1 Overview
The tropicalization process transforms algebraic varieties into piecewise polyhedral objects. While losing part of the geometry, some of the invariants, such as dimension and degree, of the original variety can still be computed from its tropicalization. For the complement of a hyperplane arrangement, Zharkov shows that the tropical cohomology of the tropicalization computes the usual cohomology of the variety [Reference Zharkov49]. Moreover, Hacking relates the top-weight mixed Hodge structure of a variety to the homology of its tropicalization [Reference Hacking18]. We are interested in determining for which varieties the tropicalization remembers the cohomology. Part of our motivation to study this question comes from the work of Deligne [Reference Deligne12] in which he gives a Hodge-theoretic characterization of maximal degenerations of complex algebraic varieties. We are more specifically interested in determining how tropicalization is related to maximal degenerations, as this question is intimately related to open problems on large-scale limits of families of complex algebraic varieties; see Section 8.4 for a discussion.
We introduce the relevant concepts and notation before stating our results. Let N be a lattice of rank n, M the dual lattice, and
${\mathbf {T}} = \operatorname {\mathrm {Spec}}(\mathbb {C}[M]) \cong (\mathbb {C}^*)^{n}$
the corresponding torus. We let
$N_{\mathbb {R}}$
and
$N_{\mathbb {Q}}$
denote
$N \otimes _{\mathbb {Z}} \mathbb {R}$
and
$N \otimes _{\mathbb {Z}} \mathbb {Q}$
, respectively. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a non-singular subvariety of
${\mathbf {T}}$
and denote by
$X = \mathrm {trop}({\mathbf {X}})$
its tropicalization [Reference Maclagan and Sturmfels35, Reference Mikhalkin and Rau34]. A unimodular fan
$\Sigma $
in
$N_{\mathbb {R}}$
with support X gives rise to a complex toric variety
$\mathbb {CP}_\Sigma $
and a tropical toric variety
$\mathbb {TP}_\Sigma $
. Taking the closures of
${\mathbf {X}}$
and X in
$\mathbb {CP}_\Sigma $
and
$\mathbb {TP}_\Sigma $
, respectively, gives compactifications 
and 
. We note that the compactifications depend on the choice of the fan
$\Sigma $
whose support is
$\mathrm {trop}(X)$
; however, we have chosen not to indicate it in the notation for 
or 
. Here and elsewhere in the paper, we use bold letters for algebraic varieties and regular letters for tropical varieties.
For a complex variety
${\mathbf {Z}}$
, we denote by
$H^{\bullet }({\mathbf {Z}})$
the cohomology ring of
${\mathbf {Z}}$
with coefficients in
$\mathbb {Q}$
. For a tropical variety Z, the k-th tropical cohomology group of Z can be defined as 
, where
$H^{p,q}(Z)$
is the
$(p,q)$
-th tropical cohomology group with
$\mathbb {Q}$
-coefficients introduced in [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20]; see Section 2.6. The tropical cohomology groups together form a ring
$H^{\bullet }(Z) = \bigoplus _{k} H^k(Z)$
, the product structure being induced by the cup product in cohomology [Reference Mikhalkin and Zharkov36, Reference Gross and Shokrieh17]. We note that the tropical cohomology of Z depends only on Z. In particular, if
$Z = \mathrm {trop}({\mathbf {Z}})$
, no information about
${\mathbf {Z}}$
beyond
$\mathrm {trop}({\mathbf {Z}})$
goes into the recipe for computing
$H^{\bullet }(\mathrm {trop}({\mathbf {Z}}))$
.
The question addressed in this paper can be informally stated as follows: Under which conditions can the cohomology of
${\mathbf {X}}$
be related to the tropical cohomology of
$\mathrm {trop}({\mathbf {X}})$
?
Let
${\mathbf {X}}$
,
$\Sigma $
and X be as above, with 
and 
the corresponding compactifications. We define
$\mathrm {\tau }^*$
to be the ring homomorphism 
between the cohomologies of 
and 
, defined by composing the isomorphism 
, proved in [Reference Amini and Piquerez7, Theorem 1.1], with the cycle class map
$A^k(\mathbb {CP}_\Sigma ) \to H^{2k}(\mathbb {CP}_\Sigma )$
and the pullback morphism on cohomology associated to the embedding 
. The groups 
are sent to zero by
$\mathrm {\tau }^*$
for
$p \neq q$
. We refer to Section 3 for more details.
In the following, we will use the map
$\mathrm {\tau }^*$
not only on 
and 
, but also on some of their subvarieties: the toric varieties
$\mathbb {CP}_\Sigma $
and
$\mathbb {TP}_\Sigma $
are endowed with natural stratifications induced by the cone structure of
$\Sigma $
. Each cone
$\sigma \in \Sigma $
gives rise to the torus orbits
${\mathbf {T}}^\sigma $
and
$N^\sigma _{\mathbb {R}}$
in
$\mathbb {CP}_\Sigma $
and
$\mathbb {TP}_\Sigma $
, respectively, with corresponding lattice
$N^\sigma $
. The closures in
$\mathbb {CP}_\Sigma $
and
$\mathbb {TP}_\Sigma $
of these orbits are denoted by
$\mathbb {CP}_\Sigma ^\sigma $
and
$\mathbb {TP}_\Sigma ^\sigma $
, respectively, and are isomorphic to the complex and tropical toric varieties associated to the star fan
$\Sigma ^\sigma $
of
$\sigma $
in
$\Sigma $
. Intersection with these strata induces a stratification of 
and 
. We denote by 
and 
the stratum associated to
$\sigma \in \Sigma $
, and by 
and 
their closures in 
and 
, respectively. The stratum
${\mathbf {X}}^\sigma $
is a closed subvariety of the torus
${\mathbf {T}}^\sigma $
, and its tropicalization coincides with
$X^\sigma $
. Moreover, the star fan
$\Sigma ^\sigma $
is a unimodular fan with support
$X^\sigma $
. We thus obtain a morphism 
that we also denote by
$\tau ^*$
.
Definition 1.1. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a subvariety,
$\Sigma $
a unimodular fan with support
$X = \mathrm {trop}({\mathbf {X}})$
, and 
and 
the corresponding compactifications. We say that
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma $
if the induced maps 
are isomorphisms for all
$\sigma \in \Sigma $
.
We show that the property of being cohomologically tropical for schön subvarieties of tori does not depend on the chosen unimodular fan. Recall from [Reference Tevelev44, Reference Hacking18] that a subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is schön if for some, equivalently for any, unimodular fan
$\Sigma $
of support
$\mathrm {trop}({\mathbf {X}})$
, the open strata
${\mathbf {X}}^\sigma $
,
$\sigma \in \Sigma $
, of the corresponding compactification are all non-singular. It also implies that the compactification 
is non-singular and that 
is a simple normal crossing divisor.
Theorem 4.4. Suppose that the subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is schön and let
$X = \mathrm {trop}({\mathbf {X}})$
be its tropicalization. The following are equivalent.
-
(1) There exists a unimodular fan
$\Sigma $
with support X such that
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma $
. -
(2) For any unimodular fan
$\Sigma $
with support X,
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma $
.
Such a schön subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
will be called cohomologically tropical. For example, the linear subspaces in
$\mathbb {CP}^n$
, restricted to the torus, form a family of cohomologically tropical subvarieties. These very affine varieties are complements of hyperplane arrangements; see Section 8.2. A generalization is given in [Reference Schock40] in which Schock defines quasilinear subvarieties of tori as those having a tropicalization which is quasilinear – in other words,
$\mathcal T\!$
-stable in the language of [Reference Amini and Piquerez5]. He shows that these subvarieties are necessarily schön. It follows from his results that quasilinear subvarieties of tori are cohomologically tropical.
We now introduce a class of subvarieties
${\mathbf {X}} \subseteq {\mathbf {T}}$
with cohomology amenable to a tropical description using the notion of mixed Hodge structures; see Section 2.4.
Definition 1.2. A non-singular subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
of the torus is called wunderschön with respect to a unimodular fan
$\Sigma $
with support
$\mathrm {trop}({\mathbf {X}})$
if all the open strata
${\mathbf {X}}^\sigma $
of the corresponding compactification 
are non-singular and connected, and the mixed Hodge structure on
$H^k({\mathbf {X}}^\sigma )$
is pure of weight
$2k$
for each k.
In particular, a point in the torus is wunderschön. It follows from the preceding discussion that if
${\mathbf {X}}$
is wunderschön, it is schön. Therefore, if 
is the compactification with respect to a unimodular fan
$\Sigma $
, the boundary 
is a strict normal crossing divisor.
We prove that the property of being wunderschön is independent of the fan, and that the cohomology of a wunderschön variety is divisorial in the sense of Section 5.
Theorem 4.5. Suppose that the subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is wunderschön with respect to some unimodular fan. Then
${\mathbf {X}}$
is wunderschön with respect to any unimodular fan with support
$X =\mathrm {trop}({\mathbf {X}})$
.
Theorem 5.1. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a wunderschön subvariety. Let 
be the compactification of
${\mathbf {X}}$
with respect to a unimodular fan
$\Sigma $
with support
$X = \mathrm {trop}({\mathbf {X}})$
. Then the cohomology of 
is divisorial and generated by irreducible components of 
.
A tropical variety X is called a tropical homology manifold if any open subset in X verifies tropical Poincaré duality. For a tropical variety which is the support of a tropical fan, this amounts to the property that for some, equivalently for any, rational unimodular fan
$\Sigma $
of support X, the corresponding open strata
$X^\sigma $
verify tropical Poincaré duality for all
$\sigma \in \Sigma $
. In particular, this implies that, for any unimodular fan
$\Sigma $
of support X, any open subset of the corresponding tropical compactification 
verifies tropical Poincaré duality.
A tropical fanfold X is called Kähler if for some, equivalently for any, quasi-projective unimodular fan
$\Sigma $
with support X, and for any
$\sigma \in \Sigma $
, the Chow ring
$A^{\bullet }(\Sigma ^\sigma )$
verifies the Kähler package – that is, Poincaré duality, hard Lefschetz theorem and Hodge-Riemann bilinear relations. Here, for a unimodular fan
$\Sigma $
, the Chow ring
$A^{\bullet }(\Sigma )$
coincides with the Chow ring of the corresponding toric variety
$\mathbb {CP}_\Sigma $
.
We have the following main theorem on characterization of cohomologically tropical subvarieties of tori.
Theorem 6.1. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a schön subvariety with support
$X = \mathrm {trop}({\mathbf {X}})$
. Then the following statements are equivalent.
-
(1)
${\mathbf {X}}$
is wunderschön and X is a tropical homology manifold. -
(2)
${\mathbf {X}}$
is cohomologically tropical.
Moreover, if any of these statements holds, then X is Kähler.
We deduce the following result from the above theorem.
Theorem 6.2 (Isomorphism of cohomology on open strata)
Suppose that
${\mathbf {X}} \subseteq {\mathbf {T}}$
is schön and cohomologically tropical. Let
$\Sigma $
be any unimodular fan with support
$X =\mathrm {trop}({\mathbf {X}})$
. Then we obtain isomorphisms
for all
$\sigma \in \Sigma $
and all k.
Going beyond cohomologically tropical subvarieties of tori, and following the work of [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20], one can ask the following question. Which families
${\mathfrak {X}}_t$
of complex projective varieties over the complex disk degenerating at
$t=0$
have the property that the tropical cohomology of their tropical limit captures the Hodge numbers of a generic fiber in the family?
