2.1 Notations for (derived) stacks

All the spaces $\mathscr {X}$ considered are quasi-smooth (derived) stacks over $\mathbb {C}$ , see [Reference TodaTod24a, Subsection 3.1] for references. The classical truncation of $\mathscr {X}$ is denoted by $\mathscr {X}^{\mathrm {{cl}}}$ . We denote by $\mathbb {L}_{\mathscr {X}}$ the cotangent complex of $\mathscr {X}$ .

For G an algebraic group and X a dg-scheme with an action of G, denote by $X/G$ the corresponding quotient stack. When X is affine, we denote by $X/\!\!/ G$ the quotient dg-scheme with dg-ring of regular functions $\mathcal {O}_X^G$ . For a morphism $f \colon X\to Y$ and for a closed point $y \in Y$ , we denote by $\widehat {X}_y$ the following base change

$$ \begin{align*} \widehat{X}_y := X \times_{Y} \operatorname{Spec} \left(\widehat{\mathcal{O}}_{Y, y}\right). \end{align*} $$

We call $\widehat {X}_y$ the formal fiber, though it is a scheme over a complete local ring rather than a formal scheme. When X is a G-representation, $f\colon X\to Y:=X/\!\!/ G$ , and $y=0$ , we omit the subscript y from the above notation.

We use the terminology of good moduli spaces of Alper, see [Reference AlperAlp13, Section 8] for examples of stacks with good moduli spaces.

2.2 DG-categories

For $\mathscr {X}$ a quasi-smooth stack, we denote by $D^b(\mathscr {X})$ the bounded derived category of coherent sheaves on $\mathscr {X}$ and by $\mathrm {Perf}(\mathscr {X})$ the subcategory of perfect complexes on $\mathscr {X}$ , see Subsection 2.6 for more details and for more categories of (quasi)coherent sheaves.

2.2.2 Semiorthogonal decompositions

Let R be a partially ordered countable set with binary relation ${\leqslant }$ . Let $\mathbb {T}$ be a pretriangulated dg-category. We will construct semiorthogonal decompositions

(2.1) $$ \begin{align} \mathbb{T}=\langle \mathbb{A}_i \mid i \in R \rangle \end{align} $$

with summands pretriangulated subcategories $\mathbb {A}_i$ indexed by $i\in R$ such that, if $i,j\in R$ and there exist objects $\mathcal {A}_i\in \mathbb {A}_i$ , $\mathcal {A}_j\in \mathbb {A}_j$ with $\operatorname {Hom}_{\mathbb {T}}(\mathcal {A}_i,\mathcal {A}_j)\neq 0$ , then $i{\leqslant } j$ .

Let $\pi \colon \mathscr {X}\to S$ be a morphism from a quasi-smooth stack to a scheme S and assume $\mathbb {T}$ is a subcategory of $D^b(\mathscr {X})$ . We say the decomposition (2.1) is S-linear if $\mathbb {A}_i\otimes \pi ^*\mathrm {Perf}(S)\subset \mathbb {A}_i$ .

2.3 Graded matrix factorizations

References for this subsection are [Reference TodaTod24b, Section 2.2], [Reference TodaTod24a, Section 2.2], [Reference Ballard, Favero and KatzarkovBFK19, Section 2.3], [Reference Polishchuk and VaintrobPV11, Section 1]. Let G be an algebraic group and let Y be a smooth affine scheme with an action of G. Let $\mathscr {Y}=Y/G$ be the corresponding quotient stack and let f be a regular function

$$\begin{align*}f \colon \mathscr{Y}\to \mathbb{C}.\end{align*}$$

Assume that there exists an extra action of $\mathbb {C}^{\ast }$ on Y which commutes with the action of G on Y, trivial on $\mathbb {Z}/2 \subset \mathbb {C}^{\ast }$ , and f is weight two with respect to the above $\mathbb {C}^{\ast }$ -action.

Consider the category of graded matrix factorizations

$$ \begin{align*}\mathrm{MF}^{\mathrm{gr}}(\mathscr{Y}, f). \end{align*} $$

Its objects are pairs $(P, d_P)$ with P a $G\times \mathbb {C}^*$ -equivariant coherent sheaf on Y and $d_P \colon P\to P(1)$ a $G\times \mathbb {C}^*$ -equivariant morphism satisfying $d_P^2=f$ . Here $(1)$ is the twist by the character $\mathrm {pr}_2 \colon G \times \mathbb {C}^{\ast } \to \mathbb {C}^{\ast }$ . Note that as the $\mathbb {C}^{\ast }$ -action is trivial on $\mathbb {Z}/2$ , we have the induced action of $\mathbb {C}^{\star }=\mathbb {C}^{\ast }/(\mathbb {Z}/2)$ on Y and f is weight one with respect to the above $\mathbb {C}^{\star }$ -action. The objects of $\mathrm {MF}^{\mathrm {gr}}(\mathscr {Y}, f)$ can be alternatively described as tuples

(2.2) $$ \begin{align} (E, F, \alpha \colon E\to F(1)', \beta \colon F\to E), \end{align} $$

where E and F are $G\times \mathbb {C}^{\star }$ -equivariant coherent sheaves on Y, $(1)'$ is the twist by the character $G \times \mathbb {C}^{\star } \to \mathbb {C}^{\star }$ , and $\alpha $ and $\beta $ are $\mathbb {C}^{\star }$ -equivariant morphisms such that $\alpha \circ \beta $ and $\beta \circ \alpha $ are multiplication by f.

For a pretriangulated subcategory $\mathbb {M}$ of $D^b(\mathscr {Y})$ , define $\mathrm {MF}^{\mathrm {{gr}}}(\mathbb {M}, f)$ as the full subcategory of $\mathrm {MF}^{\mathrm {{gr}}}(\mathscr {Y}, f)$ with objects totalizations of pairs $(P, d_{P})$ with $P \in \mathbb {M}$ equipped with $\mathbb {C}^{\ast }$ -equivariant structure, see [Reference PădurariuPT24, Subsection 2.6.2]. If $\mathbb {M}$ is generated by a set of vector bundles $\{\mathscr {V}_i\}_{i\in I}$ on $\mathscr {Y}$ , then $\mathrm {MF}^{\mathrm {{gr}}}(\mathbb {M}, f)$ is generated by matrix factorizations whose factors are direct sums of vector bundles from $\{\mathscr {V}_i\}_{i\in I}$ , see [Reference PădurariuPT24, Lemma 2.3].

Functoriality of categories of graded matrix factorizations for pullback and proper pushforward is discussed in [Reference Polishchuk and VaintrobPV11]. In Subsection 7.2, we will also consider the category $D^{\mathrm {{gr}}}(Y)$ for a possibly singular affine variety Y with a $\mathbb {C}^{\ast }$ -action as above. It consists of objects (2.2) with $f=0$ , so its definition is the same as $\mathrm {MF}^{\mathrm {{gr}}}(Y, 0)$ , but when Y is singular an object (2.2) may not be isomorphic to the one such that $E, F$ are locally free of finite rank. See [Reference Efimov and PositselskiEP15] for factorization categories over possibly singular varieties. Note that if the $\mathbb {C}^{\ast }$ -action on Y is trivial, then $D^{\mathrm {{gr}}}(Y)=D^b(Y)$ .

2.4 The Koszul equivalence

Let Y be a smooth affine scheme with an action of an algebraic group G, let $\mathscr {Y}=Y/G$ , and let V be a G-equivariant vector bundle on Y. We always assume that Y is either of finite type over $\mathbb {C}$ or is a formal fiber of a map $X \to X/\!\!/ H$ for a finite type affine scheme X and an algebraic group H as in Subsection 2.1. Let $\mathbb {C}^*$ act on the fibers of V with weight $2$ and consider a section $s\in \Gamma (Y, V)$ . It induces a map $\partial :=s^{\vee } \colon V^{\vee } \to \mathcal {O}_Y$ . Let $s^{-1}(0)$ be the derived zero locus of s with dg-algebra of regular functions

(2.3) $$ \begin{align} \mathcal{O}_{s^{-1}(0)}:=(\cdots \to \wedge^2 V^{\vee} \stackrel{\partial}{\to} V \stackrel{\partial}{\to} \mathcal{O}_Y) \end{align} $$

where the right-hand side is the Koszul complex associated with s. Consider the quotient (quasi-smooth) stack

(2.4) $$ \begin{align} \mathscr{P}:=s^{-1}(0)/G. \end{align} $$

We call $\mathscr {P}$ the Koszul stack associated with $(Y, V, s, G)$ . There is a natural inclusion

$$\begin{align*}j\colon \mathscr{P}\hookrightarrow\mathscr{Y}.\end{align*}$$

The section s also induces the regular function

(2.5) $$ \begin{align} f\colon \mathscr{V}^{\vee}:=\text{Tot}_Y\left(V^{\vee}\right)/G\to\mathbb{C} \end{align} $$

defined by $f(y,v)=\langle s(y), v \rangle $ for $y\in Y(\mathbb {C})$ and $v\in V^{\vee }|_y$ . Consider the category of graded matrix factorizations $\text {MF}^{\text {gr}}\left (\mathscr {V}^{\vee }, f\right )$ with respect to the $\mathbb {C}^*$ -action mentioned above. The Koszul equivalence, also called dimensional reduction in the literature, says the following:

Theorem 2.1. [Reference IsikIsi13, Reference HiranoHir17, Reference TodaTod24a]

There is an equivalence

(2.6) $$ \begin{align} \Theta \colon D^b(\mathscr{P}) \stackrel{\sim}{\to} \mathrm{MF}^{\mathrm{gr}}(\mathscr{V}^{\vee}, f) \end{align} $$

given by $\Theta (-)=\mathcal {K}\otimes _{\mathcal {O}_{\mathscr {P}}}(-)$ , where $\mathcal {K}$ is the Koszul factorization, see [Reference TodaTod24a, Theorem 2.3.3].

We will use the following lemma:

2.5 Window categories

2.5.1 Attracting stacks

Let Y be an affine variety with an action of a reductive group G. Let $\lambda $ be a cocharacter of G. Let $G^\lambda $ and $G^{\lambda {\geqslant } 0}$ be the Levi and parabolic groups associated to $\lambda $ . Let $Y^\lambda \subset Y$ be the closed subvariety of $\lambda $ -fixed points. Consider the attracting variety

$$\begin{align*}Y^{\lambda{\geqslant} 0}:=\{y\in Y|\,\lim_{t\to 0}\lambda(t)\cdot y\in Y^\lambda\}\subset Y.\end{align*}$$

Consider the attracting and fixed stacks

(2.7) $$ \begin{align} \mathscr{Z}:=Y^\lambda/G^\lambda \xleftarrow{q}\mathscr{S}:=Y^{\lambda{\geqslant} 0}/G^{\lambda{\geqslant} 0}\xrightarrow{p}\mathscr{Y}:=Y/G. \end{align} $$

The map p is proper. Kempf-Ness strata are connected components of certain attracting stacks $\mathscr {S}$ , and the map p restricted to a Kempf-Ness stratum is a closed immersion, see [Reference Halpern-LeistnerHL15, Section 2.1]. The attracting stacks also appear in the definition of Hall algebras [Reference PădurariuPăd23] (for Y an affine space), where the Hall product is induced by the functor

(2.8) $$ \begin{align} \ast:=p_*q^*\colon D^b(\mathscr{Z})\to D^b(\mathscr{Y}). \end{align} $$

In this case, the map p may not be a closed immersion.

Let $T \subset G$ be a maximal torus and let $\lambda $ be a cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T$ . For a G-representation Y, the attracting variety $Y^{\lambda {\geqslant } 0} \subset Y$ coincides with the sub T-representation generated by weights which pair non-negatively with $\lambda $ . We may abuse notation and denote by $\langle \lambda , Y^{\lambda {\geqslant } 0} \rangle := \langle \lambda , \det Y^{\lambda {\geqslant } 0} \rangle $ , where $\det Y^{\lambda {\geqslant } 0}$ is the sum of T-weights of $Y^{\lambda {\geqslant } 0}$ .

2.5.2 The definition of window categories

Let Y be an affine variety with an action of a reductive group G and a linearization $\ell $ . Consider the stacks

$$\begin{align*}j\colon\mathscr{Y}^{\ell\text{-ss}}:=Y^{\ell\text{-ss}}/G\hookrightarrow\mathscr{Y}.\end{align*}$$

We review the construction of window categories of $D^b(\mathscr {Y})$ which are equivalent to $D^b(\mathscr {Y}^{\ell \text {-ss}})$ via the restriction map, due to Segal [Reference SegalSeg11], Halpern-Leistner [Reference Halpern-LeistnerHL15], and Ballard–Favero–Katzarkov [Reference Ballard, Favero and KatzarkovBFK19]. We follow the presentation from [Reference Halpern-LeistnerHL15].

