto-equivalence of
$\mathcal{K}u(X)$
that switches these classes up to sign. Hence, we only need to consider the case
$(a,b)=(1,1)$
. According to [Reference Li, Pertusi and ZhaoLPZ23, Theorem 1.1], the moduli space for
$(a,b)=(1,1)$
on a cubic fourfold is the Fano variety of lines. On the other hand, the moduli space for
$(a,b)=(1,1)$
on a cubic threefold is the Fano surface of lines (cf. [Reference Pertusi and YangPY22]). We will see in [Reference Feyzbakhsh, Guo, Liu and ZhangFGLZ25] that Theorem 5.8 yields a Lagrangian covering family, aligning with the classical construction in [Reference VoisinVoi86]. -
–
$(a,b)=(2,\pm 1), (1, \pm 2)$
. In this case, the functor
${\boldsymbol {\mathrm{L}}}_{{\mathcal{O}}_X}\circ (-\otimes {\mathcal{O}}_X(H))$
also switches these classes up to sign, allowing us to focus solely on
$(a,b)=(2,1)$
. By [Reference Li, Pertusi and ZhaoLPZ23, Theorem 1.2], the moduli space for
$(a,b)=(2,1)$
on a cubic fourfold is the LLSvS eightfold constructed in [Reference Lehn, Lehn, Sorger and van StratenLLSvS17]. On the other hand, the moduli space for
$(a,b)=(2,1)$
on a cubic threefold is considered in [Reference Bayer, Beentjes, Feyzbakhsh, Hein, Martinelli, Rezaee and SchmidtBBF+24]. We will see in [Reference Feyzbakhsh, Guo, Liu and ZhangFGLZ25] that Theorem 5.8 provides a Lagrangian covering family, consistent with the classical construction in [Reference Shinder and SoldatenkovSS17].
Acknowledgements
It is our pleasure to thank Arend Bayer, Sasha Kuznetsov, and Qizheng Yin for very useful discussions on the topics of this project. We would like to thank Enrico Arbarello, Marcello Bernardara, Lie Fu, Yong Hu, Grzegorz Kapustka, Chunyi Li, Zhiyuan Li, Laurent Manivel, Kieran O’Grady, Alexander Perry, Laura Pertusi, Richard Thomas, Claire Voisin, Ruxuan Zhang, Yilong Zhang, and Xiaolei Zhao for helpful conversations. We thank Ruxuan Zhang for pointing out a mistake in the statement of Proposition 6.6 in the first version of our paper.
Conflicts of interest
None.
Financial support
SF acknowledges the support of EPSRC postdoctoral fellowship EP/T018658/1 and the Royal Society URF/ R1/231191. HG is supported by NSFC grant (12121001, 12171090 and 12425105). ZL is partially supported by NSFC Grant 123B2002. SZ is supported by the ERC Consolidator Grant WallCrossAG, no. 819864, ANR project FanoHK, grant ANR-20-CE40-0023 and partially supported by GSSCU2021092. SZ is also supported by the NSF under grant no. DMS-1928930, while he is residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California. Part of the work was finished during the junior trimester program of SF, ZL, and SZ funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. We would like to thank the Hausdorff Research Institute for Mathematics for their hospitality.
Journal information
Compositio Mathematica is owned by the Foundation Compositio Mathematica and published by the London Mathematical Society in partnership with Cambridge University Press. All surplus income from the publication of Compositio Mathematica is returned to mathematics and higher education through the charitable activities of the Foundation, the London Mathematical Society and Cambridge University Press.
Open access