In Theorem 7.1, we weaken the condition given in [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20] by showing that it suffices to ask the open components of the central fiber to be cohomologically tropical and schön. By Theorem 6.1, this is equivalent to asking the maximal dimensional strata to be wunderschön and their tropicalizations to be tropical homology manifolds.
More precisely, let
$\pi : {\mathfrak {X}} \to D^*$
be an algebraic family of non-singular algebraic subvarieties in
$\mathbb {CP}^n$
parameterized by a punctured disk
$D^*$
and with fiber
${\mathfrak {X}}_t$
over
$t \in D^*$
. Let
$Z\subseteq \mathbb {TP}^n$
be the extended tropicalization of the family (see Section 7).
By Mumford’s proof of the semistable reduction theorem [Reference Kempf, Knudsen, Mumford and Saint-Donat25], we find a triangulation of Z (possibly after a base change) such that the extended family
$\pi : \overline {\mathfrak {X}} \to D$
is regular and the fiber over zero
${\mathfrak {X}}_0$
is reduced and a simple normal crossing divisor. Note that since the extended family is obtained by taking the closure in a toric degeneration of
$\mathbb {CP}^n$
, each open stratum in
${\mathfrak {X}}_0$
will be naturally embedded in an algebraic torus.
Theorem 7.1. Let
$\pi : {\mathfrak {X}} \to D^*$
be an algebraic family of subvarieties in
$\mathbb {CP}^n$
parameterized by the punctured disk and let
$\pi : \overline {\mathfrak {X}} \to D$
be a semistable extension. If the tropicalization
${Z\subseteq \mathbb {TP}^n}$
is a tropical homology manifold and all the open strata in
${\mathfrak {X}}_0$
are wunderschön, then
$H^{p,q}(Z)$
is isomorphic to the associated graded piece
$W_{2p}/W_{2p-1}$
of the weight filtration in the limiting mixed Hodge structure
$H_{\lim }^{p+q}$
. The odd weight graded pieces in
$H_{\lim }^{p+q}$
are all vanishing.
Moreover, for
$t \in D^*$
, we have
$\dim H^{p,q}({\mathfrak {X}}_t) = \dim H^{p,q}(Z)$
, for all nonnegative integers p and q.
Degenerations appearing in the previous theorem are necessarily maximal in the sense of [Reference Deligne12]; see Section 8.4 for more discussion of this connection.
Theorem 7.2. Notations as above, the family
${\mathfrak {X}} \to D^*$
is maximally degenerate.
We refer to the earlier work of Gross-Siebert [Reference Gross and Siebert16] on integral affine manifolds with singularities and degenerations to arrangements of complete toric varieties, the work of Ruddat [Reference Ruddat38] on non-necessarily maximal degenerations of Calabi-Yau varieties, and [Reference Helm and Katz19, Reference Katz and Stapledon28, Reference Katz and Stapledon29, Reference Ruddat39] for other interesting results connecting the topology of tropicalizations to the Hodge theory of nearby fibers.
2 Preliminaries
2.1 Subvarieties of the torus and tropicalization
We briefly recall the tropicalization of subvarieties of tori. Let N be a lattice of rank n, M its dual,
$N_{\mathbb {R}} = N \otimes \mathbb {R}$
, and
${\mathbf {T}}={\mathbf {T}}_N =\operatorname {\mathrm {Spec}}(\mathbb {C}[M]) \cong (\mathbb {C}^*)^{n}$
. Let
${\mathbf {X}}$
be a d-dimensional subvariety of the torus
${\mathbf {T}}$
, so that
${\mathbf {X}}= {\mathbf {V}}(I)$
for an ideal
$I \subseteq \mathbb {C}[M]$
. The tropicalization of
${\mathbf {X}}$
can be described using initial ideals; see, for example, [Reference Maclagan and Sturmfels35, Section 3.2],
$$\begin{align*}\mathrm{trop}({\mathbf{X}}) = \{ w \in N_{\mathbb{R}} \ | \ \mathrm{in}_w(I) \neq \langle 1 \rangle \}. \end{align*}$$
A d-dimensional fan
$\Sigma $
is weighted if it comes equipped with a weight function
${\mathbf {wt} } : \Sigma _d \to \mathbb {Z}$
, where
$\Sigma _d$
denotes the d-dimensional cones. A tropical fan is a weighted fan which is pure dimensional and which satisfies the balancing condition in tropical geometry [Reference Maclagan and Sturmfels35, Section 3.3]. A fanfold is a subset of
$N_{\mathbb {R}}$
which is the support of a rational fan, and it is a tropical fanfold if it is the support of a tropical fan.
The tropicalization 
is a tropical fanfold, and any fan structure on
$\mathrm {trop}({\mathbf {X}})$
is equipped with a weight function
${\mathbf {wt} }_{{\mathbf {X}}}$
induced by
${\mathbf {X}}$
. If
$\Sigma $
is a rational fan in
$N_{\mathbb {R}}$
of support X and
$\eta $
is some facet of
$\Sigma $
, then for a generic point w in the relative interior of
$\eta $
, the variety
${\mathbf {V}}(\mathrm {in}_w(I))$
is a union of translates of torus orbits. Then,
${\mathbf {wt} }_{{\mathbf {X}}}(\eta )$
is equal to the number of such torus orbit translates counted with multiplicity. This number is invariant for generic choices of points in the relative interior of
$\eta $
. The tropicalization endowed with the weight function
${\mathbf {wt} }_{{\mathbf {X}}}$
satisfies the balancing condition and thus is a tropical fanfold in
$N_{\mathbb {R}}$
[Reference Maclagan and Sturmfels35, Section 3.4].
2.2 Tropical compactifications of complex varieties
We now briefly review the notion of tropical compactifications introduced in [Reference Tevelev44]. Let
$\Sigma $
be a fan in
$N_{\mathbb {R}}$
, and
$\mathbb {CP}_\Sigma $
the associated toric variety. There is a bijection between cones in
$\Sigma $
and torus orbits in
$\mathbb {CP}_\Sigma $
. For each
$\sigma \in \Sigma $
, we denote by
${\mathbf {T}}^\sigma $
the corresponding torus orbit. The closure 
is the disjoint union
$\bigsqcup _{\gamma \supseteq \sigma } {\mathbf {T}}^\gamma $
, for cones
$\gamma \in \Sigma $
containing
$\sigma $
. For
${\mathbf {X}} \subseteq {\mathbf {T}}$
a subvariety and its closure 
in
$\mathbb {CP}_\Sigma $
, we have 
. We denote the stratum 
by
${\mathbf {X}}^\sigma $
, and its closure by 
. Note that 
.
For
$\Sigma $
a unimodular fan with support equal to
$X=\mathrm {trop}({\mathbf {X}})$
, the closure 
of
${\mathbf {X}}$
in
$\mathbb {CP}_\Sigma $
is compact, giving a tropical compactification [Reference Tevelev44, Proposition 2.3]. Moreover, for such a
$\Sigma $
, the compactification 
of
${\mathbf {X}}$
in
$\mathbb {CP}_\Sigma $
is said to be schön if the torus action 
is non-singular and surjective, in which case 
is non-singular, and the boundary 
is a simple normal crossing divisor [Reference Tevelev44, Theorem 1.2]. The compactification 
is schön if and only if
${\mathbf {X}}^\sigma $
is non-singular for each
$\sigma \in \Sigma $
[Reference Hacking18, Lemma 2.7]. If
${\mathbf {X}}$
admits a schön compactification, then any unimodular fan with support equal to X will provide a schön compactification [Reference Luxton and Qu31, Theorem 1.5], and in this case, we will say that
${\mathbf {X}}$
is schön.
Example 2.1. For
$f= \sum _{I\in \Delta (f)} a_I x^I \in \mathbb {C}[x_1^{\pm }, \dots , x_n^{\pm }]$
a Laurent polynomial, it is pointed out in [Reference Tevelev44, p. 1088] that the very affine hypersurface
${\mathbf {X}}=V(f)$
being schön is equivalent to the condition that f is nondegenerate (with respect to its Newton Polytope), a concept studied in [Reference Varchenko46, Reference Varchenko45] and [Reference Kouchnirenko26]. For each face
$\gamma \in \Delta (f)$
of the Newton Polytope of f, one defines
$f_\gamma = \sum _{I\in \gamma } a_I \mathbf {x}^I$
. Then f is nondegenerate if, for all
$\gamma \in \Delta (f)$
, the polynomials
$$\begin{align*}x_1 \frac{\partial f_\gamma}{\partial x_1}, \dots, x_n \frac{\partial f_\gamma}{\partial x_n} \end{align*}$$
share no common zero in
$(\mathbb {C}^*)^n$
. This implies that
${\mathbf {X}}=V(f)$
is schön by, for instance, [Reference Varchenko46, Lemma 10.3].
2.3 Canonical compactifications of tropical varieties
Let
$\Sigma $
be a rational fan in
$N_{\mathbb {R}}$
. The dimension of a cone
$\sigma $
will be denoted by 
, and we denote by
$\Sigma _k$
the set of cones of
$\Sigma $
of dimension k. The unique cone of dimension
$0$
is denoted
${\underline 0}$
. Let
$\gamma , \delta $
be two faces of
$\Sigma $
. We write
$\gamma \preccurlyeq \delta $
if
$\gamma $
is a face of
$\delta $
. For
$\delta \in \Sigma $
a cone, the saturated sublattice parallel to
$\delta $
is denoted
$N_\delta $
, and the quotient lattice
$N/{N_\delta }$
is denoted
$N^\delta $
, with quotient map
$\pi ^\delta : N \to N^\delta $
. By a slight abuse of terminology, we also denote by
$\pi ^\delta $
the map of real vector spaces
$N_{\mathbb {R}} \to N_{\mathbb {R}}^\delta $
. Furthermore, the star at
$\delta $
is the fan
$\Sigma ^\delta $
in
$N^\delta _{\mathbb {R}}$
whose cones are given by
$ \left \lbrace {\pi ^\delta (\sigma ) \mid \delta \preccurlyeq \sigma } \right \rbrace $
.
We briefly review the construction of tropical toric varieties, referring to [Reference Maclagan and Sturmfels35, Chapter 6.2] for a detailed construction. Let
$\mathbb {T} = \mathbb {R} \cup \{+\infty \}$
denote the tropical semi-field. Denote by
$\sigma ^\vee $
the semigroup of element of
$M_{\mathbb {R}}$
which are nonnegative on
$\sigma $
. For each
$\sigma \in \Sigma $
, one defines 
, which can be identified with the set
$\bigsqcup _{\delta \preccurlyeq \sigma } N_{\mathbb {R}}^\delta $
. We equip
$U_\sigma ^{\mathrm {trop}}$
with the subset topology of the product topology on the infinite product
$\mathbb {T}^{\sigma ^\vee \cap M}$
. For
$\sigma $
unimodular,
$U_\sigma ^{\mathrm {trop}}$
is isomorphic to 
. For
$\delta \preccurlyeq \sigma $
, the inclusion identifies
$U_\delta ^{\mathrm {trop}}$
as an open subset of
$U_\sigma ^{\mathrm {trop}}$
. The tropical toric variety
$\mathbb {TP}_{\Sigma }$
associated to
$\Sigma $
is the space given by gluing the
$U_\sigma ^{\mathrm {trop}}$
along common faces, with underlying set
$\bigsqcup _{\sigma \in \Sigma } N_{\mathbb {R}}^\sigma $
.
