# Tropical geometry, the motivic nearby fiber, and limit mixed Hodge numbers of hypersurfaces

- Eric Katz
^{1}Email author and - Alan Stapledon
^{2, 3}

**3**:10

https://doi.org/10.1186/s40687-016-0058-9

© The Author(s) 2016

**Received: **2 December 2015

**Accepted: **14 January 2016

**Published: **5 June 2016

## Abstract

A complex variety, degenerating over a punctured disk, carries a limit mixed Hodge structure on its cohomology, encoding the action of monodromy. The associated limit mixed Hodge–Deligne polynomial can be expressed in terms of the motivic nearby fiber. Using techniques from tropical geometry, we present a new formula for the motivic nearby fiber and concentrate on the case of degenerating families of complex hypersurfaces, generalizing work of Danilov and Khovanskiĭ and Batyrev and Borisov. If these families satisfy a natural smoothness condition, called schönness, their limit mixed Hodge–Deligne numbers can be expressed in terms of new, combinatorial invariants of a polyhedral subdivision of the associated Newton polytope. These invariants are multivariable, Ehrhart-theoretic extensions of Stanley’s invariants of subdivisions and are situated in his theory in a companion combinatorial paper.

## Keywords

## 1 Background

Let \(\mathcal {O}\) be the ring of germs of analytic functions in \(\mathbb {C}\) in a neighborhood of the origin, and let \(\mathbb {K}\) be its field of fractions. A variety *X* over \(\mathbb {K}\) is naturally interpreted as a family of complex varieties \(f:X\rightarrow \mathbb {D}^*\) where \(\mathbb {D}^*\) is a small punctured disk about the origin over which *X* is defined. After possibly shrinking \(\mathbb {D}^*\), we may assume that \(X \rightarrow \mathbb {D}^*\) is a locally trivial fibration, and we fix a nonzero fiber \(X_{{{\mathrm{gen}}}} := f^{-1}(t)\) for some \(t \in \mathbb {D}^*\).

Our goal is to compute an important invariant of *X* called the **motivic nearby fiber**
\(\psi _X = \psi _f\), which was introduced by Denef and Loeser [23] and contains information about the extension of *f* to a family over the whole complex disk \(\mathbb {D}\). Moreover, the motivic nearby fiber specializes to the limit Hodge–Deligne polynomial of *X* and to both the \(\chi _y\)-characteristic and Euler characteristic of \(X_{{{\mathrm{gen}}}}\).

*k*, the

**Grothendieck ring**\(K_0({{\mathrm{Var}}}_k)\) of algebraic varieties over

*k*is the free abelian group generated by isomorphism classes [

*V*] of varieties

*V*over

*k*, modulo the relation

*U*is an open subvariety of

*V*. Multiplication is defined by

**motivic invariant**over

*k*is a ring homomorphism \(K_0({{\mathrm{Var}}}_k) \rightarrow R\), for some ring

*R*. The motivic nearby fiber is a ring homomorphism

*k*has characteristic zero, then \(K_0({{\mathrm{Var}}}_k)\) is generated by the classes of smooth, proper varieties. If

*X*is smooth and proper, then by [42] there exists a

**semi-stable reduction**of

*X*. That is, after possibly pulling back the family \(f: X \rightarrow \mathbb {D}^*\) by a map \(\mathbb {D}^*\rightarrow \mathbb {D}^*\) ramified over the puncture, there exists an extension of

*f*defined over \(\mathbb {D}\) such that the central fiber is a reduced, simple normal crossings divisor with irreducible components \(\{ D_i \}_{i \in \{ 1, \ldots , r \}}\). If for every non-empty subset \(I \subseteq \{ 1, \ldots , r \}\), we set \(D_I^\circ = \cap _{i \in I} D_i {\backslash } \cup _{j \notin I} D_j\), then