By also fixing a Weyl-invariant norm on the cocharacter lattice, the unstable locus $\mathscr {Y}\setminus \mathscr {Y}^{\ell \text {-ss}}$ has a stratification in Kempf-Ness strata $\mathscr {S}_i$ for $i\in I$ a finite ordered set:

$$\begin{align*}\mathscr{Y}\setminus \mathscr{Y}^{\ell\text{-ss}}=\bigsqcup_{i\in I}\mathscr{S}_i.\end{align*}$$

A Kempf-Ness stratum $\mathscr {S}_i$ is the attracting stack in $\mathscr {Y} \setminus \bigsqcup _{j<i}\mathscr {S}_j$ for a cocharacter $\lambda _i$ , with the fixed stack $\mathscr {Z}_i:=\mathscr {S}_i^{\lambda _i}$ . Let $N_{\mathscr {S}_i/\mathscr {Y}}$ be the normal bundle of $\mathscr {S}_i$ in $\mathscr {Y}$ . Define the width of the window categories

$$\begin{align*}\eta_i:=\left\langle \lambda_i, \det\left(N^{\vee}_{\mathscr{S}_i/\mathscr{Y}}|_{\mathscr{Z}_i}\right) \right\rangle.\end{align*}$$

For a choice of real numbers $m_{\bullet }=(m_i)_{i\in I}\in \mathbb {R}^I$ , define the category

(2.9) $$ \begin{align} \mathbb{G}^\ell_{m_{\bullet}}:=\{\mathcal{F}\in D^b(\mathscr{Y})\text{ such that } \mathrm{wt}_{\lambda_i}(\mathcal{F}|_{\mathscr{Z}_i})\subset [ m_i, m_i+\eta_i) \text{ for all }i\in I\}. \end{align} $$

In the above, $\mathrm {wt}_{\lambda _i}(\mathcal {F}|_{\mathscr {Z}_i})$ is the set of $\lambda _i$ -weights on $\mathcal {F}|_{\mathscr {Z}_i}$ . Then [Reference Halpern-LeistnerHL15, Theorem 2.10] says that the restriction functor $j^*$ induces an equivalence of categories:

(2.10) $$ \begin{align} j^*\colon \mathbb{G}^\ell_{m_{\bullet}}\xrightarrow{\sim} D^b\big(\mathscr{Y}^{\ell\text{-ss}}\big) \end{align} $$

for any choice of real numbers $m_{\bullet }=(m_i)_{i\in I}\in \mathbb {R}^I$ .

2.6 Quasi-smooth derived stacks

2.6.1 Derived categories of (quasi-)coherent sheaves

Let $\mathfrak {M}$ be a derived Artin stack over $\mathbb {C}$ and let $\mathcal {M}$ be its classical truncation. Let $\mathbb {L}_{\mathfrak {M}}$ be the cotangent complex of $\mathfrak {M}$ . The stack $\mathfrak {M}$ is called quasi-smooth if for all closed points $x\to \mathcal {M}$ , the restriction $\mathbb {L}_{\mathfrak {M}}|_x$ has cohomological amplitude in $[-1, 1]$ . By [Reference Ben-Bassat, Brav, Bussi and JoyceBBBBJ15, Theorem 2.8], the stack $\mathfrak {M}$ is quasi-smooth if and only if it is a $1$ -stack and any point of $\mathfrak {M}$ lies in the image of a $0$ -representable smooth morphism

(2.11) $$ \begin{align} \alpha \colon \mathscr{U}\to\mathfrak{M} \end{align} $$

for a Koszul scheme $\mathscr {U}$ as in (2.3). Let $D_{\mathrm {{qc}}}(\mathscr {U})$ be the derived category of dg-modules over $\mathcal {O}_{\mathscr {U}}$ and let $D^b(\mathscr {U}) \subset D_{\mathrm {{qc}}}(\mathscr {U})$ be the subcategory of objects with bounded coherent cohomologies. Further, let $\operatorname {Ind} D^b(\mathscr {U})$ be the ind-completion of $D^b(\mathscr {U})$ [Reference GaitsgoryGai13]. For a quasi-smooth stack $\mathfrak {M}$ , the dg-categories $D_{\mathrm {{qc}}}(\mathfrak {M})$ , $D^b(\mathfrak {M})$ , and $\operatorname {Ind} D^b(\mathfrak {M})$ are defined to be limits in the $\infty $ -category of smooth morphisms (2.11), see [Reference TodaTod24a, Subsection 3.1.1], [Reference GaitsgoryGai13]:

$$ \begin{align*} D_{\mathrm{{qc}}}(\mathfrak{M})=\lim_{\mathscr{U} \to \mathfrak{M}} D_{\mathrm{{qc}}}(\mathscr{U}), \ D^b(\mathfrak{M})=\lim_{\mathscr{U} \to \mathfrak{M}} D^b(\mathscr{U}), \ \operatorname{Ind} D^b(\mathfrak{M})=\lim_{\mathscr{U} \to \mathfrak{M}} \operatorname{Ind} D^b(\mathscr{U}). \end{align*} $$

The category $\operatorname {Ind} D^b(\mathfrak {M})$ is a module over $D_{\mathrm {{qc}}}(\mathfrak {M})$ via the tensor product. For $\mathcal {E}_1, \mathcal {E}_2 \in \operatorname {Ind} D^b(\mathfrak {M})$ , there exists an internal homomorphism, see [Reference Drinfeld and GaitsgoryDG13, Remark 3.4.5]

$$ \begin{align*} \mathcal{H}om(\mathcal{E}_1, \mathcal{E}_2) \in D_{\mathrm{{qc}}}(\mathfrak{M}), \end{align*} $$

such that for any $\mathcal {A} \in D_{\mathrm {{qc}}}(\mathfrak {M})$ we have

$$ \begin{align*} \operatorname{Hom}_{D_{\mathrm{{qc}}}(\mathfrak{M})}(\mathcal{A}, \mathcal{H}om(\mathcal{E}_1, \mathcal{E}_2)) \cong \operatorname{Hom}_{\operatorname{Ind} D^b(\mathfrak{M})}(\mathcal{A} \otimes \mathcal{E}_1, \mathcal{E}_2). \end{align*} $$

If $\mathfrak {M}$ is QCA (quasi-compact and with affine automorphism groups) [Reference Drinfeld and GaitsgoryDG13, Definition 1.1.8], then $\operatorname {Ind} D^b(\mathfrak {M})$ is compactly generated with compact objects $D^b(\mathfrak {M})$ , see [Reference Drinfeld and GaitsgoryDG13, Theorem 3.3.5].

2.6.2 Étale and formal local structures along good moduli spaces

Let $\mathfrak {M}$ be a quasi-smooth stack over $\mathbb {C}$ and let $\mathcal {M}$ be its classical truncation. Suppose that $\mathcal {M}$ admits a good moduli space map

$$\begin{align*}\pi\colon \mathcal{M} \to M,\end{align*}$$

see [Reference AlperAlp13] for the notion of a good moduli space. In particular, M is an algebraic space and $\pi $ is a quasi-compact morphism. For each point in M, there are an étale neighborhood $U \to M$ and Cartesian squares:

(2.12)

where each vertical arrow is étale and $\mathfrak {M}_U$ is equivalent to a Koszul stack $\mathscr {P}=s^{-1}(0)/G$ as in (2.4), see [Reference TodaTod24a, Subsection 3.1.4], [Reference Halpern-LeistnerHLa, Theorem 4.2.3], [Reference Alper, Hall and DavidAHD20]. Similarly, for each closed point $y \in M$ , there exist Cartesian squares, see [Reference TodaTod24a, Subsection 3.1.4]:

(2.13)

2.6.3 $(-1)$ -shifted cotangent stacks

Let $\mathfrak {M}$ be a quasi-smooth stack. Let $\mathbb {T}_{\mathfrak {M}}$ be the tangent complex of $\mathfrak {M}$ , which is the dual complex to the cotangent complex $\mathbb {L}_{\mathfrak {M}}$ . We denote by $\Omega _{\mathfrak {M}}[-1]$ the (-1)-shifted cotangent stack of $\mathfrak {M}$ :

$$ \begin{align*} \Omega_{\mathfrak{M}}[-1]:=\mathrm{Spec}_{\mathfrak{M}}\left(\mathrm{Sym}(\mathbb{T}_{\mathfrak{M}}[1])\right). \end{align*} $$

Consider the projection map

(2.14) $$ \begin{align} p_0\colon \mathcal{N}:=\Omega_{\mathfrak{M}}[-1]^{\mathrm{{cl}}}\to \mathfrak{M}. \end{align} $$

For a Koszul stack $\mathscr {P}$ as in (2.4), recall the function f from (2.5) and consider the critical locus $\mathrm {Crit}(f)\subset \mathscr {V}^{\vee }$ . In this case, the map $p_0$ is the natural projection

(2.15) $$ \begin{align} p_0\colon \mathrm{Crit}(f)=\Omega_{\mathscr{P}}[-1]^{\mathrm{{cl}}}\to \mathscr{P}. \end{align} $$

For an object $\mathcal {F} \in D^b(\mathfrak {M})$ , Arinkin–Gaitsgory [Reference Arinkin and GaitsgoryAG15] defined the notion of singular support denoted by

$$ \begin{align*} \mathrm{Supp}^{\mathrm{{sg}}}(\mathcal{F}) \subset \mathcal{N}. \end{align*} $$

The definition is compatible with maps $\alpha $ as in (2.11), see [Reference Arinkin and GaitsgoryAG15, Section 7]. Consider the group $\mathbb {C}^*$ scaling the fibers of the map $p_0$ . A closed substack $\mathscr {Z}$ of $\mathcal {N}$ is called conical if it is closed under the action of $\mathbb {C}^*$ . The singular support $\mathrm {Supp}^{\mathrm {{sg}}}(\mathcal {F})$ of $\mathcal {F}$ is a conical subset $\mathscr {Z}$ of $\mathcal {N}$ . For a given conical closed substack $\mathscr {Z} \subset \mathcal {N}$ , we denote by $\mathcal {C}_{\mathscr {Z}} \subset D^b(\mathfrak {M})$ the subcategory of objects whose singular supports are contained in $\mathscr {Z}$ .

Consider a Koszul stack $\mathscr {P}$ as in (2.4) and recall the Koszul equivalence $\Theta $ from (2.6). Under $\Theta $ , the singular support of $\mathcal {F}\in D^b(\mathscr {P})$ corresponds to the support $\mathscr {Z}$ of the matrix factorization $\Theta (\mathcal {F})$ , namely the maximal closed substack $\mathscr {Z}\subset \mathrm {Crit}(f)$ such that $\mathcal {F}|_{\mathscr {V}^{\vee }\setminus \mathscr {Z}}=0$ in $\mathrm {MF}^{\mathrm {gr}}(\mathscr {V}^{\vee }\setminus \mathscr {Z}, f)$ , see [Reference TodaTod24a, Subsection 2.3.9].

2.7 The window theorem for quasi-smooth stacks

We review the theory of window categories for singular support quotients of quasi-smooth stacks [Reference TodaTod24a, Chapter 5], which itself is inspired by Halpern-Leistner’s theory of window categories for $0$ -shifted symplectic derived stacks [Reference Halpern-LeistnerHLa]. We continue with the notation from the previous subsection.

Let $\mathfrak {M}$ be a quasi-smooth stack and assume throughout this subsection that its classical truncation $\mathcal {M}$ admits a good moduli space $\mathcal {M} \to M.$ Let $\ell $ be a line bundle on $\mathcal {M}$ and let $b \in H^4(\mathcal {M}, \mathbb {Q})$ be a positive definite class, see [Reference Halpern-LeistnerHLb, Definition 3.7.6]. We also use the same symbols $(\ell , b)$ for $p_0^{\ast }\ell \in \mathrm {Pic}(\mathcal {N})$ and $p_0^{\ast }b \in H^4(\mathcal {N}, \mathbb {Q})$ . Then there is a $\Theta $ -stratification with respect to $(\ell , b)$ :

(2.16) $$ \begin{align} \mathcal{N}=\mathscr{S}_1 \sqcup \cdots \sqcup \mathscr{S}_N\sqcup \mathcal{N}^{\ell\text{-ss}} \end{align} $$

with centers $\mathscr {Z}_i\subset \mathscr {S}_i$ , see [Reference Halpern-LeistnerHLb, Theorem 5.2.3, Proposition 5.3.3]. In the above situation, an analogue of the window theorem is proved in [Reference TodaTod, Theorem 1.1], [Reference TodaTod24a, Theorem 5.3.13] (which generalizes [Reference Halpern-LeistnerHLa, Theorem 3.3.1] in the case that $\mathfrak {M}$ is $0$ -shifted symplectic):

Theorem 2.3. [Reference TodaTod24a, Reference TodaTod]

In addition to the above, suppose that $\mathcal {M} \to M$ satisfies the formal neighborhood theorem, see below. Then for each $m_{\bullet }=(m_i)_{i=1}^N\in \mathbb {R}^N$ , there is a subcategory $\mathbb {W}(\mathfrak {M})^\ell _{m_{\bullet }} \subset D^b(\mathfrak {M})$ such that the composition

$$ \begin{align*} \mathbb{W}(\mathfrak{M})^\ell_{m_{\bullet}} \subset D^b(\mathfrak{M}) \twoheadrightarrow D^b(\mathfrak{M})/\mathcal{C}_{\mathscr{Z}} \end{align*} $$

is an equivalence, where $\mathscr {Z}:=\mathcal {N} \setminus \mathcal {N}^{\ell \text {-ss}}$ .

Remark 2.5. Suppose that $\mathbb {L}_{\mathfrak {M}}$ is self-dual, for example, $\mathfrak {M}$ is 0-shifted symplectic. In this case, we have $\mathcal {N}^{\ell \text {-ss}}=\Omega _{\mathfrak {M}^{\ell \text {-ss}}}[-1]^{\mathrm {{cl}}}$ , which easily follows from [Reference Halpern-LeistnerHLa, Lemma 4.3.22]. Then we have the equivalence, see [Reference TodaTod24a, Lemma 3.2.9]:

$$ \begin{align*} D^b(\mathfrak{M})/\mathcal{C}_{\mathscr{Z}} \stackrel{\sim}{\to} D^b(\mathfrak{M}^{\ell\text{-ss}}). \end{align*} $$

We now explain the meaning of “the formal neighborhood theorem” in the statement of Theorem 2.3, see [Reference TodaTod24a, Definition 5.2.3]. For a closed point $y \in M$ , denote also by $y \in \mathcal {M}$ the unique closed point in the fiber of $\mathcal {M} \to M$ at y. Set $G_y:=\mathrm {Aut}(y)$ , which is a reductive algebraic group. Let $\widehat {\mathcal {M}}_y$ be the formal fiber along with $\mathcal {M} \to M$ at y. Let $\widehat {\mathcal {H}}^0(\mathbb {T}_{\mathcal {M}}|_{y})$ be the formal fiber of at the origin, and define $\widehat {\mathcal {H}}^0(\mathbb {T}_{\mathfrak {M}}|_{y})$ similarly, see also the convention from Subsection 2.1. Then the formal neighborhood theorem says that there is a $G_y$ -equivariant morphism

$$ \begin{align*} \kappa_y \colon \widehat{\mathcal{H}}^0(\mathbb{T}_{\mathcal{M}}|_{y}) \to \mathcal{H}^1(\mathbb{T}_{\mathcal{M}}|_{y}) \end{align*} $$

such that, by setting $\mathcal {U}_y$ to be the classical zero locus of $\kappa _y$ , there is an isomorphism $\widehat {\mathcal {M}}_y \cong \mathcal {U}_y/G_y$ . Let $\mathfrak {U}_y$ be the derived zero locus of $\kappa _y$ . Then, by replacing $\kappa _y$ if necessary, $\widehat {\mathfrak {M}}_y$ is equivalent to $\mathfrak {U}_y/G$ , see [Reference TodaTod24a, Lemma 5.2.5].