Let
$\Sigma $
be a fan with support X. The canonical compactification 
of X relative to the fan
$\Sigma $
is the closure of X as a subset of its tropical toric variety
$\mathbb {TP}_\Sigma $
. Furthermore, 
has a cellular structure, which we denote
$\overline {\Sigma }$
; see Section 2.6 and [Reference Amini and Piquerez7, Section 2] for details. For any cone
$\sigma \in \Sigma $
, we denote by
$X^\sigma $
the fanfold associated to
$\Sigma ^\sigma $
. The canonical compactification 
of
$X^\sigma $
is canonically isomorphic to the closure of
$X^\sigma $
when considered as a subset in
$N_{\mathbb {R}}^\sigma \subseteq \mathbb {TP}_\Sigma $
, and we will denote this compactified fanfold by 
, when
$\Sigma $
is understood from the context. Moreover, there is an inclusion of canonical compactifications 
for
$\delta \preccurlyeq \sigma $
.
When
$X = \mathrm {trop}({\mathbf {X}})$
, the tropical canonical compactification 
relative to any fan
$\Sigma $
with support X is the same as the extended tropicalization of the closure 
in the sense of [Reference Kajiwara23, Reference Payne37, Section 3].
2.4 Mixed Hodge structures
Keeping the notation from Section 2.2, let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a non-singular subvariety, and
$\Sigma $
a unimodular fan supported on the tropicalization
$X = \operatorname {\mathrm {Trop}}({\mathbf {X}})$
, so that we obtain a tropical compactification 
of
${\mathbf {X}}$
. Moreover, suppose that the boundary 
is a simple normal crossing divisor. We have that 
.
By [Reference Deligne11, Section 3], the logarithmic de Rham complex 
induces an isomorphism
for each k. Moreover, there is a weight filtration
$W_\bullet $
on the logarithmic de Rham complex, which gives a mixed Hodge structure on
$H^k({\mathbf {X}})$
. This is given by the Deligne weight spectral sequence
which degenerates on the
$E_2$
-page. Below, we display the rows
$E_1^{\bullet ,2k+1}$
and
$E_1^{\bullet , 2k}$
, where the rightmost elements are in position
$(0,2k+1)$
and
$(0,2k)$
, respectively.
All the differentials are sums of Gysin homomorphisms with appropriate signs. Recall that, given a unimodular fan
$\Sigma $
, and a pair of faces
$\sigma ,\delta \in \Sigma $
such that
$\delta $
is a codimension one face of
$\sigma $
, the inclusion map 
induces a restriction map in cohomology 
, with dual map 
. Applying the Poincaré duality for both 
and 
gives a map 
, called the Gysin homomorphism and denoted
$\operatorname {\mathrm {Gys}}_{\sigma \succ \delta }$
.
Since the Deligne spectral sequence degenerates at the
$E_2$
page, the cohomology of the rows
$E_1^{\bullet ,2k+1}$
and
$E_1^{\bullet , 2k}$
yields the following associated graded elements
where 
and
${\operatorname {\mathrm {Gr}}_{l}^W(H^{k})}$
denotes the weight l part of the mixed Hodge structure on
$H^k$
.
Recall that a mixed Hodge structure H is pure of weight n if
$\operatorname {\mathrm {Gr}}_i^W (H)=0$
for
$i\neq n$
. A mixed Hodge structure H is Hodge-Tate if
$\operatorname {\mathrm {Gr}}_k^W(H)$
is of type
$(l,l)$
if
$k=2l$
and
$0$
for k odd; see, for example, [Reference Deligne12, p. 689].
2.5 Wunderschön varieties
We now consider wunderschön varieties
${\mathbf {X}} \subseteq {\mathbf {T}}$
as introduced in Definition 1.2. As we noted previously, wunderschön varieties are schön. In addition, we have the following.
Proposition 2.2. If a non-singular subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is wunderschön with respect to
$\Sigma $
, then the weight function of the tropicalization
${\mathbf {wt} }_{{\mathbf {X}}}$
is equal to one on all top dimensional faces
$\eta $
of
$\Sigma $
.
Proof. The weight
${\mathbf {wt} }_{{\mathbf {X}}}(\eta ) $
is equal to the intersection multiplicity of 
with the toric stratum
$\mathbb {CP}_{\Sigma ^\eta }$
. In other words, it is the number of points in the variety 
counted with multiplicities. Since
${\mathbf {X}}$
is wunderschön, the variety 
must consist of a single point. Hence, for all facets
$\eta $
, we have
${\mathbf {wt} }_{{\mathbf {X}}}(\eta ) = 1$
.
A consequence of the wunderschön property is that, for each
$\sigma \in \Sigma $
, the even rows of the
$E_2 = E_\infty $
-page for
${\mathbf {X}}^\sigma $
, taking a priori the form shown in (1), are in fact zero except in the leftmost position, which implies that
$H^k({\mathbf {X}}^{\sigma }) = \operatorname {\mathrm {Gr}}_{2k}^W(H^{k}({\mathbf {X}}^\sigma ))$
. Moreover, the odd rows of the
$E_1$
-page are all identically zero by the following lemma.
Lemma 2.3. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a wunderschön variety with respect to
$\Sigma $
. Then 
for 
and all
$\sigma \in \Sigma $
.
Proof. The property is true for a wunderschön point. By induction on dimension, we have 
for 
and all cones
$\sigma $
except the central vertex
${\underline 0}$
, so that it remains to prove that 
for 
. For each such k, the
$(2k-1)$
-th row of the
$E_2$
-page of the Deligne spectral sequence is given by 
and all other terms are
$0$
. Since
${\mathbf {X}}$
is wunderschön,
$\operatorname {\mathrm {Gr}}_{2k-1}^W(H^{2k-1}({\mathbf {X}}))=0$
, and so 
.
Since the
$E_2$
-page is the cohomology of the
$E_1$
-page, this proves the following lemma.
Lemma 2.4. For
${\mathbf {X}} \subseteq {\mathbf {T}}$
a wunderschön variety with respect to
$\Sigma $
, and for each cone
$\sigma \in \Sigma $
and each k, we have the following exact sequences
where
$\operatorname {\mathrm {res}}$
denotes the logarithmic residue map and
$\operatorname {\mathrm {Gys}}$
denotes a signed sum of suitable Gysin maps.
Example 2.5 (Wunderschön curves are rational)
We classify wunderschön curves
${{\mathbf {X}}\subseteq {\mathbf {T}}}$
. A tropical compactification 
consists of adding points to
${\mathbf {X}}$
. Points have pure mixed Hodge structure on their cohomology. Thus, for
${\mathbf {X}}$
to be wunderschön with respect to a fan
$\Sigma $
, it is necessary that each stratum
$X^\zeta $
for
$\zeta \in \Sigma _1$
be connected (i.e., consists of a single point). The Deligne weight spectral sequence degenerates on the
$E_2$
page, and is shown in Figure 1 and Figure 2. Note that
$H^2({\mathbf {X}})$
is trivial. Moreover, if
${\mathbf {X}}$
is wunderschön, then 
must be trivial.
$E_1$
-page from Example 2.5.
$E_2$
-page from Example 2.5.
Therefore, a non-singular curve
${\mathbf {X}} \subseteq {\mathbf {T}}$
is wunderschön if and only if the curve 
is isomorphic to
$\mathbb {C}\mathbb {P}^1$
and it meets each toric boundary divisor of
$\mathbb {CP}_{\Sigma }$
in only one point. We conclude that the only wunderschön (open) curves are complements of a finite set of points in a non-singular rational curve.
2.6 Tropical homology and cohomology
We now briefly sketch the theory of tropical homology and cohomology, and refer to [Reference Jell, Rau and Shaw21, Reference Jell, Shaw and Smacka22, Reference Amini and Piquerez4, Reference Amini and Piquerez7, Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20, Reference Aksnes2, Reference Gross and Shokrieh17] for details. We work with
$\mathbb {Q}$
-coefficients.
Let
$\Sigma $
be a rational fan in
$N_{\mathbb {R}}$
with support X. Let 
be the closure of X inside the tropical toric variety
$\mathbb {TP}_\Sigma $
. The closure 
has a cellular structure
$\overline {\Sigma }$
where the cells of
$\overline {\Sigma }$
consist of the closures of the cones in
$\Sigma ^\sigma $
for all
$\sigma \in \Sigma $
. In particular, each face of
$\overline {\Sigma }$
is indexed by a pair of cones
$\sigma , \gamma \in \Sigma $
satisfying
$\gamma \succcurlyeq \sigma $
, and denoted
$C_\gamma ^\sigma \in \overline \Sigma $
. For each face
$C_\gamma ^\sigma \in \overline {\Sigma }$
, the p-th multi-tangent space
$\mathcal {F}_p(C_\gamma ^\sigma )$
(with
$\mathbb {Q}$
-coefficients) is defined as
Moreover, for
$\alpha \preccurlyeq \beta $
two faces of
$\overline {\Sigma }$
, there is a map
$\iota _{\beta \succcurlyeq \alpha } : \mathcal {F}_p(\beta ) \to \mathcal {F}_p(\alpha )$
, which is an inclusion if both faces lie in the same subfan
$\Sigma ^\sigma $
for some
$\sigma $
, or if
$\alpha = C_\eta ^\sigma $
and
$\beta = C_\eta ^{\sigma '}$
with
$\eta \succcurlyeq \sigma \succ \sigma '$
, then
$\iota _{\beta \succcurlyeq \alpha }$
is induced by the projection
$N^{\sigma '} \to N^\sigma $
. Generally, the map
$\iota _{\beta \succcurlyeq \alpha }$
is defined as compositions of such inclusions and projections. Furthermore, by dualizing, we obtain the p-th multi-cotangent spaces
$\mathcal {F}^p(\alpha )$
and reversed morphisms.
By selecting orientations for each of the cones
$\alpha \in \overline {\Sigma }$
, we obtain relative compatibility signs
$\mathrm {sign}(\alpha ,\beta ) \in \left \lbrace {\pm 1} \right \rbrace $
for
$\alpha \prec \beta $
with 
. We may thus use the multi-tangent spaces to define a chain complex
that is, summing over faces
$\alpha $
of dimension q in
$\overline \Sigma $
, with differentials 
defined component-wise as the maps
$\mathrm {sign}(\alpha ,\beta )\iota _{\beta \succ \alpha }$
when
$\alpha \prec \beta $
and 
, and defined to be
$0$
, otherwise. Similarly, by dualizing everything, we obtain a cochain complex
$C^{p,q}(\overline {\Sigma })$
for the multi-cotangent spaces.
The homology groups 
of the complex
$C_{p,\bullet }(\overline {\Sigma })$
are invariants of the canonically compactified support 
of the support X of the fan
$\Sigma $
. Therefore, we define the tropical homology of 
as the homology 
of the complex
$C_{p,\bullet }(\overline {\Sigma })$
. The tropical cohomology of 
is 
.