A result of Luxton and Qu [43, Theorem 6.11] that was conjectured by Tevelev in [60] states that every variety *X* over \(\mathbb {K}\) contains an open, very affine subvariety \(X^\circ \) that is **schön** in the sense of Tevelev [60, Definition 1.1]. Here \(X^\circ \) being very affine means that it can be embedded as a closed subvariety of \((\mathbb {K}^*)^n\) for some *n*, defined by an ideal \(I \subseteq \mathbb {K}[x_{1}^{\pm 1},\ldots , x_n^{\pm 1}]\). In this case, \(X^\circ \) being schön means that for every \(w \in \mathbb {R}^n\), the corresponding initial degeneration \({{\mathrm{in}}}_w X^\circ \) defined by the ideal \({{\mathrm{in}}}_w I := ( {{\mathrm{in}}}_w(f) \mid f \in I) \subseteq \mathbb {C}[x_{1}^{\pm 1},\ldots , x_n^{\pm 1}]\) of initial degenerations is a smooth subvariety of \((\mathbb {C}^*)^n\) [33, Prop 3.9]. For a more geometric description of \({{\mathrm{in}}}_w X^\circ \), we refer the reader to Sect. 2. The notion of schönness of a hypersurface of a complex torus was introduced by Khovanskiĭ in [41] as a hypersurface *non-degenerate* with respect to its Newton polytope.

Luxton and Qu’s result immediately implies that the Grothendieck ring \(K_0({{\mathrm{Var}}}_\mathbb {K})\) is generated by schön subvarieties of tori. In particular, to describe the motivic nearby fiber, in principle, we may reduce to the case of a schön subvariety of a torus. *In what follows, we will always assume that*
\(X^\circ \subseteq (\mathbb {K}^*)^n\)
*is schön.*

###
**Definition 1.1**

The tropical variety \({{\mathrm{Trop}}}(X^\circ )\) can be given a rational polyhedral structure \(\Sigma \) such that the initial degeneration at \(w \in {{\mathrm{Trop}}}(X^\circ )\) only depends on the cell containing *w* in its relative interior (this follows from [43, Theorem 1.5]). Hence for every cell \(\gamma \) of \(\Sigma \), we may define \([{{\mathrm{in}}}_{\gamma } X^\circ ] := [{{\mathrm{in}}}_w X^\circ ] \in K_0({{\mathrm{Var}}}_\mathbb {C})\) for any \(w \in \mathbb {R}^n\) in the relative interior of \(\gamma \). Our main result is as follows:

###
**Theorem 1.2**

We provide a proof of Theorem 1.2 in Sect. 2. The theorem immediately gives expressions for the motivic nearby fiber of various partial compactifications of \(X^\circ \) that are not smooth in general (Corollary 2.4). In the case of smooth compactifications of \(X^\circ \), this result generalizes [38, Theorem 5.1] (see Theorem 2.3, Remark 2.5 below). In particular, in contrast to the above theorem, [38, Theorem 5.1] requires that \(\Sigma \) has a unimodular recession fan.

A key point is that there exist explicit algorithms to compute both the initial degenerations of \(X^\circ \) and its tropical variety with a choice of rational polyhedral structure. Moreover, there is a range of available software that implements these algorithms [36, 37]. Hence given any variety over \(\mathbb {K}\), if one is able to produce a stratification into locally closed, very affine schön subvarieties, as guaranteed by Luxton and Qu’s result, then the above theorem gives a practical approach to computing the motivic nearby fiber.