Below we give a formal local description of $\mathbb {W}(\mathfrak {M})_{m_{\bullet }}^\ell $ . Consider the pair of a smooth stack and a regular function $(\mathscr {X}_y, f_y)$ :

$$ \begin{align*} \mathscr{X}_y:=\left(\widehat{\mathcal{H}}^0(\mathbb{T}_{\mathfrak{M}}|_{y}) \times \mathcal{H}^1(\mathbb{T}_{\mathfrak{M}}|_{y})^{\vee}\right)/G_y \stackrel{f_y}{\to} \mathbb{C}, \end{align*} $$

where $f_y(u, v)=\langle \kappa _y(u), v \rangle $ . From (2.15), the critical locus of $f_y$ is isomorphic to the classical truncation of the $(-1)$ -shifted cotangent stack over $\widehat {\mathfrak {M}}_y$ , so it is isomorphic to the formal fiber $\widehat {\mathcal {N}}_y$ of $\mathcal {N} \to \mathcal {M} \to M$ at y. The pull-back of the $\Theta $ -stratification (2.16) to $\widehat {\mathcal {N}}_y$ gives a Kempf-Ness stratification

$$ \begin{align*} \widehat{\mathcal{N}}_y=\widehat{\mathscr{S}}_{1, y} \sqcup \cdots \sqcup \widehat{\mathscr{S}}_{N, y} \sqcup \widehat{\mathcal{N}}_y^{\ell\text{-ss}} \end{align*} $$

with centers $\widehat {\mathscr {Z}}_{i, y}\subset \widehat {\mathscr {S}}_{i, y}$ and one parameter subgroups $\lambda _i \colon \mathbb {C}^{\ast } \to G_y$ . By the Koszul equivalence, see Theorem 2.1, there is an equivalence:

$$ \begin{align*} \Theta_y \colon D^b(\widehat{\mathfrak{M}}_y) \stackrel{\sim}{\to} \mathrm{MF}^{\text{gr}}(\mathscr{X}_y, f_y). \end{align*} $$

Then the subcategory $\mathbb {W}(\mathfrak {M})_{m_{\bullet }}^\ell $ in Theorem 2.3 is characterized as follows: an object $\mathcal {E} \in D^b(\mathfrak {M})$ is an object of $\mathbb {W}(\mathfrak {M})^\ell _{m_{\bullet }}$ if and only if, for any closed point $y \in M$ , we have

(2.17) $$ \begin{align} \Theta_{y}(\mathcal{E}|_{\widehat{\mathfrak{M}}_y}) \in \mathrm{MF}^{\text{gr}}(\mathbb{G}^\ell_{m_{\bullet}'}, f_y), \ m_i'=m_i-\left\langle \lambda_i, \det\left(\mathcal{H}^1(\mathbb{T}_{\mathfrak{M}}|_{y})^{\lambda_i>0}\right) \right\rangle. \end{align} $$

The category $\mathbb {G}^{\ell }_{m^{\prime }_{\bullet }}$ is the window category (2.9) for the weights $(m^{\prime }_i)_{i=1}^N$ and the line bundle $\ell $ . The difference between $m_i$ and $m_i'$ is due to the discrepancy of categorical Hall products on $\mathfrak {M}_y$ and $\mathscr {X}_y$ , see [Reference PădurariuPăd23, Proposition 3.1].

2.8 Intrinsic window subcategory

We continue to consider a quasi-smooth derived stack $\mathfrak {M}$ whose classical truncation $\mathcal {M}$ admits a good moduli space $\mathcal {M} \to M$ . We say that $\mathfrak {M}$ is symmetric if for any closed point $y \in \mathfrak {M}$ , the $G_y:=\mathrm {Aut}(y)$ -representation

$$ \begin{align*} \mathcal{H}^0(\mathbb{T}_{\mathfrak{M}}|_{y}) \oplus \mathcal{H}^1(\mathbb{T}_{\mathfrak{M}}|_{y})^{\vee} \end{align*} $$

is a self-dual $G_y$ -representation. In this subsection, we assume that $\mathfrak {M}$ is symmetric. Let $\delta \in \mathrm {Pic}(\mathfrak {M})_{\mathbb {R}}$ . We now define a different kind of window categories, called intrinsic window subcategories $\mathbb {W}(\mathfrak {M})_{\delta }^{\mathrm {{int}}} \subset D^b(\mathfrak {M})$ , see [Reference TodaTod24a, Definition 5.2.12, 5.3.12]. These categories are the quasi-smooth version of “magic window categories” from [Reference Špenko and Van den BerghŠVdB17, Reference Halpern-Leistner and SamHLS20].

First, assume that $\mathfrak {M}$ is a Koszul stack associated with $(Y, V, s, G)$ as in (2.4)

(2.18) $$ \begin{align} \mathfrak{M}=s^{-1}(0)/G. \end{align} $$

Consider the quotient stack $\mathscr {Y}=Y/G$ , the closed immersion $j \colon \mathfrak {M} \hookrightarrow \mathscr {Y}$ , and let $\mathscr {V} \to \mathscr {Y}$ be the total space of $V/G \to Y/G$ . In this case, we define $\mathbb {W}(\mathfrak {M})_{\delta }^{\mathrm {{int}}} \subset D^b(\mathfrak {M})$ to be consisting of $\mathcal {E} \in D^b(\mathfrak {M})$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathfrak {M}$ we have

$$ \begin{align*} \mathrm{wt}(\nu^{\ast}j^{\ast}j_{\ast}\mathcal{E}) \subset \left[\frac{1}{2}\mathrm{wt}\left(\det \nu^{\ast}(\mathbb{L}_{\mathscr{V}}|_{\mathscr{Y}})^{\nu<0} \right), \frac{1}{2}\mathrm{wt}\left(\det \nu^{\ast}(\mathbb{L}_{\mathscr{V}}|_{\mathscr{Y}})^{\nu>0}\right) \right] +\mathrm{wt}(\nu^{\ast}\delta). \end{align*} $$

The above subcategory $\mathbb {W}(\mathfrak {M})_{\delta }^{\mathrm {{int}}}$ is intrinsic to $\mathfrak {M}$ , that is, independent of a choice of a presentation $\mathfrak {M}$ as (2.18) for $(Y, V, s, G)$ , see [Reference TodaTod24a, Lemma 5.3.14].

In general, the intrinsic window subcategory is defined as follows (which generalizes the magic window category in [Reference Halpern-LeistnerHLa, Definition 4.3.5] considered when $\mathbb {L}_{\mathfrak {M}}$ is self-dual):

Definition 2.6. [Reference TodaTod24a, Definition 5.3.12]

We define the subcategory

$$ \begin{align*} \mathbb{W}(\mathfrak{M})_{\delta}^{\mathrm{{int}}} \subset D^b(\mathfrak{M}) \end{align*} $$

to be consisting of objects $\mathcal {E}$ such that, for any étale morphism $\iota _U\colon U \to M$ such that $\mathfrak {M}_U$ is of the form $s^{-1}(0)/G$ as in (2.18) and $\iota _U$ induces an étale morphism $\iota _U \colon \mathfrak {M}_U \to \mathfrak {M}$ , we have $\iota _U^{\ast }\mathcal {E} \in \mathbb {W}(\mathfrak {M}_U)_{\iota _U^{\ast }\delta }^{\mathrm {{int}}} \subset D^b(\mathfrak {M}_U)$ .

3.1 Moduli stacks of representations of quivers

3.1.1 Moduli stacks

Let $Q=(I, E)$ be a quiver, where I is the set of vertices and E is the set of edges. For a dimension vector $\boldsymbol {d}=(d^{(a)})_{a \in I} \in \mathbb {N}^{I}\subset \mathbb {Z}^I$ , we denote by

(3.1) $$ \begin{align} \mathscr{X}(\boldsymbol{d})=R_Q(\boldsymbol{d})/G(\boldsymbol{d}) \end{align} $$

the moduli stack of Q-representations of dimension $\boldsymbol {d}$ . Here, the affine space $R_Q(\boldsymbol {d})$ and the reductive group $G(\boldsymbol {d})$ are defined by

$$ \begin{align*} R_Q(\boldsymbol{d})=\bigoplus_{(a \to b) \in E} \operatorname{Hom}(V^{(a)}, V^{(b)}), \ G(\boldsymbol{d})=\prod_{a \in I}GL(V^{(a)}), \end{align*} $$

where $V^{(a)}$ is a $\mathbb {C}$ -vector space of dimension $d^{(a)}$ . We denote by $\mathfrak {g}(\boldsymbol {d})$ the Lie algebra of $G(\boldsymbol {d})$ .

3.1.2 Doubled quivers

Let $Q^{\circ }=(I, E^{\circ })$ be a quiver. Let $E^{\circ \ast }$ be the set of edges $e^{\ast }=(b \to a)$ for each $e=(a \to b)$ in $E^{\circ }$ . Consider the doubled quiver of $Q^\circ $ :

$$ \begin{align*} Q^{\circ, d}=(I, E^{\circ, d}), \ E^{\circ, d}=E^{\circ} \sqcup E^{\circ \ast}. \end{align*} $$

Let $\mathscr {I}$ be the quadratic relation $\sum _{e \in E^{\circ }} [e, e^{\ast }]\in \mathbb {C}[Q^{\circ , d}]$ . For a dimension vector $\boldsymbol {d}=(d^{(a)})_{a \in I}$ , the relation $\mathscr {I}$ induces a moment map:

(3.2) $$ \begin{align} \mu \colon R_{Q^{\circ, d}}(\boldsymbol{d})=T^*R_{Q^\circ}(\boldsymbol{d}) \to \mathfrak{g}(\boldsymbol{d}). \end{align} $$

The derived zero locus

(3.3) $$ \begin{align} \mathscr{P}(\boldsymbol{d}):=\mu^{-1}(0)/G(\boldsymbol{d}) \stackrel{j}{\hookrightarrow} \mathscr{Y}(\boldsymbol{d}):=R_{Q^{\circ, d}}(\boldsymbol{d})/G(\boldsymbol{d}) \end{align} $$

is the derived moduli stack of $(Q^{\circ , d}, \mathscr {I})$ -representations of dimension vector $\boldsymbol {d}$ . Note that a $(Q^{\circ , d}, \mathscr {I})$ -representation is the same as a representation of the preprojective algebra $\Pi _{Q^\circ }:=\mathbb {C}[Q^{\circ , d}]/(\mathscr {I})$ of $Q^\circ $ , and we will use these two names interchangeably.

3.1.3 Tripled quivers

Consider a quiver $Q^{\circ }=(I, E^{\circ })$ . For $a\in I$ , let $\omega _a$ be a loop at a. The tripled quiver of $Q^\circ $ is:

$$ \begin{align*} Q=(I, E), \ E=E^{\circ, d}\sqcup \{\omega_a\}_{a \in I}. \end{align*} $$

The tripled potential W of Q is:

$$ \begin{align*} W=\left(\sum_{a \in I}\omega_a \right) \left( \sum_{e \in E^{\circ}}[e, e^{\ast}] \right). \end{align*} $$

Consider the stack (3.1) of representations of dimension $\boldsymbol {d}$ for the tripled quiver Q:

$$\begin{align*}\mathscr{X}(\boldsymbol{d}):=R_{Q}(\boldsymbol{d})/G(\boldsymbol{d}):=R_{Q^{\circ, d}}(\boldsymbol{d}) \oplus \mathfrak{g}(\boldsymbol{d})/G(\boldsymbol{d}).\end{align*}$$

The potential W induces the regular function:

(3.4) $$ \begin{align} \mathrm{Tr} W\colon \mathscr{X}(\boldsymbol{d})=R_{Q}(\boldsymbol{d})/G(\boldsymbol{d}) \to \mathbb{C}. \end{align} $$

We have the Koszul duality equivalence, see Theorem 2.1:

(3.5) $$ \begin{align} \Theta \colon D^b(\mathscr{P}(\boldsymbol{d})) \stackrel{\sim}{\to} \mathrm{MF}^{\mathrm{{gr}}}(\mathscr{X}(\boldsymbol{d}), \mathrm{Tr} W). \end{align} $$

3.2 The weight lattice

Let $Q=(I, E)$ be a quiver. For a dimension vector $\boldsymbol {d} \in \mathbb {N}^I$ , let $T(\boldsymbol {d}) \subset G(\boldsymbol {d})$ be the standard maximal torus and let $M(\boldsymbol {d})$ be the character lattice for $T(\boldsymbol {d})$ :

$$ \begin{align*} M(\boldsymbol{d})=\bigoplus_{a\in I} \bigoplus_{1{\leqslant} i{\leqslant} d^{(a)}} \mathbb{Z} \beta_i^{(a)}. \end{align*} $$