In fact, tropical homology and cohomology can be defined for any rational polyhedral space. Moreover, there are various equivalent descriptions of tropical (co)homology in terms of cellular, singular and sheaf theoretic terms [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20, Reference Mikhalkin and Zharkov36, Reference Gross and Shokrieh17]. For any rational polyhedral space Z, we set
For example, for a fanfold X, the tropical homology is
$H_{p,q}(X)= \mathcal {F}_p({\underline 0})$
if
$q=0$
and
$0$
otherwise, and the tropical cohomology of X is
$H^{p,q}(X)= \mathcal {F}^p({\underline 0})$
if
$q=0$
and
$0$
otherwise [Reference Jell, Shaw and Smacka22, Proposition 3.11].
If X is a tropical fanfold, the balancing condition implies the existence of a fundamental class 
, which induces a cap product 
for each
$p,q\in \left \lbrace {0,\dots ,d} \right \rbrace $
. When these maps are isomorphisms for all p and q, the variety 
is said to satisfy tropical Poincaré duality.
Definition 2.6. A tropical fanfold X is called a tropical homology manifold if one of the three following equivalent conditions hold:
-
• There exists a unimodular fan
$\Sigma $
with support equal to X such that each of the canonical compactifications satisfies tropical Poincaré duality, for all cones
$\sigma \in \Sigma $
. -
• For any unimodular fan
$\Sigma $
with support equal to X, each of the canonical compactifications satisfies tropical Poincaré duality, for all cones
$\sigma \in \Sigma $
. -
• Any open subset U of X satisfies tropical Poincaré duality – that is, the tropical Poincaré duality induces an isomorphism between the tropical cohomology and the tropical Borel-Moore homology of U (see [Reference Jell, Shaw and Smacka22, Reference Jell, Rau and Shaw21] for details).
satisfies tropical Poincaré duality, for all cones 
satisfies tropical Poincaré duality, for all cones 
A tropical variety Z is called a tropical homology manifold if any open subset U of Z verifies tropical Poincaré duality. here
This definition corresponds to the notion of homological smoothness in [Reference Amini and Piquerez6] and to local tropical Poincaré duality spaces in [Reference Aksnes2]. The equivalence of the three statements for tropical fanfolds is nontrivial and follows from Theorems 1.2 and 1.3 in [Reference Amini and Piquerez6], and Theorem 1.8 of the article [Reference Amini and Piquerez7].
2.7 Chow rings of fans
We now recall some facts about the Chow ring of a fan; see, for instance, [Reference Amini and Piquerez5] for more details.
Let
$\Sigma $
be a unimodular fan in a vector space
$N_{\mathbb {R}}$
. The Chow ring
$A^{\bullet }(\Sigma )$
is the quotient ring
with a variable
$\mathrm {x}_\zeta $
for each ray
$\zeta \in \Sigma _1$
. Here, I is the ideal generated by all monomials
$\mathrm {x}_{\zeta _1}\!\cdots \mathrm {x}_{\zeta _l}$
such that the rays
$\zeta _1,\dots , \zeta _l$
do not form a cone of
$\Sigma $
, and J is the ideal generated by the expressions
$\sum _{\zeta \in \Sigma _1} \langle m, {\mathfrak e}_\zeta \rangle \mathrm {x}_\zeta $
, where
${\mathfrak e}_\zeta \in N$
is the primitive vector of the ray
$\zeta $
and m ranges over elements of the dual lattice M.
For
$\sigma \in \Sigma $
, we define 
, where
$\zeta _1, \dots , \zeta _k$
are the rays of
$\sigma $
. As a vector space,
$A^{\bullet }(\Sigma )$
is generated by
$\mathrm {x}_\sigma $
,
$\sigma \in \Sigma $
. For a pair of cones
$\delta \preccurlyeq \sigma $
, there is a Gysin map 
. This map is defined by mapping
$\mathrm {x}_{\eta '} \in \Sigma ^\sigma $
to
$\mathrm {x}_{\eta '}\mathrm {x}_{\zeta _1}\!\cdots \mathrm {x}_{\zeta _r}$
, where
$\eta '$
is a face of
$\Sigma ^\sigma $
,
$\eta $
is the corresponding face in
$\Sigma ^\delta $
, and
$\zeta _1,\dots , \zeta _r$
are the rays of
$\sigma $
not in
$\delta $
.
Since
$\Sigma $
is unimodular, there is an isomorphism of rings 
from the Chow ring of
$\Sigma $
to the Chow ring of the toric variety
$\mathbb {CP}_\Sigma $
; see, for example, [Reference Brion9, Section 3.1]. Furthermore, the cycle class map
$\mathrm {cyc}_\Sigma : A^{\bullet }(\mathbb {CP}_\Sigma ) \to H^{2\bullet }(\mathbb {CP}_\Sigma )$
gives a graded ring homomorphism to cohomology; see [Reference Fulton14, Corollary 19.2]. Consider a subvariety
${\mathbf {X}}$
of the torus, and assume that the support of
$\Sigma $
is
$\operatorname {\mathrm {Trop}}({\mathbf {X}})$
. Let 
be the corresponding compactification. There is the restriction map of rings 
. Composing all these homomorphisms gives a morphism of rings 
which maps
$\mathrm {x}_\sigma $
to the class of 
.
In the tropical world, there is a similar map. Let X be the support of
$\Sigma $
and let 
be the corresponding compactification. One can consider the composition
mapping
$\mathrm {x}_\sigma $
to the class of 
. By [Reference Amini and Piquerez7, Theorem 1.1], this composition induces an isomorphism of rings 
. We define the inverse map 
by mapping
$(p,q)$
-classes to zero if
$p \neq q$
. Here, by convention,
$A^{k/2}(\Sigma )$
is trivial for odd k. If
$\Sigma $
is a tropical homology manifold,
$\Psi $
is an isomorphism by [Reference Amini and Piquerez7, Theorem 1.3]; that is, 
is trivial for
$p\neq q$
.
Kähler package
We recall the Kähler package for Chow rings of fans; see [Reference Amini and Piquerez5]. Assume
$\Sigma $
is tropical and quasi-projective; that is, there exists a conewise linear function f on
$\Sigma $
which is strictly convex in the following sense. For any
$\sigma \in \Sigma $
, there exists a linear map
$m \in M$
such that
$f-m$
is zero on
$\sigma $
and strictly positive on
$U\smallsetminus \sigma $
for some open neighborhood U of the relative interior of
$\sigma $
. To such an f, one can associate the element 
. These elements coming from strictly convex functions are called ample classes. Since
$\Sigma $
is tropical, the degree map
$\deg : A^d(\Sigma ) \to \mathbb {Q}$
mapping
$\mathrm {x}_\eta $
to
${\mathbf {wt} }(\eta )$
for any facet
$\eta $
of
$\Sigma $
is a well-defined morphism.
The Chow ring
$A^{\bullet }(\Sigma )$
is said to verify the Kähler package if the following holds:
-
• (Poincaré duality) the pairing
is perfect for any k;
$$\begin{align*}\begin{array}{ccl} A^k(\Sigma) \times A^{d-k}(\Sigma) &\to& \mathbb{Q}, \\ a,b\phantom{-k\!} &\mapsto& \deg(ab), \end{array} \end{align*}$$
-
• (Hard Lefschetz theorem) for any ample class L, the multiplication by
$L^{d-2k}$
induces an isomorphism between
$A^k(\Sigma )$
and
$A^{d-k}(\Sigma )$
for all
$k \leq d/2$
; -
• (Hodge-Riemann bilinear relations) for any
$k\leq d/2$
and any ample class L, the bilinear map is positive definite on
$$\begin{align*}\begin{array}{ccl} A^k(\Sigma) \times A^k(\Sigma) &\to& \mathbb{Q}, \\ a, b &\mapsto& (-1)^k\deg(L^{d-2k}ab), \end{array} \end{align*}$$
$\ker (\ \cdot L^{d-2k+1}: A^k(\Sigma ) \to A^{d-k+1}(\Sigma ))$
.
A tropical fanfold X is called Kähler if it is a tropical homology manifold and there exists a quasi-projective unimodular fan of support X such that
$A^{\bullet }(\Sigma ^\sigma )$
verifies the Kähler package for any
$\sigma \in \Sigma $
. In such a case, any quasi-projective unimodular fan
$\Sigma $
on X verifies the previous property (cf. [Reference Amini and Piquerez5, Reference Amini and Piquerez6]).
2.8 Tropical Deligne resolution
Let
$\Sigma $
be a unimodular fan on some tropical homology manifold X. Let
$\delta \preccurlyeq \sigma $
be two faces of
$\Sigma $
. The inclusion 
of canonically compactified fanfolds, both satisfying tropical Poincaré duality, gives a homomorphism 
. Applying the tropical Poincaré duality for both 
and 
, this gives a map 
, called the tropical Gysin homomorphism and denoted
$\operatorname {\mathrm {Gys}}_{\sigma \succcurlyeq \delta }^{\mathrm {trop}}$
.
In [Reference Amini and Piquerez6, Theorem 1.6], it is shown that for a fanfold X which is a tropical homology manifold and a simplicial fan
$\Sigma $
with support X, there are tropical Deligne resolutions, that is, exact sequences for any k,
where the first nonzero morphism is given by integration (that is, by the evaluation of the element
$\alpha \in H^k(X)$
at the canonical multivector of each face
$\sigma \in \Sigma _k$
), and all subsequent maps are given by the tropical Gysin homomorphisms (with appropriate signs [Reference Amini and Piquerez6, Section 3]).
3 The induced morphism on cohomology by tropicalization
The aim of this section is to define a map relating tropical cohomology to classical cohomology, as well as to prove Proposition 3.2, which relates Gysin maps in tropical and classical cohomology.
Definition 3.1. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a subvariety and
$\Sigma $
a unimodular fan with support
$X=\mathrm {trop}({\mathbf {X}})$
, and 
and 
be the compactifications of
${\mathbf {X}}$
and X with respect to
$\Sigma $
. We define
to be the ring homomorphism defined as the composition of the maps 
with 
from Section 2.7.
The map
$\mathrm {\tau }^*$
is the morphism comparing the tropical and classical cohomology in order to define cohomologically tropical varieties in Definition 1.1.
We will now relate the classical and tropical Gysin maps through the map
$\mathrm {\tau }^*$
. This will be useful later for comparing Deligne sequences.
Proposition 3.2. Let
$X = \operatorname {\mathrm {Trop}}({\mathbf {X}})$
the be tropicalization of a subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
,
$\Sigma $
a unimodular fan with support X, with
$\sigma ,\delta \in \Sigma $
such that
$\delta $
is a face of
$\sigma $
of codimension one, giving inclusion maps 
and 
. Then the following diagram commutes:
Proof. Expanding the definition of
$\mathrm {\tau }^*$
, we obtain the following diagram:
The first square is commutative by [Reference Amini and Piquerez4, Remark 3.15], in light of [Reference Amini and Piquerez7, Theorem 1.1]. The commutativity of the second square follows from the functoriality of the cycle class map in light of [Reference Brion9, Section 3.2] and [Reference Fulton14, Section 19.2].
Remark 3.3. Let
${\mathbf {X}} \subseteq {\mathbf {T}}_N$
and
${\mathbf {X}}'\subseteq {\mathbf {T}}_{N'}$
be two non-singular subvarieties of tori associated to two lattices N and
$N'$
, with X and
$X'$
the corresponding tropicalizations, and two unimodular fans
$\Sigma $
and
$\Sigma '$
with supports X and
$X'$
, respectively.