###
*Example 1.3*

*t*be a local coordinate on \(\mathbb {D}\), and let

*v*, and at each bounded edge is isomorphic to \(\mathbb {A}^1\) minus 6, 2, and 1 point, respectively. Theorem 1.2 then implies that

*X*over \(\mathbb {K}\), the polynomial \(E(X_\infty ;u,v) := E(\psi ([X]))\) is called the

**limit Hodge–Deligne polynomial**of

*X*and encodes information on the variation of mixed Hodge structures of the family \(X \rightarrow \mathbb {D}^*\). The specialization obtained by setting \(v = 1\) is the \(\chi _y\)-

**characteristic**\(E(X_{{{\mathrm{gen}}}};u,1) = E(X_\infty ;u,1)\) of \(X_{{{\mathrm{gen}}}}\) and encodes information about the Hodge filtration on the cohomology with compact supports of \(X_{{{\mathrm{gen}}}}\). Finally, the specialization \(e(X_{{{\mathrm{gen}}}}) = E(X_{{{\mathrm{gen}}}};1,1)\) is the familiar

**topological Euler characteristic**of \(X_{{{\mathrm{gen}}}}\). Theorem 1.2 immediately provides formulas for these invariants in the case when \(X^\circ \) is schön.

###
**Corollary 1.4**

Here the last equality follows from the fact that the Euler characteristic \(e({{\mathrm{in}}}_\gamma X^\circ )\) is zero unless \(\gamma \) is a vertex of \(\Sigma \) (see (3)).

As discussed above, this corollary provides a strategy to compute any of these invariants. For example, if one wants to compute the Euler characteristic of a complex variety *V* and then if one can find a stratification of *V* into locally closed pieces, each of which can be realized as the general fiber of a schön degeneration, then the above corollary reduces the problem to finding the Euler characteristic of a set of ‘simpler’ complex varieties.

Before presenting our main application, we introduce a new motivic invariant over \(\mathbb {K}\) (see Sect. 3 for details). Given a variety *X* over \(\mathbb {K}\), consider the complex cohomology with compact supports \(H^m_c (X_{{{\mathrm{gen}}}})\) of the fiber \(X_{{{\mathrm{gen}}}}\). Then \(H^m_c (X_{{{\mathrm{gen}}}})\) admits three natural filtrations. Firstly, since it is a complex variety, it admits a decreasing filtration \(F^\bullet \) called the **Hodge filtration** and an increasing filtration \(W_\bullet \) called the **Deligne weight filtration**. Secondly, the monodromy map \(T: H^m_c (X_{{{\mathrm{gen}}}}) \rightarrow H^m_c (X_{{{\mathrm{gen}}}})\) can be written as \(T = T_sT_u\), where \(T_s\) is semi-simple and \(T_u\) is unipotent, and we may consider the action of the nilpotent operator \(N = \log T_u\) on \(H^m_c (X_{{{\mathrm{gen}}}})\). A result of Steenbrink and Zucker [59] and El Zein [24] states that \(H^m_c (X_{{{\mathrm{gen}}}})\) admits an increasing filtration \(M_\bullet \) called the **monodromy weight filtration**, such that the filtration induced by \(M_\bullet \) on the quotient \(Gr_r^W H_c^m (X_{{{\mathrm{gen}}}})\) encodes the Jordan block structure of the induced action of *N* on \(Gr_r^W H_c^m (X_{{{\mathrm{gen}}}})\). Note that \(M_\bullet \) is not the filtration encoding the Jordan block structure of the induced action of *N* on the whole of \(H_c^m (X_{{{\mathrm{gen}}}})\), but rather some kind of convolution of this filtration with \(W_\bullet \).

**refined limit mixed Hodge numbers**. Summing over

*q*or

*r*recovers the (usual) mixed Hodge numbers and the limit mixed Hodge numbers of \(H_c^m (X_{{{\mathrm{gen}}}})\), respectively (see (6) and (7)). We define a polynomial called the

**refined limit Hodge–Deligne polynomial**by

**Hodge–Deligne polynomial**of \(X_{{{\mathrm{gen}}}}\). One may think that every successive specialization forgets about a filtration in the following sense: The invariants \(E(X_\infty ; u,v,w), E(X_\infty ; u,v), E(X_{{{\mathrm{gen}}}}; u,w)\) and \(E(X_{{{\mathrm{gen}}}}; u,1)\) encode information about the filtrations \((F^\bullet , W_\bullet , M_\bullet ), (F^\bullet , M_\bullet ), (F^\bullet , W_\bullet )\) and \(F^\bullet \), respectively.