Here $\beta _1^{(a)}, \ldots , \beta _{d^{(a)}}^{(a)}$ are the weights of the standard representation of $GL(V^{(a)})$ for $a\in I$ . In the case that I consists of one element, we omit the superscript $(a)$ . We denote by $\rho \in M(\boldsymbol {d})_{\mathbb {Q}}$ half of the sum of the positive roots of $\mathfrak {g}(\boldsymbol {d})$ . Let W be the Weyl group of $G(\boldsymbol {d})$ and let $M(\boldsymbol {d})_{\mathbb {R}}^W \subset M(\boldsymbol {d})_{\mathbb {R}}$ be the Weyl-invariant subspace. There is a decomposition:

$$\begin{align*}M(\boldsymbol{d})_{\mathbb{R}}^W=\bigoplus_{a \in I} \mathbb{R}\sigma^{(a)},\end{align*}$$

where $\sigma ^{(a)}:=\sum _{i=1}^{d^{(a)}}\beta _i^{(a)}$ . There is a natural pairing:

$$ \begin{align*} \langle -, -\rangle \colon M(\boldsymbol{d})_{\mathbb{R}}^W \times \mathbb{R}^I \to \mathbb{R}, \ \langle \sigma^{(a)}, e^{(b)} \rangle=\delta^{ab}. \end{align*} $$

Here $\{e^{(b)}\}_{b\in I}$ is the standard basis of $\mathbb {R}^I$ . We denote by $\iota \colon M(\boldsymbol {d})_{\mathbb {R}} \to \mathbb {R}$ the linear map sending $\beta _i^{(a)}$ to $1$ , and its kernel by $M(\boldsymbol {d})_{0, \mathbb {R}}$ . An element $\ell \in M(\boldsymbol {d})_{0, \mathbb {R}}^W$ is written as

$$ \begin{align*} \ell=\sum_{a\in I}\ell^{(a)}\sigma^{(a)}, \ \langle \ell, \boldsymbol{d} \rangle= \sum_{a}\ell^{(a)}d^{(a)}=0 \end{align*} $$

that is, $\ell $ is an $\mathbb {R}$ -character of $G(\boldsymbol {d})$ which is trivial on the diagonal torus $\mathbb {C}^{\ast } \subset G(\boldsymbol {d})$ . Denote by $\underline {\boldsymbol {d}}:=\sum _{a\in I}d^{(a)}$ the total dimension. Define the following Weyl-invariant weight:

$$ \begin{align*} \tau_{\boldsymbol{d}} :=\frac{1}{\underline{\boldsymbol{d}}} \cdot \sum_{a\in I, 1{\leqslant} i{\leqslant} d^{(a)}}\beta_i^{(a)}. \end{align*} $$

Define the polytope:

(3.6) $$ \begin{align} \textbf{W}(\boldsymbol{d}):=\frac{1}{2}\mathrm{sum}[0, \beta]\subset M(\boldsymbol{d})_{\mathbb{R}}, \end{align} $$

where the Minkowski sum is after all $T(\boldsymbol {d})$ -weights $\beta $ of $R(\boldsymbol {d})$ .

Note that the set of generic weights is a dense open subset in $M(\boldsymbol {d})^W_{0,\mathbb {R}}$ .

3.3 Quasi-BPS categories for stacks of representations of preprojective algebras

Let $Q^{\circ }=(I, E^{\circ })$ be a quiver and let $\mathscr {P}(\boldsymbol {d})$ be the derived moduli stack of representations of its preprojective algebra (3.3). For $\delta \in M(\boldsymbol {d})_{\mathbb {R}}^W$ , define the quasi-BPS category to be the intrinsic window subcategory in Definition 2.6:

(3.7) $$ \begin{align} \mathbb{T}(\boldsymbol{d})_\delta:=\mathbb{W}(\mathscr{P}(\boldsymbol{d}))_{\delta}^{\mathrm{{int}}} \subset D^b(\mathscr{P}(\boldsymbol{d})), \ \mathbb{T}(\boldsymbol{d})_w :=\mathbb{T}(\boldsymbol{d})_{\delta=w \tau_{\boldsymbol{d}}}. \end{align} $$

An alternative description is as follows, where recall the map $j\colon \mathscr {P}(\boldsymbol {d})\hookrightarrow \mathscr {Y}(\boldsymbol {d})$ and choose a dominant chamber $M(\boldsymbol {d})^+\subset M(\boldsymbol {d})$ , for example, the one in [Reference PădurariuPTc, Subsection 2.2.2]:

Lemma 3.2. [Reference PădurariuPTc, Corollary 3.20]

The subcategory $\mathbb {T}(\boldsymbol {d})_{\delta }$ in (3.7) consists of objects $\mathcal {E} \in D^b(\mathscr {P}(\boldsymbol {d}))$ such that $j_{\ast }\mathcal {E}$ is classically generated by the vector bundle $\mathcal {O}_{\mathscr {Y}(d)} \otimes \Gamma _{G(\boldsymbol {d})}(\chi )$ , where $\chi $ is a dominant weight such that

(3.8) $$ \begin{align} \chi+\rho-\delta \in \mathbf{W}(\boldsymbol{d}). \end{align} $$

Here, $\Gamma _{G(\boldsymbol {d})}(\chi )$ is the irreducible representation of $G(\boldsymbol {d})$ with highest weight $\chi $ , and $\mathbf {W}(\boldsymbol {d})$ is the polytope (3.6) for the tripled quiver Q of $Q^\circ $ .

For $\ell \in M(\boldsymbol {d})_{0, \mathbb {R}}^W$ , let $\mathscr {P}(\boldsymbol {d})^{\ell \text {-ss}} \subset \mathscr {P}(\boldsymbol {d})$ be the open substack of $\ell $ -semistable locus. The quasi-BPS category for $\ell $ -semistable locus is defined to be

$$ \begin{align*} \mathbb{T}^{\ell}(\boldsymbol{d})_\delta :=\mathbb{W}(\mathscr{P}(\boldsymbol{d})^{\ell\text{-ss}})_{\delta}^{\mathrm{{int}}} \subset D^b(\mathscr{P}(\boldsymbol{d})^{\ell\text{-ss}}), \ \mathbb{T}^\ell(\boldsymbol{d})_{w} :=\mathbb{T}^\ell(\boldsymbol{d})_{\delta=w\tau_{\boldsymbol d}}. \end{align*} $$

Consider the restriction functor

(3.9) $$ \begin{align} \mathrm{res} \colon D^b(\mathscr{P}(\boldsymbol{d})) \twoheadrightarrow D^b(\mathscr{P}(\boldsymbol{d})^{\ell\text{-ss}}). \end{align} $$

We recall a wall-crossing equivalence proved in [Reference PădurariuPTc]:

Theorem 3.3. [Reference PădurariuPTc, Corollary 3.19, Remark 3.12]

For generic $\ell _+, \ell _- \in M(\boldsymbol {d})_{0, \mathbb {R}}^W$ , let $\delta '=\varepsilon _{+} \cdot \ell _{+} +\varepsilon _{-} \cdot \ell _{-}$ for general $0<\varepsilon _{\pm } \ll 1$ . Let $\delta \in M(\boldsymbol {d})^W_{\mathbb {R}}$ and let $\delta "=\delta +\delta '$ . Then the restriction functor (3.9) induces equivalences:

$$ \begin{align*} \mathbb{T}(\boldsymbol{d})_{\delta"} \stackrel{\sim}{\to} \mathbb{T}^{\ell_{\pm}}(\boldsymbol{d})_{ \delta"}. \end{align*} $$

In particular, there is an equivalence $\mathbb {T}^{\ell _{+}}(\boldsymbol {d})_{\delta "} \simeq \mathbb {T}^{\ell _{-}}(\boldsymbol {d})_{ \delta "}$ .

3.4 Semiorthogonal decompositions for preprojective algebras of quivers with one vertex

In the remaining of this section, we focus on the case of the g-loop quiver $Q^{\circ }=Q_g$ with loops $X_1, \ldots , X_g$ . In this case, we write the dimension vector by $\boldsymbol {d}=d \in \mathbb {N}$ . The doubled quiver is $Q^{\circ , d}=Q_{2g}$ with loops $X_1, \ldots , X_g, Y_1, \ldots , Y_g$ and the relation $\mathscr {I}$ is given by $\sum _{i=1}^g[X_i, Y_i]\in \mathbb {C}[Q_{2g}]$ . The map (3.2) in this case is

$$ \begin{align*} \mu \colon \mathfrak{gl}(d)^{\oplus 2g} \to \mathfrak{gl}(d), \ (x_1, \ldots, x_g, y_1, \ldots, y_g) \mapsto \sum_{i=1}^g [x_i, y_i]. \end{align*} $$

Then the derived stack in (3.3) is

$$ \begin{align*}\mathscr{P}(d)=\mu^{-1}(0)/GL(d) \hookrightarrow \mathscr{Y}(d)=\mathfrak{gl}(d)^{\oplus 2g}/GL(d). \end{align*} $$

For a partition $d=d_1+\cdots +d_k$ , let $\mathscr {P}(d_1, \ldots , d_k)$ be the derived moduli stack of filtrations

$$ \begin{align*} 0=R_0\subset R_1 \subset \cdots \subset R_k \end{align*} $$

of $(Q^{\circ , d}, \mathscr {I})$ -representations such that $R_i/R_{i-1}$ has dimension $d_i$ . Explicitly, let $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ be an antidominant cocharacter corresponding to the decomposition $d=d_1+\cdots +d_k$ and set

$$ \begin{align*} \mu^{{\geqslant} 0} \colon \left(\mathfrak{gl}(d)^{\oplus 2g}\right)^{\lambda {\geqslant} 0} \to \mathfrak{gl}(d)^{\lambda {\geqslant} 0} \end{align*} $$

to be the restriction of $\mu $ . Then

$$ \begin{align*} \mathscr{P}(d_1, \ldots, d_k)=\left(\mu^{{\geqslant} 0}\right)^{-1}(0)/GL(d)^{\lambda {\geqslant} 0}. \end{align*} $$

Consider the evaluation morphisms

$$ \begin{align*} \times_{i=1}^k \mathscr{P}(d_i) \stackrel{q}{\leftarrow} \mathscr{P}(d_1, \ldots, d_k) \stackrel{p}{\to} \mathscr{P}(d). \end{align*} $$

The map q is quasi-smooth and the map p is proper. Consider the categorical Hall product for the preprojective algebra of $Q^\circ $ [Reference Porta and SalaPS23, Reference Varagnolo and VasserotVV22]:

(3.10) $$ \begin{align} p_{\ast}q^{\ast} \colon \boxtimes_{i=1}^k D^b(\mathscr{P}(d_i)) \to D^b(\mathscr{P}(d)). \end{align} $$

We recall a result from [Reference PădurariuPTc]:

Theorem 3.4. [Reference PădurariuPTc, Theorem 4.20, Example 4.21]

There is a semiorthogonal decomposition

(3.11) $$ \begin{align} D^b(\mathscr{P}(d))= \left\langle \boxtimes_{i=1}^k \mathbb{T}(d_i)_{w_i+(g-1)d_i(\sum_{i>j}d_j-\sum_{i<j}d_j)}\right\rangle \end{align} $$

where the right-hand side is after all partitions $(d_i)_{i=1}^k$ of d and weights $(w_i)_{i=1}^k \in \mathbb {Z}^k$ such that

(3.12) $$ \begin{align} \frac{w_1}{d_1}<\cdots<\frac{w_k}{d_k}. \end{align} $$

The fully faithful functor

$$ \begin{align*} \boxtimes_{i=1}^k \mathbb{T}(d_i)_{w_i+(g-1)d_i(\sum_{i>j}d_j-\sum_{i<j}d_j)} \to D^b(\mathscr{P}(d)) \end{align*} $$

is given by the categorical Hall product (3.10). The order is as in [Reference PădurariuPT24, Subsection 3.5], [Reference PădurariuPTc, Subsection 4.6].

Remark 3.5. The semiorthogonal decomposition (3.11) is obtained from that of $(2g+1)$ -loop quiver and applying Koszul equivalence. The order of summands in (3.11) is the same one for the $(2g+1)$ -loop quiver from [Reference PădurariuPTc, Subsection 4.6], see also [Reference PădurariuPT24, Subsection 3.5] for the case $g=1$ , where it was described by (repeatedly) comparing certain ratios associated to a weight in $M(d)$ . We now state an equivalent combinatorial formulation, which is not difficult to be compared with the one from loc. cit, but we postpone writing full details to future work, which will discuss similar semiorthogonal decompositions for arbitrary reductive groups G (not necessarily of type A).

Let R be the set of convex lattice paths from the origin to $(w,d)$ . The set R is in bijection with the summands with the semiorthogonal decomposition (3.11). For example, to such a summand, we associate the convex path

$$\begin{align*}p: (0,0), (w_1,d_1), (w_1+w_2,d_1+d_2),(w_1+w_2+w_3,d_1+d_2+d_3),\ldots, (w,d).\end{align*}$$

We define the following partial order on R. For p and $p'$ convex paths in R, we have that $p'{\leqslant } p$ if and only if $p'$ “lies below” p. More explicitly, the second coordinates of $p'$ and p are functions $\gamma ', \gamma : [0,w]\to \mathbb {R}$ , and $p'$ “lies below” p means that $\gamma '(x){\leqslant } \gamma (x)$ for all $x\in [0,w]$ .