Assume there exists a morphism of lattices
$\phi : N \to N'$
which takes cones of
$\Sigma $
to cones of
$\Sigma '$
such that the induced map 
is surjective. This makes the induced morphism of toric varieties
$f : \mathbb {CP}_\Sigma \to \mathbb {CP}_{\Sigma '}$
proper [Reference Fulton and William15, Section 2.4]. We denote by
$f^{\mathrm {trop}}: \mathbb {TP}_\Sigma \to \mathbb {TP}_{\Sigma '}$
the induced morphism on tropical toric varieties.
Furthermore, suppose that
$f({\mathbf {X}})={\mathbf {X}}'$
. Since 
is compact, we have that 
. This also gives 
for the canonical compactifications of X and
$X'$
with respect to
$\Sigma $
and
$\Sigma '$
. One can then prove the commutativity of the following diagram:
here
Proposition 3.4. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a subvariety of complex dimension d and
$\Sigma $
a unimodular fan with support
$X=\mathrm {trop}({\mathbf {X}})$
, and 
and 
be the compactifications of
${\mathbf {X}}$
and X with respect to
$\Sigma $
. Suppose 
satisfies tropical Poincaré duality and 
is non-singular. Then 
is injective.
Proof. Both maps 
and 
commute with the corresponding degree maps. Now for both tropical and classical cohomology, the fact that the products induce perfect pairings implies that
$\mathrm {\tau }^*$
is injective.
4 Irrelevance of fan
To be schön, wunderschön, cohomologically tropical, Kähler or a tropical homology manifold are all properties of the form ‘there exists a fan
$\Sigma $
such that a specific property holds’ with some restriction on the fan, as unimodularity for instance. Informally, we say that such a property is fan irrelevant if we can replace ‘there exists a unimodular fan’ by ‘for any unimodular fan’ (this is strongly linked with the notion of
$\mathcal T\!$
-stability for properties of tropical fans in [Reference Amini and Piquerez5]). It is already known that to be schön, Kähler or a tropical homology manifold is fan irrelevant. In this section, we prove Theorems 4.4 and 4.5 about the fan irrelevance of being cohomologically tropical and wunderschön. We begin with a lemma.
Lemma 4.1. Suppose a schön subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is cohomologically tropical. Then, the tropicalization
$X =\mathrm {trop}({\mathbf {X}})$
is a tropical homology manifold.
Proof. Let
$\Sigma $
be a unimodular fan whose support is
$\mathrm {trop}({\mathbf {X}})$
. It follows that the cohomology groups 
are all isomorphic to the cohomology groups 
, and so they verify Poincaré duality. We infer that X is a tropical homology manifold.
Let
${\mathbf {X}}$
be a schön subvariety of the torus which is cohomologically tropical. It follows from the previous lemma and the fan irrelevance of being a tropical homology manifold that all the cohomology groups 
are vanishing provided that
$p\neq q$
, for the canonical compactification 
of X with respect to any unimodular fan with support X.
Let
$\Sigma $
be a unimodular fan with support the fanfold X, and let
$\sigma $
be a cone in
$\Sigma $
of dimension at least two. Let
$\Sigma '$
be the barycentric star subdivision of
$\Sigma $
obtained by star subdividing
$\sigma $
; see, for example, [Reference Amini and Piquerez5, Reference Włodarczyk47]. Denote by
$\rho $
the new ray in
$\Sigma '$
. Let 
and 
be the compactifications of X with respect to
$\Sigma $
and
$\Sigma '$
, respectively.
The following theorem provides a description of the Chow ring of
$\Sigma '$
in terms of the Chow rings of
$\Sigma $
and
$\Sigma ^\sigma $
.
Theorem 4.2 (Keel’s lemma)
Let
$\mathfrak {J}$
be the kernel of the surjective map
${\mathrm {i}}^*_{{\underline 0}\preccurlyeq \sigma }: A^{\bullet }(\Sigma )\to A^{\bullet }(\Sigma ^{\sigma })$
and let
There is an isomorphism of Chow groups given by the map
which sends T to
$-\mathrm {x}_\rho $
and which verifies
$$\begin{align*}\forall \zeta \in \Sigma_1, \qquad \chi(\mathrm{x}_\zeta) = \begin{cases} \mathrm{x}_\zeta+\mathrm{x}_\rho & \textrm{if }\zeta \prec \sigma,\\ \mathrm{x}_\zeta & \textrm{otherwise}. \end{cases} \end{align*}$$
In particular, this gives a vector space decomposition of
$A^{\bullet }(\Sigma ')$
as
In addition, if X is the tropicalization of a schön subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
, and 
and 
are compactifications of
${\mathbf {X}}$
with respect to
$\Sigma $
and
$\Sigma '$
, respectively, then we have an isomorphism
and the decomposition
Here, by an abuse of notation, the variable T denotes the image of
$-\mathrm {x}_\rho $
in 
for the induced map 
,
$\mathfrak {J}$
is the kernel of 
, and
$P(T)$
is the image of 
in 
under the map 
.
Decomposition (1), for instance, means that for any 
, we have a natural injective map
The piece
$A^{\bullet }(\Sigma ^\sigma )T^k$
in the above decomposition then denotes the image of the above map. We refer to [Reference Keel24] and [Reference Amini and Piquerez5] for more details and the proof.
Two unimodular fans with the same support are called elementary equivalent if one can be obtained from the other by a barycentric star subdivision. The weak equivalence between unimodular fans with the same support is then defined as the transitive closure of the elementary equivalence relation. We will need the weak factorization theorem, stated as follows.
Theorem 4.3 (Weak factorization theorem [Reference Włodarczyk48, Reference Morelli33])
Two unimodular fans with the same support are always weakly equivalent.
We are now in a position to prove the independence of being cohomologically tropical from the chosen fan for schön varieties.
Theorem 4.4. Suppose that the subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is schön and let
$X = \mathrm {trop}({\mathbf {X}})$
be its tropicalization. The following are equivalent.
-
(1) There exists a unimodular fan
$\Sigma $
with support X such that
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma $
. -
(2) For any unimodular fan
$\Sigma $
with support X,
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma $
.
Proof. Suppose that the subvariety
${\mathbf {X}}$
of the torus
${\mathbf {T}}$
is schön. Let
$X= \mathrm {trop}({\mathbf {X}})$
. Let
$\Sigma $
be a unimodular fan with support X such that
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma $
. Let
$\Sigma '$
be a second unimodular fan with support X. We need to prove that
${\mathbf {X}}$
is cohomologically tropical with respect to
$\Sigma '$
. By the weak factorization theorem, it will be enough to assume that
$\Sigma $
and
$\Sigma '$
are elementary equivalent.
We consider the compactifications 
and 
of
${\mathbf {X}}$
and X with respect to
$\Sigma '$
, and those with respect to
$\Sigma $
by 
and 
.
Consider first the case where
$\Sigma '$
is obtained as a barycentric star subdivision of
$\Sigma $
. Denote by
$\sigma $
the cone of
$\Sigma $
which has been subdivided and by
$\rho $
the new ray of
$\Sigma '$
.
We start by explaining the proof of the isomorphism 
. We use the notation preceding Theorem 4.3. By Keel’s lemma, we get
with
$\mathfrak {J}$
and
$P(T)$
as in Theorem 4.2.
By [Reference Amini and Piquerez7, Theorem 1.1] (see Section 2.7), we have isomorphisms 
and 
for each p. Moreover, since
${\mathbf {X}}$
is cohomologically tropical by Lemma 4.1, all the cohomology groups 
and 
are vanishing for
$p\neq q$
.
The isomorphism 
now follows from the commutativity of the diagram in Remark 3.3, the isomorphisms 
and 
, and the compatibility of the decompositions in Keel’s lemma in the tropical and algebraic settings with respect to these isomorphisms.
Consider now an arbitrary cone
$\delta $
of
$\Sigma '$
and denote by
$\eta $
the smallest cone of
$\Sigma $
which contains
$\delta $
. The star fan
$\Sigma ^{\prime \delta }$
of
$\delta $
in
$\Sigma '$
is isomorphic to a product of two fans
$\Delta \times \Theta $
with
$\Delta $
a unimodular fan living in
$N^\eta _{\mathbb {R}}$
and
$\Theta $
a unimodular fan living in
$N_{\sigma , \mathbb {R}}\big /N_{\delta \cap \sigma ,\mathbb {R}}$
. In the case
$\eta \not \preccurlyeq \sigma $
, the first fan
$\Delta $
coincides with the star fan
$\Sigma ^\eta $
of
$\eta $
in
$\Sigma $
. Otherwise, when
$\eta \preccurlyeq \sigma $
,
$\Delta $
is the fan obtained from
$\Sigma ^\eta $
by subdividing the cone
$\sigma /\eta $
. The other fan
$\Theta $
is
${\underline 0}$
unless
$\delta $
contains the ray
$\rho $
in which case,
$\Theta $
is the fan of the projective space of dimension 
. Similarly, 
admits a decomposition into a product
${\mathbf {Y}} \times {\mathbf {Z}}$
, where 
in the case
$\eta \not \preccurlyeq \sigma $
, and
${\mathbf {Y}}$
is the blow-up of 
in 
in the other case
$\eta \preccurlyeq \sigma $
. And
${\mathbf {Z}}$
is
$\mathbb {CP}^0$
– that is a point – unless
$\delta $
contains
$\rho $
in which case 
.
The isomorphism 
for
$\delta $
can be then obtained from the above description, and by observing that when
$\sigma $
is face of
$\eta $
and
$\Delta $
is the subdivision of
$\eta /\sigma $
in
$\Sigma ^\eta $
, we can apply the argument used in the first treated case above to 
and 
to conclude.
Consider now the case where
$\Sigma $
is obtained as a barycentric star subdivision of
$\Sigma '$
. We only discuss the isomorphism 
; the other isomorphisms 
for
$\delta \in \Sigma '$
can be obtained by using the preceding discussion. The cohomology of 
appears as a summand of the cohomology of 
according to the decomposition in Keel’s lemma. Similarly, the cohomology of 
is a summand of the cohomology of 
. Using the compatibility of the decompositions in the Keel’s lemma, the isomorphism 
induces an isomorphism 
between the two summands.
Theorem 4.5. Suppose that the subvariety
${\mathbf {X}} \subseteq {\mathbf {T}}$
is wunderschön with respect to some unimodular fan. Then
${\mathbf {X}}$
is wunderschön with respect to any unimodular fan with support
$X =\mathrm {trop}({\mathbf {X}})$
.
Proof. The proof of this theorem is similar to the proof given above for Theorem 4.4. We omit the details.
5 Divisorial cohomology
In this section, we prove Theorem 5.1, which states that the cohomology of a wunderschön variety is divisorial.
The cohomology of a non-singular algebraic variety
${\mathbf {Z}}$
is divisorial if there is a surjective ring homomorphism
$\mathbb {Q}[ \mathrm {x}_1, \dots , \mathrm {x}_s ] \to H^{\bullet }({\mathbf {Z}})$
such that the image of each
$\mathrm {x}_i$
is
$[{\mathbf {D}}_i] \in H^2({\mathbf {Z}})$
, the Poincaré dual of some divisor
${\mathbf {D}}_i$
of
${\mathbf {Z}}$
. Similarly, the Chow ring
$A^{\bullet }({\mathbf {Z}})$
is divisorial if there is a surjective ring homomorphism
$\mathbb {Q}[ \mathrm {x}_1, \dots , \mathrm {x}_s ] \to A^{\bullet }({\mathbf {Z}})$
such that the image of each
$\mathrm {x}_i$
is the class of a divisor
${\mathbf {D}}_i$
of
${\mathbf {Z}}$
. In this case, we also say that the (Chow) cohomology of
${\mathbf {Z}}$
is generated by the divisors
${\mathbf {D}}_1, \dots , {\mathbf {D}}_s$
. Notice that if
${\mathbf {Z}}$
is projective and its cohomology is divisorial, then all its cohomology is generated by algebraic cycles and the Hodge structure on the cohomology is Hodge-Tate.