For the remainder of the introduction, we assume that \(X^\circ \subseteq (\mathbb {K}^*)^n\) is a schön hypersurface. In this case, the Hodge–Deligne polynomial \(E(X^\circ _{{{\mathrm{gen}}}}; u,w)\) encodes precisely the (usual) mixed Hodge numbers of \(X^\circ _{{{\mathrm{gen}}}}\), and its computation is a classical problem. Indeed, an algorithm to compute the mixed Hodge numbers of a schön hypersurface of a complex torus was given by Danilov and Khovanskiĭ in [20]. Much later, using deep results from intersection cohomology, a combinatorial formula was given by Batyrev and Borisov and was the key technical result in their construction of mirror Calabi–Yau varieties in [8]. A cleaner combinatorial formula was later given by Borisov and Mavlyutov in [15]. Finally, a combinatorial proof of the Borisov–Mavlyutov formula was given by the second author in [56], as part of work giving a representation-theoretic generalization.

Our main application is a combinatorial formula for the refined limit mixed Hodge numbers of the schön hypersurface \(X^\circ \). In this case, this is equivalent to giving a combinatorial formula for the refined limit Hodge–Deligne polynomial \(E(X^\circ _\infty ; u,v,w)\). In particular, by specializing, we obtain a combinatorial formula for the limit mixed Hodge numbers of \(X^\circ \). Our result also specializes to give the Borisov–Mavlyutov formula for the usual mixed Hodge numbers of \(X^\circ _{{{\mathrm{gen}}}}\). Although we make use of the strategy of the Danilov–Khovanskiĭ algorithm, our proof is self-contained and only relies on Theorem 1.2 together with some new combinatorics. In particular, in Sect. 5.2, using the theory of valuations of polytopes (see, e.g., [45]), we present a new proof of a formula of Danilov–Khovanskiĭ [20, Section 4] for the \(\chi _y\)-characteristic of \(X^\circ _{{{\mathrm{gen}}}}\). Since the necessary combinatorial results are involved and we expect them to be of outside interest, we will only quote them as needed and defer all proofs and discussion to [39]. We will mention that some of these results build on the work of Stanley [55], together with recent work of Athanasiadis and Savvidou [2, 3], and Nill and Schepers [47].

As explained in Sect. 5.1, we may associate with \(X^\circ \) its corresponding Newton polytope *P* together with a corresponding regular, lattice polyhedral subdivision \(\mathcal {S}\). In [39, Section 9], we introduce a combinatorial invariant \(h^*(P,\mathcal {S}; u,v,w) \in \mathbb {Z}[u,v,w]\) called the **refined limit mixed**
\(h^*\)-**polynomial** of \((P,\mathcal {S})\), which only depends on the poset structure of \(\mathcal {S}\), together with the number of lattice points in all dilates of all cells of \(\mathcal {S}\). This invariant has several interesting specializations. In particular, \(h^*(P,\mathcal {S}; u,1,1) = h^*(P; u) \) is the usual \(h^*\)-polynomial of *P*, encoding the number of lattice points in all dilates of *P* [10].

###
**Theorem 1.5**

As discussed above, Theorem 1.5 immediately gives explicit combinatorial formulas for the refined limit mixed Hodge numbers and limit mixed Hodge numbers of \(X^\circ \) (see Corollary 5.11). In particular, we deduce that these invariants only depend on the pair \((P,\mathcal {S})\), and not on the specific choice of \(X^\circ \). Our results allow one to compute the refined limit Hodge–Deligne polynomial of various compactifications of \(X^\circ \) but not necessarily the refined limit mixed Hodge numbers as we elaborate in Remark 5.12. In Example 5.13, we apply our results to obtain formulas for **stringy invariants** associated with families of Calabi–Yau varieties.