3.5 Semiorthogonal decompositions on formal fibers

We have the following diagram:

(3.13)

Here, the vertical arrows are good moduli space morphisms and horizontal arrows are closed immersions. Consider a closed point $p\in P(d)$ which corresponds to a semisimple $(Q^{\circ , d}, \mathscr {I})$ -representation

(3.14) $$ \begin{align} R_p=\bigoplus_{i=1}^m W^{(i)} \otimes R^{(i)}, \end{align} $$

where $R^{(i)}$ is a simple $(Q^{\circ , d}, \mathscr {I})$ -representation of dimension $r^{(i)}$ and $W^{(i)}$ is a finite-dimensional $\mathbb {C}$ -vector space. We denote by $\widehat {\mathscr {Y}}(d)_p$ the formal fiber of the right vertical arrow in (3.13) at p. By the étale slice theorem, we have

$$ \begin{align*} \widehat{\mathscr{Y}}(d)_p=\widehat{\operatorname{Ext}}_{Q^{\circ, d}}^1(F_p, F_p)/G_p, \end{align*} $$

where $G_p=\mathrm {Aut}(R_p)=\prod _{i=1}^m GL(W^{(i)})$ , and see Subsection 2.1 for the notation. We denote by

(3.15) $$ \begin{align} j_p \colon \widehat{\mathscr{P}}(d)_p \hookrightarrow \widehat{\mathscr{Y}}(d)_p \end{align} $$

the natural inclusion of the derived zero locus of $\mu $ restricted to $\widehat {\mathscr {Y}}(d)_p$ .

Remark 3.6. Let $\kappa $ be the morphism

$$ \begin{align*} \kappa \colon \operatorname{Ext}^1_{(Q^{\circ, d}, \mathscr{I})}(R_p, R_p) \to \operatorname{Ext}^2_{(Q^{\circ, d}, \mathscr{I})}(R_p, R_p) \end{align*} $$

given by $x \mapsto [x, x]$ . By the formality of polystable objects in CY2 category, see [Reference DavisonDavc, Corollary 4.9], the derived stack $\widehat {\mathscr {P}}(d)_p$ is equivalent to the formal fiber of $\kappa ^{-1}(0)/G_p$ at $0 \in \kappa ^{-1}(0)^{\mathrm {cl}}/\!\!/ G_p$ .

We define

(3.16) $$ \begin{align} \mathbb{T}_p(d)_w :=\mathbb{W}(\widehat{\mathscr{P}}(d)_p)_{w\tau_d}^{\mathrm{{int}}} \subset D^b(\widehat{\mathscr{P}}(d)_p). \end{align} $$

There is a description of $\mathbb {T}_p(d)_w$ similar to Lemma 3.2, see Subsection 5.4. Consider a partition $d=d_1+\cdots +d_k$ . We have the commutative diagram:

(3.17)

where $\times _{i=1}^k \pi _{P,d_i}$ and $\pi _{P,d}$ are good moduli space maps. The base change of the categorical Hall product gives the functor

(3.18) $$ \begin{align} \bigoplus_{p_1+\cdots+p_k=p} \boxtimes_{i=1}^k D^b(\widehat{\mathscr{P}}(d_i)_{p_i}) \to D^b(\widehat{\mathscr{P}}(d)_{p}), \end{align} $$

where the sum on the left-hand side consists of the fiber of the bottom horizontal arrow $\oplus $ in (3.17), which is a finite map. Also see [Reference TodaTod, (6.12), Lemma 6.4] for the existence of base change diagram of (3.17) extended to derived stacks.

The following proposition is a formal fiber version of Theorem 3.4. The proof is technical and will be postponed to Subsection 5.4.

Proposition 3.7. There is a semiorthogonal decomposition

$$ \begin{align*} D^b(\widehat{\mathscr{P}}(d)_p) =\left\langle \bigoplus_{p_1+\cdots+p_k=p} \boxtimes_{i=1}^k \mathbb{T}_{p_i}(d_i)_{w_i+(g-1)d_i(\sum_{i>j}d_j-\sum_{i<j}d_j)} \right\rangle. \end{align*} $$

The right-hand side is after all partitions $(d_i)_{i=1}^k$ of d, all points $(p_1,\ldots , p_k)$ in the fiber over p of the addition map $\oplus \colon \times _{i=1}^k P(d_i)\to P(d)$ , and all weights $(w_i)_{i=1}^k\in \mathbb {Z}^k$ such that

$$\begin{align*}\frac{w_1}{d_1}<\cdots<\frac{w_k}{d_k}.\end{align*}$$

The order of the semiorthogonal decomposition is the same as the order of (3.11). The fully faithful functor

$$ \begin{align*} \bigoplus_{p_1+\cdots+p_k=p} \boxtimes_{i=1}^k \mathbb{T}_{p_i}(d_i)_{w_i+(g-1)d_i(\sum_{i>j}d_j-\sum_{i<j}d_j)} \to D^b(\widehat{\mathscr{P}}(d)_p) \end{align*} $$

is given by the base change of the categorical Hall product (3.18).

During the proof of Proposition 3.7, we will also obtain the following:

Corollary 3.8. The map $\iota _p \colon \widehat {\mathscr {P}}(d)_p \to \mathscr {P}(d)$ induces the functor

$$ \begin{align*} \iota_p^{\ast} \colon \mathbb{T}(d)_w \to \mathbb{T}_p(d)_w \end{align*} $$

and its image classically generates $\mathbb {T}_p(d)_w$ .

3.6 Reduced quasi-BPS categories

We continue the discussion from the previous subsection. Let $\mathfrak {gl}(d)_0 \subset \mathfrak {gl}(d)$ be the traceless Lie subalgebra, and let $\mu _0$ be the map

(3.19) $$ \begin{align} \mu_0 \colon \mathfrak{gl}(d)^{\oplus 2g} \to \mathfrak{gl}(d)_0, \ (x_1, \ldots, x_g, y_1, \ldots, y_g) \mapsto \sum_{i=1}^g [x_i, y_i]. \end{align} $$

Define the reduced stack:

$$ \begin{align*} \mathscr{P}(d)^{\mathrm{{red}}} :=\mu_0^{-1}(0)/GL(d). \end{align*} $$

We define the reduced quasi-BPS category to be

(3.20) $$ \begin{align} \mathbb{T}(d)_w^{\mathrm{{red}}}:=\mathbb{W}(\mathscr{P}(d)^{\mathrm{{red}}})^{\mathrm{{int}}}_{w\tau_d} \subset D^b(\mathscr{P}(d)^{\mathrm{{red}}}). \end{align} $$

There is a description similar to Lemma 3.2 using the embedding $\mathscr {P}(d)^{\mathrm {{red}}} \hookrightarrow \mathscr {Y}(d)$ . Denote by $\mathfrak {gl}(d)_{\mathrm {{nil}}} \subset \mathfrak {gl}(d)_0$ the subset of nilpotent elements. The categorical support lemma in [Reference PădurariuPTc] is the following:

Lemma 3.9. [Reference PădurariuPTc, Corollary 5.5]

For coprime $(d, w)\in \mathbb {N}\times \mathbb {Z}$ , any object $\mathcal {E} \in \mathbb {T}(d)_w^{\mathrm {{red}}}$ satisfies:

$$ \begin{align*} \mathrm{Supp}^{\mathrm{{sg}}}(\mathcal{E}) \subset (\mathfrak{gl}(d)^{\oplus 2g} \oplus \mathfrak{gl}(d)_{\mathrm{{nil}}})/GL(d). \end{align*} $$

For $g{\geqslant } 2$ , the derived stack $\mathscr {P}(d)^{\mathrm {{red}}}$ is classical by [Reference Kaledin, Lehn and SorgerKLS06, Proposition 3.6], in particular there is a good moduli space morphism

$$ \begin{align*} \pi_P \colon \mathscr{P}(d)^{\mathrm{{red}}}\to P(d)=\mu^{-1}(0)^{\mathrm{red}}/\!\!/ G(d). \end{align*} $$

It follows that the Hom-space between any two objects in $D^b(\mathscr {P}(d)^{\mathrm {{red}}})$ is a module over $\mathcal {O}_{P(d)}$ . The categorical support lemma is the main ingredient in the proof of the following:

Proposition 3.10. [Reference PădurariuPTc, Proposition 5.9]

For coprime $(d, w)\in \mathbb {N}\times \mathbb {Z}$ and objects $\mathcal {E}_i \in \mathbb {T}(d)_w^{\mathrm {{red}}}$ for $i=1, 2$ , the $\mathcal {O}_{P(d)}$ -module

$$\begin{align*}\bigoplus_{i\in \mathbb{Z}}\operatorname{Hom}^i_{\mathscr{P}(d)^{\mathrm{red}}}(\mathcal{E}_1, \mathcal{E}_2)\end{align*}$$

is finitely generated. In particular, we have $\operatorname {Hom}^i_{\mathscr {P}(d)^{\mathrm {red}}}(\mathcal {E}_1, \mathcal {E}_2)=0$ for $\lvert i \rvert \gg 0$ .

3.7 Relative Serre functor on reduced quasi-BPS categories

We continue the discussion from the previous subsection. We have that $\mathbb {T}:=\mathbb {T}(d)^{\mathrm {{red}}}_w$ is a subcategory of $D^b(\mathscr {P}(d)^{\mathrm {red}})$ , which is a module over $\mathrm {Perf}(\mathscr {P}(d)^{\mathrm {{red}}})$ . Thus there is an associated internal homomorphism, see Subsection 2.6:

$$ \begin{align*} \mathcal{H}om_{\mathbb{T}}(\mathcal{E}_1, \mathcal{E}_2) \in D_{\mathrm{{qc}}}(\mathscr{P}(d)^{\mathrm{{red}}}) \end{align*} $$

for $\mathcal {E}_1, \mathcal {E}_2 \in \mathbb {T}$ . Proposition 3.10 implies that $\pi _{\ast } \mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2)$ is an object of $D^b(P(d))$ .

Theorem 3.11. [Reference PădurariuPTc, Theorem 5.10]

For coprime $(d, w)\in \mathbb {N}\times \mathbb {Z}$ and $\mathcal {E}_1, \mathcal {E}_2 \in \mathbb {T}$ , there is an isomorphism:

(3.21) $$ \begin{align} \operatorname{Hom}_{P(d)}(\pi_{\ast}\mathcal{H}om_{\mathbb{T}}(\mathcal{E}_1, \mathcal{E}_2), \mathcal{O}_{P(d)}) \cong \operatorname{Hom}_{\mathbb{T}}(\mathcal{E}_2, \mathcal{E}_1). \end{align} $$

For $\mathcal {E}_1=\mathcal {E}_2=\mathcal {E}$ , the identity $\operatorname {id} \colon \mathcal {E} \to \mathcal {E}$ corresponds, under (3.21), to the morphism

$$ \begin{align*} \mathrm{tr}_{\mathcal{E}} \colon \pi_{\ast}\mathcal{H}om(\mathcal{E}, \mathcal{E}) \to \mathcal{O}_{P(d)}. \end{align*} $$

From the construction in [Reference PădurariuPTc], the above morphism coincides with the trace map determined by $(GL(d), \mathfrak {gl}(d)^{\oplus 2g}, \mathfrak {gl}(d)_0, \mu _0)$ , see Subsection 7.2 for the construction of the trace map, especially (7.7).

4.1 Generalities on K3 surfaces

Let S be a smooth projective K3 surface, that is, $K_S$ is trivial and $H^1(\mathcal {O}_S)=0$ . Let $K(S)$ be the Grothendieck group of S. Denote by $\chi (-, -)$ the Euler pairing

$$ \begin{align*} \chi(E, F)=\sum_{j}(-1)^j \mathrm{ext}^j(E, F). \end{align*} $$

Let $N(S)$ be the numerical Grothendieck group:

$$ \begin{align*} N(S):=K(S)/\equiv, \end{align*} $$

where $E_1 \equiv E_2$ in $K(S)$ if $\chi (E_1, F)=\chi (E_2, F)$ for any $F \in K(S)$ . Below for $F\in D^b(S)$ , we denote by $[F] \in N(S)$ its numerical class in $N(S)$ . There is an isomorphism by taking the Mukai vector:

(4.1) $$ \begin{align} v(-)=\operatorname{ch}(-)\sqrt{\mathrm{td}}_S \colon N(S) \stackrel{\cong}{\to} \mathbb{Z} \oplus \mathrm{NS}(S) \oplus \mathbb{Z}. \end{align} $$

Write a vector $v\in N(S)$ as $v=(r, \beta , \chi )\in \mathbb {Z}\oplus \mathrm {NS}(S)\oplus \mathbb {Z}$ via the above isomorphism. There is a symmetric bilinear pairing on $N(S)$ defined by $\langle E_1, E_2 \rangle =-\chi (E_1, E_2)$ . Under the isomorphism (4.1), we have

$$ \begin{align*} \langle E_1, E_2 \rangle=\beta_1 \beta_2-r_1 \chi_2-r_2 \chi_1, \end{align*} $$

where $v(E_i)=(r_i, \beta _i, \chi _i)$ .

We say $v\in N(S)$ is primitive if it cannot be written as $v=dv_0$ for an integer $d{\geqslant } 2$ and $v_0\in N(S)$ . Let $v\in N(S)$ and $w\in \mathbb {Z}$ . Write $v=dv_0$ for $d\in \mathbb {Z}$ and $v_0$ primitive. We define $\gcd (v, w):=\gcd (d,w)$ . Below we identify $N(S)$ with $\mathbb {Z} \oplus \mathrm {NS}(S) \oplus \mathbb {Z}$ via the isomorphism (4.1), and write an element $v \in N(S)$ as $v=(r, \beta , \chi )$ .