The Chow ring of any non-singular complex toric variety is divisorial and generated by the toric boundary divisors; see [Reference Brion9, Section 3.1] and Section 2.7. It follows, using our previous notations, that if the the map 
is a surjection, then the cohomology of 
is divisorial and generated by the irreducible components of 
.
Theorem 5.1. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a wunderschön subvariety. Let 
be the compactification of
${\mathbf {X}}$
with respect to a unimodular fan
$\Sigma $
with support
$X = \mathrm {trop}({\mathbf {X}})$
. Then the cohomology of 
is divisorial and generated by irreducible components of 
.
Proof. We proceed by induction on the dimension of
${\mathbf {X}}$
. If
${\mathbf {X}}$
is a point, then this is trivial. Notice also that if
${\mathbf {X}}$
is a wunderschön curve, then 
must be
$\mathbb {CP}^1$
, and hence, the cohomology is divisorial as
$H^{\bullet }(\mathbb {CP}^1)\cong \mathbb {Q}[\mathrm {x}]/\langle \mathrm {x}^2 \rangle $
.
We have the following commutative diagram:
where
$\rho +\zeta $
is the cone generated by the rays
$\rho $
and
$\zeta $
, the
$f_\rho $
are surjective ring homomorphisms which send
$\mathrm {x}_\zeta $
to 
, and f maps
$\mathrm {x}_\zeta $
to 
. Since
${\mathbf {X}}$
is wunderschön, the maps
from the Deligne weight spectral sequence are all surjections for
$k \geq 0$
, and we deduce that f is surjective. Therefore, the cohomology of 
is divisorial and is generated by the components of 
.
6 Proof of the main theorem
We now turn to proving Theorem 6.1.
Theorem 6.1. Let
${\mathbf {X}} \subseteq {\mathbf {T}}$
be a schön subvariety with support
$X =\mathrm {trop}({\mathbf {X}})$
. Then the following statements are equivalent.
-
(1)
${\mathbf {X}}$
is wunderschön, and X is a tropical homology manifold, -
(2)
${\mathbf {X}}$
is cohomologically tropical.
Moreover, if any of these statements holds, then X is Kähler.
Proof. We begin by assuming that
${\mathbf {X}}$
is wunderschön and that X is a tropical homology manifold, and prove that
${\mathbf {X}}$
is cohomologically tropical. We must show that the maps 
are isomorphisms for all
$\sigma \in \Sigma $
. Notice that
${\mathbf {X}}$
is non-singular since it is wunderschön.
If
${\mathbf {X}}$
is of dimension
$0$
and wunderschön, it consists of a single point. Therefore, its tropicalization is a point of weight
$1$
thus
${\mathbf {X}}$
is cohomologically tropical. We proceed by induction on the dimension of
${\mathbf {X}}$
. Therefore, we can assume that each of the
${\mathbf {X}}^\sigma $
is cohomologically tropical for all cones
$\sigma \in \Sigma $
not equal to the origin.
Since
${\mathbf {X}}$
is schön, let 
be the simple normal crossing divisor of the compactification. The Deligne weight spectral sequence for the tropical compactification 
of
${\mathbf {X}}$
abuts in the associated graded objects of the weight filtration of the cohomology of
$H^k({\mathbf {X}})$
. Since
${\mathbf {X}}$
is wunderschön, the
$E_1$
-page of Deligne spectral sequence extends to exact rows by Lemma 2.4, with the morphisms being sums of Gysin maps. In the tropical setting, since X is a tropical homology manifold, there are tropical Deligne resolutions Section 2.8, where the maps are sums of tropical Gysin maps.
Now by induction, 
is an isomorphism, and moreover, the appropriate commutative diagrams using the classical and tropical Gysin maps commute by Proposition 3.2. We may therefore identify the two exact sequences. Applying the five lemma in the cases
$k\geq 2$
, exactness gives us isomorphisms
$H^k(X)\to H^k({\mathbf {X}})$
and 
. For
$k=0$
, since
${\mathbf {X}}$
is assumed to be connected, there is an isomorphism 
, and it merely remains to show the claim for
$k=1$
.
We consider the following commutative diagram:
By induction, the middle vertical arrow is an isomorphism, and we wish to show that the rightmost vertical arrow is an isomorphism. By a diagram chase, exactness of the lower row implies that this arrow is surjective. The injectivity follows from Proposition 3.4. Therefore, the map 
is an isomorphism. Together with our induction assumption on the maps
$\mathrm {\tau }^*$
, this proves that
${\mathbf {X}}$
is cohomologically tropical.
Now assume that
${\mathbf {X}}$
is cohomologically tropical. By Lemma 4.1, we know that X is a tropical homology manifold. It remains to show that
${\mathbf {X}}$
is wunderschön. We again proceed by induction on dimension as the case for points is trivial. We equip
${\mathbf {X}}$
with the tropical compactification 
given by
$\Sigma $
, such that all open
${\mathbf {X}}^\sigma $
are wunderschön by induction, for
$\sigma $
different from the central vertex of
$\Sigma $
. We have 
by hypothesis, and 
; thus, 
is connected and so is
${\mathbf {X}}$
. It remains to show that the mixed Hodge structure on
$H^k({\mathbf {X}})$
is pure of weight
$2k$
for each k. This follows from comparing the Deligne weight spectral sequence and tropical Deligne resolution by Proposition 3.2, using that all the maps
$\mathrm {\tau }^*$
are isomorphisms. Hence, 
is wunderschön.
Finally, we prove that if
${\mathbf {X}}$
is cohomologically tropical, then X is Kähler. By Lemma 4.1, we know that X is a tropical homology manifold. There exists a unimodular fan
$\Sigma $
with support X such that
$\Sigma $
is quasi-projective. It follows that the Chow rings
$A^{\bullet /2}(\Sigma ^\sigma )$
,
$\sigma \in \Sigma $
, are isomorphic to 
. Moreover, since
$\Sigma $
is quasi-projective, and
${\mathbf {X}}$
is schön, 
is a non-singular projective variety, and so its cohomology verifies the Kähler package. We conclude that X is Kähler.
Theorem 6.2 (Isomorphism of cohomology on open strata)
Suppose that
${\mathbf {X}} \subseteq {\mathbf {T}}$
is schön and cohomologically tropical. Let
$\Sigma $
be any unimodular fan with support
$X =\mathrm {trop}({\mathbf {X}})$
. Then we obtain isomorphisms
for all
$\sigma \in \Sigma $
and all k.
Proof. It suffices to prove the statement for
${\mathbf {X}}$
, since if
${\mathbf {X}}$
is cohomologically tropical, so are all strata
${\mathbf {X}}^{\sigma }$
. It follows from the proof of Theorem 6.1 that if
${\mathbf {X}}$
is cohomologically tropical, then
${\mathbf {X}}$
is wunderschön, and hence, the
$2k$
-th row of the
$1$
st page of the Deligne weight spectral sequence provides a resolution of
$H^k({\mathbf {X}})$
for all k. Moreover, the maps 
are isomorphisms for all strata, and they commute with the tropical and complex Gysin maps. Therefore, we obtain an isomorphism of the resolutions which induces isomorphisms
$\mathrm {\tau }^* : H^k(X^\sigma ) \to H^k ({\mathbf {X}}^\sigma )$
for all
$\sigma \in \Sigma $
and all k.
7 Globalization
We discuss a natural extension of the main theorem of [Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20]. We follow the setting of that work. Let
$\pi : {\mathfrak {X}} \to D^*$
be an algebraic family of non-singular complex algebraic varieties in
$\mathbb {CP}^n$
over the punctured disk
$D^*$
. By base change, the family
${\mathfrak {X}}$
gives rise to a subvariety
${\mathfrak {X}}_\eta $
of
$\mathbb {P}^n_{\mathbb K}$
where
$\mathbb K = \mathbb {C}((t))$
is the field of formal Laurent series with complex coefficients. (Here,
$\eta $
refers to the generic fiber.) We denote by
$Z\subseteq \mathbb {TP}^n$
the tropicalization of the family, defined as the extended tropicalization of the subvariety
${\mathfrak {X}}_\eta \hookrightarrow \mathbb {P}^n_{\mathbb K}$
.
We suppose Z admits a unimodular triangulation. This is always possible after a base change of the form
$D^* \to D^*$
,
$z\mapsto z^k$
, for
$k\in \mathbb {Z}_+$
. Using the triangulation, we construct a degeneration of
$\mathbb {CP}^n$
to an arrangement of toric varieties, and taking the closure of the family
${\mathfrak {X}}$
inside this toric degeneration leads to a family
$\overline {{\mathfrak {X}}}$
extended over the full punctured disk D. By Mumford’s proof of the semistable reduction theorem, we can always find a triangulation, after a suitable base change, such that the extended family is regular and the fiber over zero is reduced and simple normal crossing. This is known as a semistable extension of the family
$\pi : {\mathfrak {X}} \to D^*$
.
Denote by
${\mathfrak {X}}_0$
the fiber at zero of the extended family. Note that since the extended family is obtained by taking the closure of the family in a toric degeneration of
$\mathbb {CP}^n$
, each open stratum in
${\mathfrak {X}}_0$
will be naturally embedded in an algebraic torus. For
$ t \in D^*$
, denote by
${\mathfrak {X}}_t$
the fiber of
$\pi $
over t.
Theorem 7.1. Let
$\pi : {\mathfrak {X}} \to D^*$
be an algebraic family of subvarieties in
$\mathbb {CP}^n$
parameterized by the punctured disk and let
$\pi : \overline {\mathfrak {X}} \to D$
be a semistable extension. If the tropicalization
$Z\subseteq \mathbb {TP}^n$
is a tropical homology manifold and all the open strata in
${\mathfrak {X}}_0$
are wunderschön, then
$H^{p,q}(Z)$
is isomorphic to the associated graded piece
$W_{2p}/W_{2p-1}$
of the weight filtration in the limiting mixed Hodge structure
$H_{\lim }^{p+q}$
. The odd weight graded pieces in
$H_{\lim }^{p+q}$
are all vanishing.
Moreover, for
$t \in D^*$
, we have
$\dim H^{p,q}({\mathfrak {X}}_t) = \dim H^{p,q}(Z)$
, for all nonnegative integers p and q.
Proof. Denote by
$\Delta $
the dual complex of
${\mathfrak {X}}_0$
. By construction,
$\Delta $
naturally lives in the tropicalization Z. For each simplex
$\delta \in \Delta $
, denote by
${\mathbf {Z}}^\delta $
the corresponding (open) stratum in
${\mathfrak {X}}_0$
and denote by
$Z^\delta $
the corresponding unimodular star fan in Z. Let 
and 
be the corresponding compactifications. Note that 
is also the closure of
${\mathbf {Z}}^\delta $
in
${\mathfrak {X}}_0$
.