###
*Example 1.6*

*M*together with a lattice polyhedral subdivision. In this case, Theorem 1.5 has the following explicit description. Let \(\partial P\) and \({{\mathrm{Int}}}(P)\) denote the boundary and interior of

*P*, respectively. Then the coefficients of \(uvw^3\) and \(uv^2w^3\) in \(h^*(P,\mathcal {S};u,v,w)\) are, respectively, given by

*X*denotes the closure of \(X^\circ \) in the toric variety over \(\mathbb {K}\) corresponding to the normal fan of

*P*, then

*X*may be viewed as a family of smooth, compact curves with

###
*Example 1.7*

*P*is the convex hull of \(a_0 = (0,0), a_1 = (4,0)\) and \(a_2 = (0,4)\). Setting \(b_0 = (1,1), b_1 = (2,1)\) and \(b_2 = (1,2)\), the lattice polyhedral subdivision \(\mathcal {S}\) has four maximal cells: \(\{ a_i, a_j, b_i, b_j \}\) for \(i \ne j\) and \(\{ b_0, b_1, b_2 \}\). By Example 1.6,

In the case when we have a family of varieties over a punctured curve, we also give an alternative approach to Theorem 1.5 via intersection cohomology making use of the pure Hodge structure on the intersection cohomology of projective varieties. By the use of the decomposition theorem of Beilinson et al. [9], one can show that for certain stratifications, intersection cohomology admits a motivic formula if one includes terms accounting for the singularities in the normal cones to strata. This idea is used in the computation of intersection cohomology of toric varieties (see, e.g., [26]), in the work of Batyrev and Borisov [8], and is developed in greater generality by Cappell et al. [18]. Here, we observe that a motivic formula holds for the refined limit Hodge–Deligne polynomials for intersection cohomology with compact support (Theorem 6.1) and deduce that the following corollary is equivalent to Theorem 1.5 (see Lemma 6.2). The degree of \(h^*(P,\mathcal {S};u,v,w)\) as a polynomial in *w* is at most \(\dim P + 1\), and we denote the coefficient of \(w^{\dim P + 1}\) by \(l^*(P, \mathcal {S}; u,v)\) and call it the **local limit mixed**
\(h^*\)-**polynomial**.

###
**Corollary 1.8**

*X*denote the closure of \(X^\circ \) in the projective toric variety over \(\mathbb {K}\) corresponding to the normal fan of

*P*. Then the refined limit Hodge–Deligne polynomial associated with the intersection cohomology of

*X*is given by

From the above corollary, one may deduce an explicit formula for the corresponding refined limit mixed Hodge numbers for intersection cohomology (see Corollary 6.3). When \(\mathbb {K}= \mathbb {C}(t)\), we also present an alternative proof of Corollary 1.8 and hence of Theorem 1.5 using intersection cohomology. This proof extends the ideas of Batyrev and Borisov’s original proof of a formula for the usual mixed Hodge numbers of \(X^\circ _{{{\mathrm{gen}}}}\) in [8].

###
*Example 1.9*

As in Example 1.6, let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface. Let \((P,\mathcal {S})\) denote the corresponding pair consisting of a lattice polytope together with a lattice polyhedral subdivision. Let *X* denote the closure of \(X^\circ \) in the toric variety over \(\mathbb {K}\) corresponding to the normal fan of *P*. When \(n = 2, X\) may be viewed as a family of compact, smooth curves, and \(E_{{{\mathrm{int}}}}(X_\infty ;u,v,w) = E(X_\infty ;u,v,w)\) is computed in Example 1.6. When \(n = 3, X\) may be viewed as a family of compact, possibly singular surfaces, and \(E_{{{\mathrm{int}}}}(X_\infty ;u,v,w)\) is given explicitly by Corollary 1.8, Example 4.13 and the computation \(E_{{{\mathrm{int}}},{{\mathrm{Lef}}}}(P;t) = 1+ \mu t + t^2\), where \(\mu + 3\) is the number of facets of *P*.