4.2 Bridgeland stability conditions on K3 surfaces

For a K3 surface S, we denote by

$$ \begin{align*} \mathrm{Stab}(S) \end{align*} $$

(the main connected component of) the space of Bridgeland stability conditions [Reference BridgelandBri07, Reference BridgelandBri08] on $D^b(S)$ . A point $\sigma \in \mathrm {Stab}(S)$ consists of a pair

$$ \begin{align*} \sigma=(Z, \mathcal{A}), \ Z \colon N(S) \to \mathbb{C}, \ \mathcal{A} \subset D^b(S), \end{align*} $$

where Z is a group homomorphism (called central charge) and $\mathcal {A}$ is the heart of a bounded t-structure satisfying some axioms, see [Reference BridgelandBri07]. One of the axioms is the following positivity property

$$ \begin{align*} Z(E) \in \{ z \in \mathbb{C} : \mathrm{Im}(z)>0 \mbox{ or } z \in \mathbb{R}_{<0}\} \end{align*} $$

for any $0\neq E \in \mathcal {A}$ . An object $E \in \mathcal {A}$ is called Z-(semi)stable if for any subobject $0\neq F \subsetneq E$ we have $\arg Z(F)<({\leqslant }) \arg Z(E)$ in $(0, \pi ]$ . An object $E \in D^b(X)$ is called $\sigma $ -(semi)stable if $E[a] \in \mathcal {A}$ is Z-semistable for some $a \in \mathbb {Z}$ .

For each $B+iH \in \mathrm {NS}(S)_{\mathbb {C}}$ such that H is ample with $H^2>2$ , there is an associated stability condition

$$ \begin{align*} \sigma_{B, H}=(Z_{B, H}, \mathcal{A}_{B, H}) \in \mathrm{Stab}(S), \end{align*} $$

where $\mathcal {A}_{B, H} \subset D^b(S)$ is the heart of a bounded t-structure obtained by a tilting of $\mathrm {Coh}(S)$ and $Z_{B, H}$ is given by

$$ \begin{align*} Z_{B, H}(E)=-\int_S e^{-B-iH} v(E) \in \mathbb{C}. \end{align*} $$

We refer to [Reference BridgelandBri08, Section 6] for the construction of the above stability conditions. A stability condition $\sigma _{B, mH}$ for $m\gg 0$ is said to be in a neighborhood of the large volume limit. Recall the following proposition about semistable objects at the large volume limit:

4.3 Moduli stacks of semistable objects on K3 surfaces

For each $\sigma =(Z, \mathcal {A}) \in \mathrm {Stab}(S)$ and $v \in N(S)$ , we denote by

$$ \begin{align*} \mathfrak{M}_S^{\sigma}(v) \end{align*} $$

the derived moduli stack of $\sigma $ -semistable objects $F \in \mathcal {A} \cup \mathcal {A}[1]$ with numerical class v. We denote by $\mathbb {F}$ the universal object

$$ \begin{align*} \mathbb{F} \in D^b(S \times \mathfrak{M}_S^{\sigma}(v)). \end{align*} $$

We also consider the reduced version of the stack $\mathfrak {M}_S^{\sigma }(v)$ . Let $v=(r, \beta , \chi )$ . Let $\mathcal {P}ic^{\beta }(S)$ be the derived moduli stack of line bundles on S with first Chern class $\beta $ . Then $\mathcal {P}ic^{\beta }(S)=\operatorname {Spec}\mathbb {C}[\varepsilon ]/\mathbb {C}^{\ast }$ , where $\varepsilon $ is of degree $-1$ . We consider the determinant morphism

$$ \begin{align*} \det \colon \mathfrak{M}_S^{\sigma}(v) \to \mathcal{P}ic^{\beta}(S)=\operatorname{Spec}\mathbb{C}[\varepsilon]/\mathbb{C}^{\ast}. \end{align*} $$

Define the reduced stack:

(4.2) $$ \begin{align} \mathfrak{M}_S^{\sigma}(v)^{\mathrm{{red}}}:= \mathfrak{M}_S^{\sigma}(v) \times_{\mathcal{P}ic^{\beta}(S)} B\mathbb{C}^{\ast}. \end{align} $$

The obstruction space of the reduced stack $\mathfrak {M}_S^{\sigma }(v)^{\mathrm {{red}}}$ at $F\in \mathcal {A}\cup \mathcal {A}[1]$ is the kernel of the trace map:

$$ \begin{align*} \operatorname{Ext}_S^2(F, F)_0:=\operatorname{Ker}\left(\operatorname{Ext}_S^2(F, F) \stackrel{\mathrm{tr}}{\twoheadrightarrow} H^2(\mathcal{O}_S)=\mathbb{C} \right). \end{align*} $$

Note that $\mathfrak {M}^\sigma _S(v)^{\mathrm {red}}$ may still not be a classical stack. There are decompositions:

$$ \begin{align*} D^b(\mathfrak{M}_S^{\sigma}(v))=\bigoplus_{w \in \mathbb{Z}}D^b(\mathfrak{M}_S^{\sigma}(v))_w, \ D^b(\mathfrak{M}_S^{\sigma}(v)^{\mathrm{{red}}})=\bigoplus_{w \in \mathbb{Z}}D^b(\mathfrak{M}_S^{\sigma}(v)^{\mathrm{{red}}})_w, \end{align*} $$

where each summand contains complexes F of weight w with respect to the scaling automorphisms $\mathbb {C}^{\ast } \subset \mathrm {Aut}(F)$ , see [Reference TodaTod24a, Subsection 3.2.4].

We denote by $\mathcal {M}_S^{\sigma }(v)$ the classical truncation of $\mathfrak {M}_S^{\sigma }(v)$ . It admits a good moduli space (cf. [Reference AlperAlp13], [Reference Alper, Halpern-Leistner and HeinlothAHLH, Example 7.26]):

(4.3) $$ \begin{align} \pi \colon \mathcal{M}_S^{\sigma}(v) \to M_S^{\sigma}(v), \end{align} $$

where $M_S^{\sigma }(v)$ is a proper algebraic space. A closed point $y \in M_S^{\sigma }(v)$ corresponds to a $\sigma $ -polystable object

(4.4) $$ \begin{align} F=\bigoplus_{i=1}^m V^{(i)} \otimes F^{(i)}, \end{align} $$

where $F^{(1)}, \ldots , F^{(m)}$ are mutually nonisomorphic $\sigma $ -stable objects such that $\arg Z(F^{(i)})=\arg Z(F)$ , and $V^{(i)}$ is a finite-dimensional vector space with dimension $d^{(i)}$ for $1{\leqslant } i{\leqslant } m$ .

Let $G_y:=\mathrm {Aut}(F)=\prod _{i=1}^m GL(V^{(i)})$ and let $\widehat {\operatorname {Ext}}_S^1(F, F)$ be the formal fiber at the origin of the morphism

$$ \begin{align*} \operatorname{Ext}_S^1(F, F) \to \operatorname{Ext}_S^1(F, F)/\!\!/ G_y. \end{align*} $$

By the formality of the dg-algebra $\mathrm {RHom}(F, F)$ , see [Reference DavisonDavc, Corollary 4.9], there are equivalences

(4.5) $$ \begin{align} \widehat{\mathfrak{M}}_S^{\sigma}(v)_y \simeq \widehat{\kappa}^{-1}(0)/G_y, \ \widehat{\mathfrak{M}}_S^{\sigma}(v)_y^{\mathrm{{red}}} \simeq \widehat{\kappa}_0^{-1}(0)/G_y, \end{align} $$

where $\kappa $ , $\kappa _0$ are the maps

$$ \begin{align*} \kappa\colon \operatorname{Ext}_S^1(F, F) \to \operatorname{Ext}_S^2(F, F), \ \kappa_0 \colon \operatorname{Ext}_S^1(F, F) \to \operatorname{Ext}_S^2(F, F)_0 \end{align*} $$

given by $x \mapsto [x, x]$ , and $\widehat {\kappa }$ , $\widehat {\kappa }_0$ are their restrictions to $\widehat {\operatorname {Ext}}_S^1(F, F)$ .

There is a wall-chamber structure on $\mathrm {Stab}(S)$ such that $\mathcal {M}_S^{\sigma }(v)$ is constant if $\sigma $ lies in a chamber, but may change when $\sigma $ crosses a wall. Locally, a wall is defined by the equation

$$ \begin{align*} \frac{Z(v_1)}{Z(v_2)} \in \mathbb{R}_{>0}, \ v=v_1+v_2 \end{align*} $$

such that $v_1$ and $v_2$ are not proportional, see [Reference BridgelandBri08, Proposition 9.3].

A stability condition $\sigma \in \mathrm {Stab}(S)$ is generic if $\sigma $ is not on a wall. If $\sigma $ is generic, then for a polystable object (4.4) each numerical class $[F^{(i)}]$ of $F^{(i)}$ is proportional to v. Let $v=dv_0$ for a primitive $v_0$ . Then we have

$$ \begin{align*}[F^{(i)}]=r^{(i)}v_0, \ d^{(1)}r^{(1)}+\cdots+d^{(m)}r^{(m)}=d. \end{align*} $$

The good moduli space $M_S^{\sigma }(v)$ has a stratification indexed by data $(d^{(i)}, r^{(i)})_{i=1}^m$ , and the deepest stratum corresponds to $m=1$ , $\dim V^{(1)}=d$ and $d^{(1)}=1$ .

Let $v=dv_0$ for a primitive $v_0$ with $\langle v_0, v_0\rangle =2g-2$ , and $Q^{\circ , d}=Q_{2g}$ be the quiver with one vertex and $2g$ -loops with relation $\mathscr {I}$ as in Subsection 3.3. Recall the stacks:

$$ \begin{align*} \mathscr{P}(d)=\mu^{-1}(0)/GL(d), \ \mathscr{P}(d)^{\mathrm{{red}}}=\mu_0^{-1}(0)/GL(d) \end{align*} $$

where $\mu \colon \mathfrak {gl}(d)^{\oplus 2g} \to \mathfrak {gl}(d)$ and $\mu _0 \colon \mathfrak {gl}(d)^{\oplus 2g} \to \mathfrak {gl}(d)_0$ are moment maps.

Lemma 4.3. For any closed point $y \in M_S^{\sigma }(v)$ , there is a point $p \in P(d)$ which is sufficiently close to $0\in P(d)$ such that we have equivalences

(4.6) $$ \begin{align} \widehat{\mathfrak{M}}_S^{\sigma}(v)_y \simeq \widehat{\mathscr{P}}(d)_p, \ \widehat{\mathfrak{M}}_S^{\sigma}(v)_y^{\mathrm{{red}}} \simeq \widehat{\mathscr{P}}(d)_p^{\mathrm{{red}}}. \end{align} $$

If y lies in the deepest stratum, we can take $p=0$ .

Proof. Let y correspond to a direct sum (4.4) such that $[F^{(i)}]=r^{(i)} v_0$ , and let $R^{(i)}$ be a simple $(Q^{\circ , d}, \mathscr {I})$ -representation with dimension $r^{(i)}$ . Such $R^{(i)}$ exists by a straightforward dimension count argument, for example see the proof of [Reference PădurariuPTc, Lemma 5.7 (i)]. Let

$$ \begin{align*} R=\bigoplus_{i=1}^m V^{(i)} \otimes R^{(i)} \end{align*} $$

be a semisimple $(Q^{\circ , d}, \mathscr {I})$ -representation and let $p\in P(d)$ be the corresponding point. Note that

$$ \begin{align*} \chi(R^{(i)}, R^{(j)})=\chi(F^{(i)}, F^{(j)})=r^{(i)}r^{(j)}(2-2g). \end{align*} $$

By the CY2 property of $(Q^{\circ , d}, \mathscr {I})$ -representations (cf. see [Reference KellerKel11], [Reference DavisonDavc, Proposition 7.1]), and the fact that $\hom (R^{(i)}, R^{(j)})=\hom (F^{(i)}, F^{(j)})=\delta _{ij}$ , we have an isomorphism

$$ \begin{align*} \operatorname{Ext}^{\ast}(R^{(i)}, R^{(j)})\cong \operatorname{Ext}^{\ast}(F^{(i)}, F^{(j)}). \end{align*} $$

Then by the formality of polystable objects CY2 categories [Reference DavisonDavc, Corollary 4.9], there is an isomorphism of dg-algebras $\mathrm {RHom}(R, R) \cong \mathrm {RHom}(F, F)$ . Therefore we have equivalences (4.6), see Remark 3.6.

There is an action of $\mathbb {C}^{\ast }$ on the moduli of $Q^{\circ , d}$ -representations which scales the linear maps corresponding to each edge of $Q^{\circ , d}$ , which induces an action on $P(d)$ . The above $\mathbb {C}^{\ast }$ -action preserves the type of the semisimplification, and any point $p \in P(d)$ satisfies $\lim _{t\to 0} (t \cdot p)=0$ . Therefore we can take p to be sufficiently close to $0$ . By the above construction, we can take $p=0$ if y lies in the deepest stratum.