Since Z is a tropical homology manifold, the local fanfolds appearing in the tropical variety Z are all tropical homology manifolds. Moreover, for each
$\delta \in \Delta $
, the Chow ring of the star fan
$Z^\delta $
is also the tropical cohomology ring of 
. Since
${\mathbf {Z}}^\delta $
is cohomologically tropical, these rings are as well isomorphic to the cohomology ring of 
.
The Steenbrink spectral sequence [Reference Steenbrink43] for
${\mathfrak {X}}$
degenerates at page two and gives the weight b part of the limit mixed Hodge structure
$H^d_{\lim }$
in each cohomological degree d. The first page of the Steenbrink sequence is given by
where
The differential
$\mathrm {d} : {\textbf {ST}}_1^{a,b} \to {\textbf {ST}}^{a+1, b}$
is given by a signed sum of Gysin and restriction maps. (The correct formulation involves taking the Tate twist 
in
${\textbf {ST}}^{a,b,s}$
so that all the terms have weight b; we omit them for sake of simplification.)
Denoting the weight filtration in the limit mixed Hodge structure
$H^d_{\lim }$
by
$W_{\bullet }H^d_{\lim }$
, and setting
we have
$$\begin{align*}\mathrm{gr}^w_b H^{d}_{\lim} \simeq H^{d-b}({\textbf{ST}}^{\bullet,b}). \end{align*}$$
The wunderschön assumption implies that the cohomology of
${\mathbf {Z}}^\delta $
is of Hodge-Tate type for all
$\delta \in \Delta $
. By the Deligne weight spectral sequence, this implies that the same holds for the cohomology of 
. Therefore, the odd cohomology groups of 
all vanish. Since
$a-s$
is even, we infer that for b odd, all the terms
${\textbf {ST}}_1^{a,b,s}$
vanish. In other words, the odd weight graded pieces of the limit mixed Hodge structures all vanish. Furthermore, the limit mixed Hodge structure will be of Hodge-Tate type.
We set
$b=2p$
and
$d=p+q$
, so that in weight
$2p$
and degree
$p+q$
, we get
$$\begin{align*}\mathrm{gr}^w_{2p} H^{p+q}_{\lim} \simeq H^{q-p}({\textbf{ST}}^{\bullet,2p}). \end{align*}$$
The tropical Steenbrink spectral sequence for the tropical homological manifold Z is given in a similar way by
with differential
$\mathrm {d} : {\textsf {ST}}_1^{a,2p} \to {\textsf {ST}}^{a+1, 2p}$
a signed sum of Gysin and restriction maps. The Steenbrink-Tropical comparison theorem proved in [Reference Amini and Piquerez4, Reference Itenberg, Katzarkov, Mikhalkin and Zharkov20] implies that
$$\begin{align*}H^{p,q}(Z) = H^{q-p}({\textsf{ST}}^{\bullet,2p}). \end{align*}$$
By assumption on the degeneration, we get isomorphisms between the cohomology groups 
. Comparing the two Steenbrink sequences
${\textbf {ST}}^{\bullet , 2p}$
and
${\textsf {ST}}^{\bullet , 2p}$
, we deduce that
$\mathrm {gr}^w_{2p} H^{p+q}_{\lim } \simeq H^{p,q}(Z)$
. This gives the first part.
To prove the second statement, note that since the limit mixed Hodge structure on
$H^d_{\lim }$
is Hodge-Tate, the p-th part of the limit Hodge filtration
$F^{\bullet }_{\lim }$
on
$H^{d}_{\lim }$
contributes only in the graded piece
$\mathrm {gr}^w_{2p} H^{p+q}_{\lim }$
. This implies that the
$H^{p,q}$
piece in the Hodge decomposition of
$H^{p+q}({\mathfrak {X}}_t)$
has dimension equal to that of
$\mathrm {gr}^w_{2p}H^{p+q}(Z)$
, finishing the proof of the theorem.
From the proof, we deduce the following statement that shows that degenerations appearing in the above theorem are all maximal.
Theorem 7.2. Notations as in Theorem 7.1, the family
${\mathfrak {X}} \to D^*$
is maximally degenerate.
Proof. Each closed stratum in
${\mathfrak {X}}_0$
has a cohomology of Hodge-Tate type. Steenbrink spectral sequence shows that the limit mixed Hodge structure is Hodge-Tate.
We discuss maximal degenerations further in Section 8.4.
8 Discussions
8.1 Examples
In this section, we give various examples of varieties verifying some but not all conditions of the main Theorem 6.1. These examples tend to demonstrate that the main theorem cannot be weakened.
8.1.1 A wunderschön variety which is not cohomologically tropical
Take
$N = \mathbb {Z}^2$
. Let
${\mathbf {X}} \subseteq {\mathbf {T}}_N$
be the conic given by the equation
$a + bz_1 + cz_2 + dz_1z_2 = 0$
for generic complex coefficients
$a,b,c$
and d. The variety
${\mathbf {X}}$
is
$\mathbb {CP}^1$
with four points removed. This is a wunderschön variety: looking at the compactification 
,
${\mathbf {X}}$
is non-singular, and the intersections with torus orbits are the points hence non-singular, so that
${\mathbf {X}}$
is schön. Moreover, each of the points removed is trivially wunderschön. Finally, the Deligne weight spectral sequence shows that
${\mathbf {X}}$
has pure Hodge structure. However, the tropicalization X of
${\mathbf {X}}$
is the union of the axes in
$\mathbb {R}^2$
, which is not uniquely balanced (i.e.,
$\dim H^2(X)=2>1$
). This means that X is not a tropical homology manifold (see [Reference Aksnes2, Theorem 4.8]). Moreover, computing the cohomology groups of 
, we obtain 
, 
and 
, which differs from the cohomology groups of the sphere 
.
8.1.2 A schön variety with pure strata, whose tropicalization is a tropical homology manifold but which is not cohomologically tropical
Let
${\mathbf {X}}$
be a generic conic in
$(\mathbb {C}^*)^2$
. The variety
${\mathbf {X}}$
is
$\mathbb {CP}^1$
with six points removed. Its tropicalization is the usual tropical line equipped with weights equal to
$2$
on all edges – hence again a tropical homology manifold by [Reference Aksnes2, Theorem 4.8]. The variety
${\mathbf {X}}$
is schön since it is non-singular, and each one of the three strata consists of two distinct points; hence, it is non-singular. The mixed Hodge structure on
${\mathbf {X}}$
is pure, as the Deligne weight spectral sequence shows that 
. However, it is not wunderschön since its strata are not connected. The map 
is an isomorphism: it maps the class of a point in 
to twice the class of a point in 
. Nevertheless,
${\mathbf {X}}$
is not cohomologically tropical since, for any ray
$\zeta $
of X, 
but 
.
8.1.3 A schön variety which is not pure nor cohomologically tropical and whose tropicalization is a tropical homology manifold
Consider the punctured elliptic curve
${\mathbf {X}}$
in
$(\mathbb {C}^*)^2$
of equation
$az_1^2+bz_2+cz_1z_2^2=0$
for generic complex coefficents
$a,b$
and c. Topologically, it is a torus punctured in three points. The tropicalization is the unimodular tropical line of weight one with rays generated by
$(2,1),(-1,1)$
and
$(-1,-2)$
, which is a tropical homology manifold. The variety
${\mathbf {X}}$
is non-singular and connected, and each of the three strata at infinity of its compactification is a point hence non-singular and connected. Hence,
${\mathbf {X}}$
is schön. The cohomology group 
is nontrivial of dimension
$2$
. However, 
is trivial. Hence,
${\mathbf {X}}$
is not cohomologically tropical. This is because
${\mathbf {X}}$
is not wunderschön. More precisely,
$H^1({\mathbf {X}})$
is not pure of weight 2. Indeed, by the Deligne weight spectral sequence, 
.
8.1.4 A non-schön variety which is cohomologically tropical
Once again, N is of dimension 2. Let
${\mathbf {X}} \subseteq {\mathbf {T}}_N$
be given by the equation
$(z_1-a)(z_2-b) = 0$
for
$a,b \neq 0$
. The variety
${\mathbf {X}}$
is a reducible nodal curve with two components both being
$\mathbb {CP}^1$
with two punctures. The tropicalization is again the union of the two coordinate axes in
$\mathbb {R}^2$
, which is not a tropical homology manifold, and the variety
${\mathbf {X}}$
is not schön as it is singular. However, for each line of the cross, the cocycle associated to this line is mapped to the cocycle associated to the corresponding sphere. This is an isomorphism between
$H^2(X)$
and
$H^2({\mathbf {X}})$
. Since
$H^0(X)$
is trivially isomorphic to
$H^0({\mathbf {X}})$
and other cohomology groups are trivial, we deduce that
${\mathbf {X}}$
is cohomologically tropical.
8.2 Hyperplane arrangement complements
We will now see that all three properties of Theorem 6.1 are satisfied for complements of projective hyperplane arrangements. We will use the de Concini-Procesi model of the complement of a projective hyperplane arrangement [Reference De Concini and Procesi13], as discussed in [Reference Maclagan and Sturmfels35, Section 4.1]. Let
$\mathcal {A}= \left \lbrace {H_i} \right \rbrace _{i=0}^n$
be an arrangement of
$n+1$
hyperplanes in
$\mathbb {P}_{\mathbb {C}}^{d}$
, not all having a common intersection point, and let
${\mathbf {X}}_{\mathcal {A}} =\mathbb {P}_{\mathbb {C}}^d \smallsetminus \bigcup _{H_i \in \mathcal {A}} H_i$
be the complement of the arrangement. For each i, let
$\ell _i$
be the homogeneous linear form such that
$H_i= \left \lbrace {z\in \mathbb {P}_{\mathbb {C}}^d\mid \ell _i(z) = 0} \right \rbrace $
. These define a map
${\mathbf {X}}_{\mathcal {A}} \to (\mathbb {C}^*)^{n}$
given by
$z \mapsto (\ell _i(z))$
in homogeneous coordinates on
$(\mathbb {C}^*)^{n}$
. This map is injective, since no
$z\in {\mathbf {X}}_{\mathcal {A}}$
lies on all hyperplanes by assumption, and induces an isomorphism of
${\mathbf {X}}_{\mathcal {A}} \cong {\mathbf {Y}}_{\mathcal {A}}$
, where
${\mathbf {Y}}_{\mathcal {A}}$
is a subvariety of
$(\mathbb {C}^*)^{n}$
; see [Reference Maclagan and Sturmfels35, Proposition 4.1.1] for details. By a theorem of Ardila and Klivans [Reference Ardila and Klivans1], the tropicalization
$Y_{\mathcal {A}} = \operatorname {\mathrm {Trop}}({\mathbf {Y}}_{\mathcal {A}})$
is the support of the Bergman fan
$\Sigma _{M_{\mathcal {A}}}$
of the matroid
$M_{\mathcal {A}}$
associated to the arrangement
$\mathcal {A}$
; see [Reference Maclagan and Sturmfels35, Sections 4.1–4.2].
First, Tevelev shows [Reference Tevelev44, Theorem 1.5] that the variety
${\mathbf {X}}_{\mathcal {A}}$
is schön, it is clearly connected, and by [Reference Shapiro42], its cohomology has a pure Hodge structure of Hodge-Tate type. Moreover, given a face
$\sigma \in \Sigma _{M_{\mathcal {A}}}$
, the star fan of
$\sigma $
corresponds to the complement of a hyperplane arrangement. By induction, this shows that complements of hyperplane arrangements are wunderschön.