We note that connections between limit mixed Hodge structures and the combinatorics of dual complexes have been studied in a number of contexts by Arapura et al. [1], Berkovich [11], Helm and the first author [33], Nicaise [46], and Payne [48].

### 1.1 Organization of the paper

This paper is structured as follows. In Sect. 2, we review necessary background from tropical geometry, introduce our invariant \(\psi _{(X^\circ ,\Sigma ,\Delta )}\) of partial compactifications of subvarieties of algebraic tori, and prove Theorem 1.2. In Sect. 3, we discuss motivic invariants and the refined limit Hodge–Deligne polynomial. Section 4 introduces combinatorial invariants whose properties are established in [39] and which are related to the refined limit Hodge–Deligne polynomial of hypersurfaces of algebraic tori in Sect. 5. In Sect. 6, we derive a formula for the limit Hodge–Deligne polynomial of the intersection cohomology of a schön subvariety and use it to give an alternative proof of Theorem 1.5.

*Notation and conventions.* If \(\mathbb {P}\) is a toric variety, then we let \(\mathbb {P}_\mathbb {C}, \mathbb {P}_\mathbb {K}\) and \(\mathbb {P}_\mathcal {O}\) denote the corresponding toric variety over \(\mathbb {C}\) and \(\mathbb {K}\), and corresponding toric scheme over \(\mathcal {O}\), respectively.

## 2 A tropical approach to the motivic nearby fiber

In this section, we present the proof of Theorem 1.2. We will continue with the notation of the introduction. In particular, \(X^\circ \subseteq (\mathbb {K}^*)^n\) is a schön subvariety, and \(\Sigma \) is a rational polyhedral structure on \({{\mathrm{Trop}}}(X^\circ )\) that extends to a polyhedral subdivision of \(\mathbb {R}^n\). Such a \(\Sigma \) exists by [43, Prop 6.8].

We first recall the following toric interpretation of the initial degenerations of \(X^\circ \), and refer the reader to [33, Section 1] for details. One can define a toric scheme \(\mathbb {P}(\Sigma )_\mathcal {O}\) over \(\mathcal {O}\) from \(\Sigma \). For a cell \(\gamma \) of \(\Sigma \), let \({{\mathrm{rec}}}(\gamma )\) denote the recession cone of \(\gamma \). That is, \({{\mathrm{rec}}}(\gamma )\) is the unique cone such that there exists a bounded polytope *Q* satisfying \(\gamma = Q + {{\mathrm{rec}}}(\gamma )\). By [17], the set of recession cones of \(\Sigma \) forms the recession fan \(\Delta \). Note that the bounded cells of \(\Sigma \) are precisely the cells whose recession cone is \(\{0\}\). The generic fiber of \(\mathbb {P}(\Sigma )_\mathcal {O}\) is the toric variety \(\mathbb {P}(\Delta )_\mathbb {K}\). For cones \(\tau \) in \(\Delta \), let \(U_\tau \) be the corresponding torus orbit of \(\mathbb {P}(\Delta )_\mathbb {K}\). Cells \(\gamma \in \Sigma \) correspond to torus orbits \(U_\gamma \) contained in the central fiber of \(\mathbb {P}(\Sigma )_\mathcal {O}\). We define \(T_\gamma \) to be the torus fixing \(U_\gamma \) pointwise.