Combining Lemma 4.3 with [Reference Kaledin, Lehn and SorgerKLS06], we have the following:

4.4 Quasi-BPS categories for K3 surfaces

Let $v\in N(S)$ and $w\in \mathbb {Z}$ . Take $a \in K(S)_{\mathbb {R}}$ such that $\chi (a\otimes v)=w \in \mathbb {Z}$ . We define the $\mathbb {R}$ -line bundle $\delta $ on $\mathfrak {M}_S^{\sigma }(v)$ to be

(4.7) $$ \begin{align} \delta=\det p_{\mathfrak{M}_{\ast}}(a \boxtimes \mathbb{F}), \end{align} $$

where $p_{\mathfrak {M}} \colon S \times \mathfrak {M}_S^{\sigma }(v) \to \mathfrak {M}_S^{\sigma }(v)$ , $p_{S} \colon S \times \mathfrak {M}_S^{\sigma }(v) \to S$ are the projections, and $a\boxtimes \mathbb {F}:=p_S^* a\otimes \mathbb {F}$ . Note that the object $p_{\mathfrak {M}\ast }(A \boxtimes \mathbb {F})$ is a perfect complex on $\mathfrak {M}_S^{\sigma }(v)$ for any $A \in D^b(S)$ , so the $\mathbb {R}$ -line bundle (4.7) is well-defined. The pull-back of $\delta $ to $\mathfrak {M}_S^{\sigma }(v)^{\mathrm {{red}}}$ is also denoted by $\delta $ . We define the (nonreduced or reduced) quasi-BPS categories to be the following intrinsic window categories from Definition 2.6:

(4.8) $$ \begin{align} &\mathbb{T}_S^{\sigma}(v)_ \delta:=\mathbb{W}(\mathfrak{M}_S^{\sigma}(v))^{\mathrm{{int}}}_{\delta}\subset D^b(\mathfrak{M}_S^{\sigma}(v))_w, \\ \notag &\mathbb{T}_S^{\sigma}(v)_ \delta^{\mathrm{{red}}}:=\mathbb{W}(\mathfrak{M}_S^{\sigma}(v)^{\mathrm{{red}}})^{\mathrm{{int}}}_{\delta}\subset D^b(\mathfrak{M}_S^{\sigma}(v)^{\mathrm{{red}}})_w. \end{align} $$

Remark 4.5. For each $y\in M_S^{\sigma }(v)$ , let $\widehat {\mathfrak {M}}_S^{\sigma }(v)_y$ be the formal fiber at y and $\delta _y$ the pull-back of $\delta $ to it. The quasi-BPS category for the formal fiber is defined in a similar way:

$$ \begin{align*}\mathbb{T}^{\sigma}_{S, y}(v)_{\delta_y} :=\mathbb{W}(\widehat{\mathfrak{M}}_S^{\sigma}(v)_y)_{\delta_y}^{\mathrm{{int}}} \subset D^b(\widehat{\mathfrak{M}}_S^{\sigma}(v)_y). \end{align*} $$

By the definition of $\mathbb {T}_S^{\sigma }(v)_{\delta }$ , an object $\mathcal {E} \in D^b(\mathfrak {M}_S^{\sigma }(v))$ is an object in $\mathbb {T}_S^{\sigma }(v)_{\delta }$ if and only if its restriction to any formal fiber is an object in $\mathbb {T}_{S, y}^{\sigma }(v)_{\delta _y}$ . There is an analogous statement for the reduced version.

Proof. Let $b \in K(S)_{\mathbb {R}}$ such that $\chi (b \otimes v)=0$ and set $a'=a+b$ . Let $\delta '$ be the $\mathbb {R}$ -line bundle defined as in (4.7) for $a'$ . By Remark 4.5, it is enough to show that $\delta _y=\delta ^{\prime }_y$ for any closed point $y \in M_S^{\sigma }(v)$ . Let y be a point which corresponds to the polystable object F as in (4.4). By the decomposition (4.4), we have $\mathrm {Aut}(F)=\prod _{i=1}^m GL(V^{(i)})$ and $\delta _y$ is the character of $\mathrm {Aut}(F)$ given by

(4.9) $$ \begin{align} \delta_y= \det\left(\sum_{i=1}^m V^{(i)} \otimes \chi(a \otimes F^{(i)})\right) =\bigotimes_{i=1}^m (\det V^{(i)})^{\chi(a\otimes F^{(i)})}. \end{align} $$

As $\sigma $ is generic, the numerical class of $F^{(i)}$ is proportional to v. Therefore $\chi (b \otimes v)=0$ implies $\chi (b \otimes F^{(i)})=0$ for $1{\leqslant } i{\leqslant } m$ , hence $\delta _y=\delta ^{\prime }_y$ .

By the above lemma, the following definition makes sense.

Definition 4.7. For $v \in N(S)$ , let $\sigma \in \mathrm {Stab}(S)$ be generic. For $w \in \mathbb {Z}$ , define

$$ \begin{align*} \mathbb{T}_S^{\sigma}(v)_{w} := \mathbb{T}_S^{\sigma}(v)_\delta, \ \mathbb{T}_S^{\sigma}(v)_{w}^{\mathrm{{red}}} := \mathbb{T}_S^{\sigma}(v)_\delta^{\mathrm{{red}}}. \end{align*} $$

Here $\delta $ is defined as in (4.7) for any $a \in K(S)_{\mathbb {R}}$ such that $\chi (a \otimes v)=w$ .

The first main result of this section is the following wall-crossing equivalence of quasi-BPS categories.

Theorem 4.8. Let $\sigma _1, \sigma _2 \in \mathrm {Stab}(S)$ be generic stability conditions. Then there exist equivalences

(4.10) $$ \begin{align} \mathbb{T}_S^{\sigma_1}(v)_{w} \stackrel{\sim}{\to} \mathbb{T}_S^{\sigma_2}(v)_{w}, \ \mathbb{T}_S^{\sigma_1}(v)_{w}^{\mathrm{{red}}}\stackrel{\sim}{\to} \mathbb{T}_S^{\sigma_2}(v)_{w}^{\mathrm{{red}}}. \end{align} $$

Proof. We only prove the first equivalence, the second one follows by the same argument. We reduce the proof of the equivalence to a local statement as in Theorem 3.3.

Consider a stability $\sigma =(Z, \mathcal {A}) \in \mathrm {Stab}(S)$ lying on a wall and consider stability conditions $\sigma _{\pm }=(Z_{\pm }, \mathcal {A}_{\pm }) \in \mathrm {Stab}(S)$ lying in adjacent chambers. Let $b \in K(S)_{\mathbb {R}}$ be an element satisfying $\chi (b \otimes v)=0$ and let $\delta ' \in \mathrm {Pic}(\mathfrak {M}_S^{\sigma }(v))_{\mathbb {R}}$ be defined as in (4.7) using b. Let $\delta \in \mathrm {Pic}(\mathfrak {M}_S^{\sigma }(v))_{\mathbb {R}}$ be as in Definition 4.7, and set $\delta "=\delta +\delta '$ . It is enough to show that there exists b as above such that the restriction functors for the open immersions $ \mathfrak {M}^{\sigma _{\pm }}_S(v) \subset \mathfrak {M}^{\sigma }_S(v)$ restrict to the equivalence

(4.11) $$ \begin{align} \mathbb{T}_S^{\sigma}(v)_{\delta"} \stackrel{\sim}{\to} \mathbb{T}_S^{\sigma_{\pm}}(v)_{\delta"}. \end{align} $$

The open substacks $\mathfrak {M}_S^{\sigma _{\pm }}(v) \subset \mathfrak {M}_S^{\sigma }(v)$ are semistable loci with respect to line bundle $\ell _{\pm }$ on $\mathfrak {M}_S^{\sigma }(v)$ and they are parts of $\Theta $ -stratifications, see [Reference Halpern-LeistnerHLa, Proposition 4.4.5]. The line bundles $\ell _{\pm }$ are constructed as follows. We may assume that $Z(v)=Z_{\pm }(v)=\sqrt {-1}$ , and write $Z_{\pm }(-)=\chi (\omega _{\pm }\otimes -)$ for $\omega _{\pm } \in K(S)_{\mathbb {C}}$ . Then set $b_{\pm } \in K(S)_{\mathbb {R}}$ to be the real parts of $\omega _{\pm }$ , which satisfy $\chi (b_{\pm } \otimes v)=0$ . The line bundles $\ell _{\pm }$ are defined by $\ell _{\pm }=\det p_{\mathfrak {M}\ast }(b_{\pm } \boxtimes \mathbb {F})$ , see [Reference Halpern-LeistnerHLb, Theorem 6.4.11]. Then we set $b=\varepsilon _{+} b_{+} +\varepsilon _{-} b_{-}$ for general elements $0< \varepsilon _{\pm } \ll 1$ .

Since $\mathfrak {M}_S^{\sigma }(v)$ is 0-shifted symplectic, from Theorem 2.3 and Remark 2.5 (see also [Reference Halpern-LeistnerHLa, Theorem 3.3.1]), there exist subcategories $\mathbb {W}(\mathfrak {M}_S^{\sigma }(v))^{\ell _{\pm }}_{m_{\bullet \pm }} \subset D^b(\mathfrak {M}_S^{\sigma }(v))$ which induce equivalences:

(4.12) $$ \begin{align} \mathbb{W}(\mathfrak{M}_S^{\sigma}(v))^{\ell_{\pm}}_{m_{\bullet\pm}} \stackrel{\sim}{\to} D^b(\mathfrak{M}_S^{\sigma_{\pm}}(v)). \end{align} $$

Moreover, there exist choices of $m_{\bullet \pm }$ such that $\mathbb {T}_S^{\sigma }(v)_{\delta "} \subset \mathbb {W}(\mathfrak {M}_S^{\sigma }(v))^{\ell _{\pm }}_{m_{\bullet \pm }}$ , see [Reference Halpern-LeistnerHLa, Lemma 4.3.10] or [Reference TodaTod, Proposition 6.15] for a choice of $m_{\bullet }$ . Therefore, by Remark 4.5, it is enough to show that, for each closed point $y\in M_S^{\sigma }(v)$ , we have the equivalences

(4.13) $$ \begin{align} \mathbb{T}_{S, y}^{\sigma}(v)_{\delta_y"} \stackrel{\sim}{\to} \mathbb{T}_{S, y}^{\sigma_{\pm}}(v)_{\delta_y"}. \end{align} $$

Here, on the right-hand side we consider the intrinsic window subcategories for the formal fibers of the morphisms $\mathcal {M}_S^{\sigma _{\pm }}(v) \subset \mathcal {M}_S^{\sigma }(v) \to M_S^{\sigma }(v)$ at y.

Let y correspond to the polystable object (4.4) and set $\boldsymbol {d}=(\dim V^{(i)})_{i=1}^m$ . Let $(Q^{\circ , d}_y, \mathscr {I}_y)$ be the Ext-quiver at y with relation $\mathscr {I}_y$ , see Remark 4.2. The quiver with relation $(Q^{\circ , d}_y, \mathscr {I}_y)$ is the double of some quiver $Q_y^{\circ }$ . Let $\mathscr {P}_y(\boldsymbol {d})$ be the derived stack of $(Q^{\circ , d}_y, \mathscr {I}_y)$ -representations with dimension vector $\boldsymbol {d}$ , see (3.3), and $P_y(\boldsymbol {d})$ the good moduli space of its classical truncation. By the equivalence (4.5), there is an equivalence

(4.14) $$ \begin{align} \widehat{\mathfrak{M}}_S^{\sigma}(v)_y \simeq \widehat{\mathscr{P}}_y(\boldsymbol{d}). \end{align} $$

Here the right-hand side is the formal fiber of $\mathscr {P}(\boldsymbol {d})$ at $0 \in P(\boldsymbol {d})$ . The line bundles $\ell _{\pm }$ restricted to $\widehat {\mathfrak {M}}_S^{\sigma }(v)$ correspond to generic elements $\ell _{\pm } \in M(\boldsymbol {d})_{\mathbb {R}}^W$ , where $M(\boldsymbol {d})_{\mathbb {R}}$ is the character lattice of the maximal torus of $G_y:=\mathrm {Aut}(y)=\prod _{i=1}^m GL(V^{(i)})$ . Moreover the $\sigma _{\pm }$ -semistable loci in the left-hand side of (4.14) correspond to $\ell _{\pm }$ -semistable $(Q^{\circ , d}_y, \mathscr {I}_y)$ -representations. Therefore the equivalences (4.13) follow from the formal fiber version of the equivalences

$$\begin{align*}\mathbb{T}(\boldsymbol{d})_{\delta_y"} \stackrel{\sim}{\to} \mathbb{T}^{l_{\pm}}(\boldsymbol{d})_{\delta_y"}\end{align*}$$

in Theorem 3.3, whose proof is identical to loc. cit.

By Lemma 4.6 and Theorem 4.8, the following definition makes sense:

Definition 4.9. For $v \in N(S)$ and $w \in \mathbb {Z}$ , define

$$ \begin{align*} \mathbb{T}_S(v)_{w}:= \mathbb{T}_S^{\sigma}(v)_{w}, \ \mathbb{T}_S(v)_{w}^{\mathrm{{red}}}:= \mathbb{T}_S^{\sigma}(v)_{w}^{\mathrm{{red}}}, \end{align*} $$

where $\sigma \in \mathrm {Stab}(S)$ is a generic stability condition.

Remark 4.11. Suppose that $g{\geqslant } 2$ and take a generic $\sigma \in \mathrm {Stab}(S)$ . Then we have $\mathfrak {M}_S^{\sigma }(v)^{\mathrm {{red}}}=\mathcal {M}_S^{\sigma }(v)$ . Let $\mathcal {M}_S^{\sigma \text {-st}}(v) \subset \mathcal {M}_S^{\sigma }(v)$ be the open substack of $\sigma $ -stable objects. Then the good moduli space morphism $\mathcal {M}_S^{\sigma \text {-st}}(v) \to M_S^{\sigma \text {-st}}(v)$ is a $\mathbb {C}^{\ast }$ -gerbe classified by $\alpha \in \mathrm {Br}(M_S^{\sigma \text {-st}}(v))$ which gives the obstruction of the existence of a universal object in $S \times M_S^{\sigma \text {-st}}(v)$ . We then have that

(4.15) $$ \begin{align} \mathbb{T}_S(v)_w^{\mathrm{{red}}}|_{M_S^{\sigma\text{-st}}(v)}=D^b(M_S^{\sigma}(v), \alpha^w), \end{align} $$

where the right hand is the derived category of $\alpha ^w$ -twisted coherent sheaves on $M_S^{\sigma \text {-st}}(v)$ , see [Reference CăldăraruCăl00, Reference LieblichLie07], and the left-hand side is the subcategory of $D^b(\mathcal {M}^{\sigma \text {-st}}(v))$ classically generated by the restriction of objects in $\mathbb {T}_S(v)_w^{\mathrm {{red}}}$ .