Furthermore, it is shown in [Reference Jell, Shaw and Smacka22, Reference Jell, Rau and Shaw21] by an inductive argument that the Bergman fan
$\Sigma _{M_{\mathcal {A}}}$
of a matroid is a tropical homology manifold. Therefore, one can apply Theorem 6.1, which gives us that
${\mathbf {Y}}_{\mathcal {A}}$
is cohomologically tropical (i.e., the map 
is an isomorphism).
In light of Theorem 6.2, this can be compared with the main result of [Reference Zharkov49], also independently proved in [Reference Shaw41], showing that
$H^{\bullet }({\mathbf {X}}_{\mathcal {A}})\cong H^{\bullet }(X_{\mathcal {A}})$
, however lacking explicit maps.
8.3 Non-matroidal examples
We say that a subvariety
${\mathbf {X}}$
of a torus
${\mathbf {T}}_N$
is non-matroidal if the tropicalization of
${\mathbf {X}}$
is not a Bergman fanfold, up to isomorphisms of the lattice N.
We present an example of
${\mathbf {X}} \subseteq {\mathbf {T}}_N$
which is not a complement of a hyperplane arrangement yet is wunderschön and cohomologically tropical, and the tropicalization
$\mathrm {trop}({\mathbf {X}})$
is a tropical homology manifold.
The variety
${\mathbf {X}}$
will be the complement of an arrangement of lines and a single conic in
$\mathbb {CP}^2$
. Let
$[z_0:z_1: z_2]$
be homogeneous coordinates on
$\mathbb {CP}^2$
. Let
${\mathbf {L}}_0$
,
${\mathbf {L}}_1$
, and
${\mathbf {L}}_2$
be the coordinate lines of
$\mathbb {CP}^2$
so that
${\mathbf {L}}_i $
is defined by
$z_i = 0$
. Let
${\mathbf {L}}_3$
be defined by the linear form
$z_0 - z_1 + z_2 = 0$
and let the conic
${\mathbf {C}}$
be defined by
$z_1^2 +z_2^2 - z_0z_1 - 2z_1z_2 = 0$
. Let
$\mathcal {A}$
denote the union of
${\mathbf {L}}_0, \dots , {\mathbf {L}}_3, {\mathbf {C}}$
.
As depicted in Figure 3, note that
${\mathbf {C}}$
is tangent to
${\mathbf {L}}_1$
at the point
$[1:0:0]$
where
${\mathbf {L}}_1$
intersects
${\mathbf {L}}_2$
. Also the conic
${\mathbf {C}}$
is tangent to
${\mathbf {L}}_0$
at the intersection point
$[0:1:1]$
with
${\mathbf {L}}_3$
. The conic also passes through the intersection point
$[1:1:0]$
of
${\mathbf {L}}_2$
and
${\mathbf {L}}_3$
.
A very-affine variety
${\mathbf {X}}$
which is not a complement of hyperplane arrangement and verifies the main theorem; see Section 8.3.
Consider the map
$\phi : \mathbb {CP}^2 \smallsetminus \mathcal {A} \to (\mathbb {C}^*)^4$
defined by
$$\begin{align*}[z_0: z_1:z_2] \mapsto (\widetilde{z} _1, \widetilde{z} _2, 1 - \widetilde{z} _1 + \widetilde{z} _2, \widetilde{z} _1^2 +\widetilde{z} _2^2 - \widetilde{z} _1 - 2\widetilde{z} _1\widetilde{z} _2 ), \quad\text{ with }\widetilde{z} _1 = \frac{z_i}{z_0}\text{ and }\widetilde{z} _2 = \frac{z_2}{z_0}.\end{align*}$$
Let
${\mathbf {X}} \subseteq (\mathbb {C}^*)^4$
denote the image of the map
$\phi $
. The space
$\mathrm {trop}({\mathbf {X}})$
is
$2$
-dimensional and is the support of the fan described below.
The fan has
$8$
rays in directions given in Figure 4. Each ray is adjacent to exactly
$3$
faces of dimension
$2$
for a total of
$12$
faces of dimension
$2$
. The structure is given in Figure 4: we draw an edge between two vertices if the there is a face between the two corresponding rays. Note that to get a unimodular subdivision, one has to add some rays – for instance, the rays
$\alpha $
and
$\beta $
of Figure 4. We denote by
$\Sigma $
this unimodular fan.
The combinatorial structure of a non-Bergman fan verifying the main theorem described in Section 8.3.
It can be verified in polymake that this fan is a tropical homology manifold and its tropical Betti numbers are
$1,0,6,0,1$
. For an alternative proof, note that the the fan
$\Sigma $
is obtained by the process of tropical modification [Reference Mikhalkin and Rau34] as follows. Let
$\Sigma _{U_{3,4}} \subseteq \mathbb {R}^3$
be the Bergman fan of the uniform matroid
$U_{3,4}$
. Its rays are the rays
$0,1,2$
and
$3$
in Figure 4, where we forget the fourth coordinate. Let
$C \subseteq \Sigma _{U_{3,4}}$
be a tropical trivalent curve with rays
$a,b,c$
(once again, we forget the last coordinate). Then
$\Sigma $
in
$\mathbb {R}^4$
is obtained by a tropical modification of
$\Sigma _{U_{3,4}}$
along C. By [Reference Jell, Rau and Shaw21], the Bergman fan
$\Sigma _{U_{3,4}}$
is a tropical homology manifold; see also Section 8.2. By [Reference Aksnes2, Theorem 4.8], the trivalent tropical curve is also a tropical homology manifold. By [Reference Amini and Piquerez6, Theorem 1.4], the modification of
$\Sigma _{U_{3, 4}} \subseteq \mathbb {R}^3$
along C is a tropical homology manifold. The tools developed in this last article also allow to compute the cohomology of 
quite easily, and to check that the fan is Kähler.
The compactification of
${\mathbf {X}}$
in
$\mathbb {CP}_{\Sigma }$
is given as follows. Consider
$\mathbb {CP}^2$
blown up in the three points whose homogeneous coordinates are
$[1:0:0], [0:1:1],$
and
$[1:1:0]$
. Then, in the blow-up, the exceptional divisor above
$[1:0:0]$
, the proper transform of
${\mathbf {C}}$
, and the proper transform of
${\mathbf {L}}_1$
all intersect in a single point. Similarly, there is a triple intersection of the exceptional divisor above
$[0:1:1]$
and the proper transforms of
${\mathbf {C}}$
and
${\mathbf {L}}_0$
. We further blow up these two intersection points to obtain a surface 
. The divisor 
consists of the five exceptional divisors and the proper transforms of all curves in
$\mathcal {A}$
. Therefore, 
and 
and 
otherwise.
We claim that
${\mathbf {X}}$
is wunderschön. Indeed, for each ray
$\zeta $
of the fan
$\Sigma $
, the variety 
is
$\mathbb {CP}^1$
with two or three marked points corresponding to the intersections with the other divisors in 
, so it is wunderschön. Moreover,
${\mathbf {X}}$
is non-singular and connected, and its cohomology is pure. Hence,
${\mathbf {X}}$
is wunderschön.
In [Reference Aksnes3], tropicalizations of complements of curve arrangements are considered more generally from the perspective of determining when they are cohomologically tropical. In particular, [Reference Aksnes3, Example 7.4] gives an infinite family of non-matroidal cohomologically tropical varieties of dimension two.
Other examples of cohomologically tropical varieties include the moduli spaces
$M(3,6)$
and
$M(3,7)$
of 6 and 7 lines in
$\mathbb {CP}^2$
, respectively; see [Reference Luxton32, Reference Schock40, Reference Corey and Luber10].
By the tropical and classical Künneth formula, the product of two cohomologically tropical varieties is again cohomologically tropical. Moreover, in a product
${\mathbf {X}}_1 \times {\mathbf {X}}_2 \subset {\mathbf {T}}_{N_1} \times {\mathbf {T}}_{N_2}$
, if one of the factors is non-matroidal, the product remains non-matroidal (because the same statement holds for a product of fanfolds, using the fact that local fanfolds of a Bergman fanfold are all Bergman).
Using the family of non-matroidal examples above, as well as the complements of hyperplane arrangements in any dimension, we obtain infinite families of non-matroidal cohomologically tropical varieties in arbitrary dimension at least two.
8.4 Maximal degenerations
Motivated by the work of Deligne [Reference Deligne12] and our results, Theorem 7.1 and Theorem 7.2, we can ask the following question.
Question 8.1. Is there a tropical geometric characterization of maximally degenerate families of complex algebraic varieties? Is it true that those families in which the open strata of special fibers have a cohomology which is pure of Hodge-Tate type are exactly those covered by our Theorem 7.1?
These questions are intimately related to open problems on large scale limits of families of complex algebraic varieties. For instance, a recent work by Yang Li [Reference Li30] reduces the metric SYZ conjecture in maximally degenerate families of complex algebraic varieties to the existence of solutions to a tropical Monge-Ampère equation, once this equiation has been properly formulated. For those degenerations appearing in Theorem 7.1, our results show that the corresponding tropical variety is Kähler in the sense of [Reference Amini and Piquerez4] and moreover recovers the geometry of the degenerate fiber as well as the limit Hodge-theoretic geometry of the family. In a forthcoming work [Reference Amini and Piquerez8], a differential calculus on tropical varieties is developed that combines Chow rings of local tropical fans with real differential forms on the variety. Tropical Hodge theory can be then used to properly formulate the Monge-Ampère equation on the tropicalization using tropical Kähler forms. This allows to formulate a tropical analogue of the Monge-Ampère equation and study its solutions.
Another instance of a large-scale asymptotic problem is a conjecture by Kontsevich and Soibelman [Reference Kontsevich and Soibelman27] that predicts the convergence of the normalized Calabi metrics in maximally degenerate families of Calabi-Yau varieties to a tropical Calabi metric (once this has been properly defined) on the limit tropical variety.
8.5
$\mathcal T\!$
-stability
It seems plausible that a framework parallel to the one in [Reference Amini and Piquerez5] can be developed for properties of tropicalization of algebraic varieties. The properties discussed in this paper concern pairs consisting of a subvariety of an algebraic torus and a fan structure on its tropicalization. Three basic operations can be conducted on these pairs: products, blow-ups and blow-downs, and taking the graph of a holomorphic function on the subvariety, restricted to the complement of its divisor. For example, the cases described in Sections 8.2 and 8.3 can both be obtained by these operations. The properties of being schön, wunderschön and cohomologically tropical should be
$\mathcal T\!$
-stable in this framework. We refer to [Reference Schock40] for some results in this direction.
Acknowledgements
The research of E. A. and K. S. is supported by the Trond Mohn Foundation project ‘Algebraic and Topological Cycles in Complex and Tropical Geometries’. O. A. thanks Math+, the Berlin Mathematics Research Center, for support. M. P. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001995).
This project was started during a visit to the Norwegian Academy of Science and Letters under the Center for Advanced Study Young Fellows Project ‘Real Structures in Discrete, Algebraic, Symplectic, and Tropical Geometries’. We thank the Centre of Advanced Study and Academy for their hospitality and wonderful working conditions. We thank as well the hospitality of the mathematics institute at TU Berlin where part of this research was carried out.
We are additionally grateful to an anonymous referee for providing helpful comments which helped us to improve this paper.
Competing interest
The authors declare no competing interests.
Data availability statement
No data was created or analysed in this work. Data sharing is not applicable.
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