Let \(\mathcal {X}\) denote the closure of \(X^\circ \) in \(\mathbb {P}(\Sigma )_\mathcal {O}\), and let \(X_{\Delta }\) and \(X_0\) denote the generic fiber and central fiber of \(\mathcal {X}\), respectively. For cones \(\tau \) in \(\Delta \), let \(X_\tau ^\circ = X_{\Delta } \cap U_\tau \), so that \(X_{\Delta }\) admits a stratification \(X_{\Delta } = \cup _{\tau \in \Delta } X^\circ _{\tau }\). Similarly, for cells \(\gamma \) in \(\Sigma \), if we let \(X_\gamma ^\circ =\mathcal {X}\cap U_\gamma \), then \(X_0 = \cup _{\gamma \in \Sigma } X^\circ _{\gamma }\).

For each cone \(\tau \) in \(\Delta \), let \(\mathbb {R}_\tau \) denote the linear span of \(\tau \) and consider the projection \(\pi _\tau : \mathbb {R}^n \rightarrow \mathbb {R}^n/\mathbb {R}_\tau \). Then \(X_\tau ^\circ \) is a schön subvariety of \(U_\tau \), and its corresponding tropical variety has a polyhedral structure \(\Sigma _\tau = \{ \pi _\tau (\gamma ) \mid \tau \subseteq {{\mathrm{rec}}}(\gamma ) \}\). In particular, the bounded cells of \(\Sigma _\tau \) correspond to the cells of \(\Sigma \) with recession cone \(\tau \), and the recession fan \(\Delta _\tau \) of \(\Sigma _\tau \) is the *star-quotient* of \(\Delta \) by \(\tau \) (see [28, Section 3.1]).

*w*in the relative interior of \(\gamma \), the initial degeneration \({{\mathrm{in}}}_w X^\circ \) depends only on \(\gamma \) because the closure of

*X*in \(\mathbb {P}(\Sigma )_\mathcal {O}\) is a tropical compactification by [43, Theorem 1.5]. Moreover, \({{\mathrm{in}}}_w X^\circ \) is invariant under the torus \(T_\gamma \). Moreover, there is a non-canonical isomorphism

###
**Lemma 2.1**

*P*be a

*n*-dimensional polytope and let

*Q*be a proper (possibly empty) face of

*P*. Let \(\mathcal {S}\) be a polyhedral subdivision of

*P*. Then

###
*Proof*

*F*disjoint from

*Q*and given by \(r_F:F{\backslash } Q\rightarrow \partial F{\backslash } Q\) for cells intersecting

*Q*where \(r_F\) is projection away from some point \(x_F\) in the relative interior of \(F\cap Q\). We define \(r:P{\backslash } Q\rightarrow P^\bullet \) to be the composition \(r_0\circ \dots \circ r_d:P{\backslash } Q=P^\bullet _{d}\rightarrow P^\bullet _0=P^\bullet .\)

The inclusion \(\partial P^\bullet \hookrightarrow \partial P{\backslash } Q\) is also a homotopy equivalence. Its inverse is defined similarly to the map above. Finally, \(\partial P{\backslash } Q\hookrightarrow P{\backslash } Q\) is a homotopy equivalence whose inverse can be given by projection from a point in the relative interior of *Q*. Since these three maps induce isomorphisms in homology, so must \(\partial P^\bullet \hookrightarrow P^\bullet \). \(\square \)

The following lemma is analogous to [38, Theorem 3.6].

###
**Lemma 2.2**

The expression \(\psi _{(X^\circ ,\Sigma ,\{0\})}\) above is independent of the choice of rational polyhedral structure \(\Sigma \) on \({{\mathrm{Trop}}}(X^\circ )\).

###
*Proof*

*H*to be an affine hyperplane such that \(P=C_\gamma \cap H\) is a polytope not containing the origin; set

*Q*to be the intersection of

*P*with \(\mathbb {R}^n\times \{ 0 \}\); and let \(\mathcal {S}\) be polyhedral subdivision of

*P*induced by the fan refinement of \(C_\gamma \) induced by \(\Sigma '\). The cells in \(\mathcal {S}\) that intersect

*Q*correspond to unbounded cells in \(\Sigma '\). \(\square \)

Steenbrink has applied a result [58, Theorem 5] similar to Lemma 2.2 to study motivic Milnor fibers of function germs on toric singularities.