If v is primitive, then $M_S^{\sigma \text {-st}}(v)=M_S^{\sigma }(v)$ and it is a nonsingular holomorphic symplectic variety deformation equivalent to the Hilbert scheme of points $S^{[n]}$ , where $n=\langle v, v\rangle /2+1$ . By (4.15), we have

(4.16) $$ \begin{align} \mathbb{T}_S(v)_w^{\mathrm{{red}}}=D^b(M_S^{\sigma}(v), \alpha^w). \end{align} $$

Let $v=dv_0$ for a primitive $v_0$ . We expect that, if $\gcd (d, w)=1$ , the category $\mathbb {T}_S(v)_w^{\mathrm {{red}}}$ is a “noncommutative hyperkähler manifold”, so that it shares several properties with $D^b(M)$ for a smooth projective hyperkähler variety of $K3^{[n]}$ -type for $n=\langle v, v \rangle /2+1$ . More precisely, we may expect the following, which we view as a categorical $\chi $ -independence phenomenon.

4.5 Quasi-BPS categories for Gieseker semistable sheaves

In Definition 4.9, we defined quasi-BPS categories for Bridgeland semistable objects. By applying the categorical wall-crossing equivalence in Theorem 4.8, we can relate the categories in Definition 4.9 with those under Hodge isometries, and with those for moduli stacks of Gieseker semistable sheaves.

Let G be the group of Hodge isometries of the Mukai lattice $H^{\ast }(S, \mathbb {Z})$ , preserving the orientation of the positive definite four-dimensional plane of $H^{\ast }(S, \mathbb {R})$ . Note that it acts on the algebraic part $\mathbb {Z} \oplus \mathrm {NS}(S) \oplus \mathbb {Z}$ . The following is a categorical analogue of derived invariance property of counting invariants for K3 surfaces [Reference TodaTod08, Reference TodaTod12b].

Corollary 4.14. For any $\gamma \in G$ , there is an equivalence

$$ \begin{align*} \mathbb{T}_S(v)_w \simeq \mathbb{T}_S(\gamma v)_w. \end{align*} $$

Proof. Let $\mathrm {Aut}_{\circ }(D^b(S))$ be the group of autoequivalences $\Phi $ of $D^b(S)$ whose action $\Phi _{\ast }$ on the space of stability conditions preserves the component $\mathrm {Stab}(S)$ . It also acts on $H^{\ast }(S, \mathbb {Z})$ , and denote the action by $\Phi _{\ast } \colon H^{\ast }(S, \mathbb {Z}) \to H^{\ast }(S, \mathbb {Z})$ . Then we have the surjective group homomorphism, see [Reference HartmannHar12, Proposition 7.9], [Reference Huybrechts, Macri and StellariHMS09, Corollary 4.10]:

(4.17) $$ \begin{align} \mathrm{Aut}_{\circ}(D^b(S)) \to G, \ \Phi \mapsto \Phi_{\ast} \end{align} $$

For $\Phi \in \mathrm {Aut}_{\circ }(D^b(S))$ , there is an equivalence of derived stacks $\phi \colon \mathfrak {M}_S^{\sigma }(v) \simeq \mathfrak {M}_S^{\Phi _{\ast }\sigma }(\Phi _{\ast }\sigma )$ given by $F \mapsto \Phi (F)$ . The above equivalence induces an equivalence

$$ \begin{align*} \phi_{\ast} \colon \mathbb{T}_S^{\sigma}(v)_{\delta} \simeq \mathbb{T}_S^{\Phi_{\ast}\sigma}(\Phi_{\ast}v)_{\delta'} \end{align*} $$

where $\delta '$ is determined by $a'=\Phi _{\ast }^{-1}a \in K(S)_{\mathbb {R}}$ which satisfies $\chi (a'\otimes v')=w$ for $v'=\Phi _{\ast }v$ . By Theorem 4.8 and the surjectivity of (4.17), we obtain the corollary.

Let H be an ample divisor on S. We denote by $\mathfrak {M}_S^H(v)$ the derived moduli stack of H-Gieseker semistable sheaves on S with Mukai vector v, and by $\mathfrak {M}_S^H(v)^{\mathrm {{red}}}$ its reduced stack. For $a \in K(S)_{\mathbb {R}}$ , we define the $\mathbb {R}$ -line bundle $\delta $ on $\mathfrak {M}_S^H(v)$ , $\mathfrak {M}_S^H(v)^{\mathrm {{red}}}$ similarly to (4.7). Then we define

$$ \begin{align*} &\mathbb{T}_S^H(v)_{\delta}:=\mathbb{W}(\mathfrak{M}_S^H(v))_{\delta}^{\mathrm{{int}}} \subset D^b(\mathfrak{M}_S^H(v)), \\ &\mathbb{T}_S^H(v)_{\delta}^{\mathrm{{red}}}:=\mathbb{W}(\mathfrak{M}_S^H(v)^{\mathrm{{red}}})_{\delta}^{\mathrm{{int}}} \subset D^b(\mathfrak{M}_S^H(v)^{\mathrm{{red}}}). \end{align*} $$

Below we consider H generic with respect to v, so that all Jordan-Hölder factors of objects in $\mathfrak {M}_S^H(v)$ have numerical class proportional to v. The following is a corollary of the wall-crossing equivalence in Theorem 4.8.

Corollary 4.15. For $v \in N(S)_{\mathbb {R}}$ and generic $\sigma \in \mathrm {Stab}(S)$ , there is $\varepsilon \in \{0, 1\}$ and $m\gg 0$ such that, by setting $v'=(-1)^{\varepsilon }v(mH)$ we have equivalences

$$ \begin{align*} \mathbb{T}_S(v)_{\delta} \simeq \mathbb{T}_S^H(v')_{\delta'}, \ \mathbb{T}_S(v)_{\delta}^{\mathrm{{red}}} \simeq \mathbb{T}_S^H(v')_{\delta'}^{\mathrm{{red}}}. \end{align*} $$

Here, $\delta '$ is a line bundle on $\mathfrak {M}_S^H(v)$ determined by $a'=(-1)^{\varepsilon }a(-mH) \in K(S)_{\mathbb {R}}$ with $\chi (a \otimes v)=\chi (a' \otimes v')=w$ . Then:

$$ \begin{align*} \mathbb{T}_S(v)_{w} \simeq \mathbb{T}_S^H(v')_{w}, \ \mathbb{T}_S(v)_{w}^{\mathrm{{red}}} \simeq \mathbb{T}_S^H(v')_{w}^{\mathrm{{red}}}. \end{align*} $$

We next mention the natural periodicity and symmetry of quasi-BPS categories:

Lemma 4.16. Let $m:=\gcd \{\chi (a \otimes v) : a \in K(S)\}$ . We have equivalences

$$ \begin{align*} \mathbb{T}_S(v)_w \simeq \mathbb{T}_S(v)_{w+m}, \ \mathbb{T}_S(v)_w \simeq \mathbb{T}_S(v)_{-w}^{\mathrm{{op}}}. \end{align*} $$

The similar equivalences also hold for $\mathbb {T}_S(v)_w^{\mathrm {{red}}}$ .

Proof. For $a \in K(S)$ , let $\delta $ be the line bundle (4.7). Then tensoring by $\delta $ induces equivalences:

$$ \begin{align*} \mathbb{T}_S^{\sigma}(v)_w \simeq \mathbb{T}^{\sigma}_S(v)_{w+\chi(a\otimes v)}, \ \mathbb{T}^{\sigma}_S(v)_w^{\mathrm{{red}}} \simeq \mathbb{T}^{\sigma}_S(v)_{w+\chi(a\otimes v)}^{\mathrm{{red}}}. \end{align*} $$

Therefore we obtain the first equivalence. The second equivalence is given by the restriction of $\mathcal {H}om(-, \mathcal {O}_{\mathfrak {M}_S^{\sigma }(v)})$ to $\mathbb {T}_S^{\sigma }(v)$ .

4.6 Conjecture 4.13 for $g=0,1$

Write the Mukai vector as $v=dv_0$ , where $d\in \mathbb {Z}_{{\geqslant } 1}$ and for $v_0$ a primitive Mukai vector with $\langle v_0, v_0 \rangle =2g-2$ and $g{\geqslant } 0$ .

The following proposition proves Conjecture 4.13 when $g=0$ .

Proposition 4.17. Suppose that $g=0$ and $\gcd (d, w)=1$ . Then we have

$$ \begin{align*} \mathbb{T}_S(dv_0)_w^{\mathrm{{red}}}=\begin{cases} D^b(\operatorname{Spec} \mathbb{C}), & d=1, \\ 0, & d>1. \end{cases} \end{align*} $$

Proof. By Corollary 4.15, we can assume that $\mathbb {T}_S(v)_w=\mathbb {T}_S^H(v)_{\delta }$ in $D^b(\mathfrak {M}_S^H(v))$ for H an ample divisor on S. It is well known that $\mathfrak {M}_S^H(v)$ consists of a single point $F^{\oplus d}$ for a spherical stable sheaf F, so we have

$$ \begin{align*} \mathfrak{M}_S^H(v)^{\mathrm{{red}}}= \operatorname{Spec} \mathbb{C}[\mathfrak{gl}(d)_0^{\vee}[1]]/GL(d). \end{align*} $$

By the definition of $\mathbb {T}_S(v)_w^{\mathrm {{red}}}$ , it consists of objects such that for the inclusion

$$\begin{align*}j \colon \mathfrak{M}_S^H(v)^{\mathrm{{red}}} \hookrightarrow BGL(d),\end{align*}$$

the object $j_{\ast }\mathcal {E}$ is generated by $\Gamma _{GL(d)}(\chi )$ for a dominant weight $\chi $ such that

$$ \begin{align*} \chi+\rho \in \frac{1}{2} \mathrm{sum}[0, \beta_i-\beta_j]+\frac{w}{d}\sum_{i=1}^d \beta_i, \end{align*} $$

where the Minkowski sum is after all $1{\leqslant } i, j{\leqslant } d$ . By [Reference TodaTod23a, Lemma 3.2], such a weight exists if and only if $d|w$ , and thus only if $d=1$ because $\gcd (d,w)=1$ . Therefore, together with (4.16) in the primitive case, the proposition follows.

We next discuss the case of $g=1$ . Then $v=dv_0$ , where $v_0$ is primitive with $\langle v_0, v_0 \rangle =0$ . For a generic $\sigma $ , set

$$ \begin{align*}S':=M_S^{\sigma}(v_0) \end{align*} $$

which is well known to be a K3 surface [Reference MukaiMuk87, Reference Bayer and MacrìBM14]. We have the good moduli space morphism $\mathcal {M}_S^{\sigma }(v_0) \to S'$ which is a $\mathbb {C}^{\ast }$ -gerbe classified by some $\alpha \in \mathrm {Br}(S')$ . There is an equivalence

(4.18) $$ \begin{align} D^b(S', \alpha) \stackrel{\sim}{\to} D^b(S) \end{align} $$

given by the Fourier-Mukai transform with kernel the universal $(1\boxtimes \alpha )$ -twisted sheaf on $S \times S'$ , see [Reference CăldăraruC02]. There is also an isomorphism given by the direct sum map

(4.19) $$ \begin{align} \mathrm{Sym}^d(S') \stackrel{\cong}{\to} M_S^{\sigma}(dv_0). \end{align} $$

Let $\mathcal {M}_S^{\sigma }(v_0, \ldots , v_0)$ be the classical moduli stack of filtrations of semistable objects on S:

$$ \begin{align*} 0=F_0 \subset F_1 \subset \cdots \subset F_d \end{align*} $$

such that $F_i/F_{i-1}$ is a $\sigma $ -semistable object with numerical class $v_0$ . We define $\mathscr {Z}_S$ and $\widetilde {\mathcal {M}}_S^{\sigma }(v_0)$ by the following diagram, where the two squares are Cartesian in the classical sense:

(4.20)

Let $T(d)=(\mathbb {C}^{\ast })^{\times d}$ . The map $\widetilde {\mathcal {M}}_S^{\sigma }(v_0) \to S'$ is a $T(d)$ -gerbe, so we have the decomposition into $T(d)$ -weights

(4.21) $$ \begin{align} D^b(\widetilde{\mathcal{M}}_S^{\sigma}(v_0)) =\bigoplus_{(w_1, \ldots, w_d)\in \mathbb{Z}^d} D^b(\widetilde{\mathcal{M}}_S^{\sigma}(v_0))_{(w_1, \ldots, w_d)} \end{align} $$

where the summand corresponding to $(w_1, \ldots , w_d)$ is equivalent to $D^b(S', \alpha ^{w_1+\cdots +w_d})$ . For $1{\leqslant } i{\leqslant } d$ , define $m_i$ by the formula

(4.22) $$ \begin{align} m_i :=\left\lceil \frac{wi}{d} \right\rceil -\left\lceil \frac{w(i-1)}{d} \right\rceil +\delta_{id} -\delta_{i1} \in \mathbb{Z}. \end{align} $$

Define the functor

$$ \begin{align*} \Phi_{d,w}\colon D^b(S', \alpha^w)\to D^b(\mathfrak{M}_{S}^{\sigma}(v)^{\mathrm{{red}}})_{w},\ \mathcal{F}\mapsto p_{S*}\left(q_S^{\ast}\circ i_{(m_1, \ldots, m_d)}\mathcal{F}\right), \end{align*} $$

where $i_{(m_1, \ldots , m_d)}$ is the inclusion of $D^b(S', \alpha ^w)$ into the weight $(m_1, \ldots , m_d)$ -part of (4.21). When $v_0=[\mathcal {O}_x]$ for a point $x \in S$ , it is proved in [Reference Pădurariu and TodaPTa, Proposition 4.7] that the image of the functor $\Phi _{d, w}$ lies in $\mathbb {T}_S^{\sigma }(v)_w$ . We now state a stronger form of Conjecture 4.13 for $g=1$ .