Since \(\psi _{(X^\circ ,\Sigma ,\{0\})}\) is independent of the choice of \(\Sigma \) by Lemma 2.2, after possible ramified base-extension of \(\mathbb {K}\), it follows from [33, Proposition 2.3] that we may choose \(\Sigma \) such that \(\mathbb {P}(\Delta )_\mathbb {K}\) is smooth. In this case, we may invoke the following result.

###
**Theorem 2.3**

[38, Theorem 5.1] With the notation above, if \(\mathbb {P}(\Delta )_\mathbb {K}\) is smooth, then the motivic nearby fiber \(\psi _{X_{\Delta }}\) of \(X_{\Delta }\) is equal to \(\psi _{(X^\circ ,\Sigma ,\Delta )}\).

Using (5), we immediately deduce the following corollary.

###
**Corollary 2.4**

###
*Remark 2.5*

Note that when \(\Delta ' = \{0\}\) above, we recover Theorem 1.2, while when \(\Delta ' = \Delta \), then we recover the statement of Theorem 2.3 without the assumption that \(\mathbb {P}(\Delta )_\mathbb {K}\) is smooth. In this way, we see that Theorem 1.2 is a generalization of Theorem 2.3. Note that \(X_{\Delta }\) is proper, and, while the assumption that \(\mathbb {P}(\Delta )_{\mathbb {K}}\) is smooth forces \(X_{\Delta }\) to be smooth, in general, \(X_{\Delta }\) and \(\mathbb {P}(\Delta )_{\mathbb {K}}\) may have singularities.

###
*Remark 2.6*

Note that Theorem 1.2 implies that if \(X^\circ \) is schön, then the expression \(\psi _{(X^\circ ,\Sigma ,\{0\})}\) is not only independent of \(\Sigma \) (Lemma 2.2), but independent of the choice of embedding \(X^\circ \subseteq (\mathbb {K}^*)^n\). We do not know a direct proof of this fact.

###
*Remark 2.7*

As in [38, Section 3], the definition of \(\psi _{(X^\circ ,\Sigma ,\{0\})}\) can be extended to the case when \(X^\circ \) is not necessarily schön, but the pair \((X^\circ ,\mathbb {P}(\Sigma )_\mathcal {O})\) is *tropical*. In this case, the proof of Lemma 2.2 holds unchanged and \(\psi _{(X^\circ ,\Sigma ,\{0\})}\) is independent of the choice of \(\Sigma \). The expression \(\psi _{(X^\circ ,\Sigma ,\Delta )}\) was called the *tropical motivic nearby fiber* in [38]. However, one cannot expect an analog of Theorem 1.2, as the following example demonstrates. We also do not know how the tropical motivic nearby fiber depends on the embedding of \(X^\circ \) into \((\mathbb {K}^*)^n\).

*G*(

*X*,

*Y*,

*Z*) be a homogeneous polynomial over \(\mathbb {C}\) of degree 3 whose zero locus

*V*(

*G*) in \(\mathbb {P}^2\) is a nodal cubic curve. Suppose further that

*G*has the following properties

- (1)
all coefficients of degree 3 monomials in

*G*are nonzero, - (2)
the node of

*V*(*G*) lies in \((\mathbb {C}^*)^2\subset \mathbb {P}^2\), and - (3)
*V*(*G*) intersects each coordinate lines in three distinct points.

*G*by applying a generic element of \({\text {Gl}}_3(\mathbb {C})\) to the equation of a nodal cubic. The tropicalization of \(V(G)^\circ \subseteq (\mathbb {C}^*)^2\) in \(\mathbb {R}^2\) consists of the origin and three rays in the directions \((1,0),(0,1),(-1,-1)\), each with multiplicity 3. Now let

*H*be a generic homogeneous polynomial of degree 3. Consider \(F=G+tH\) considered as a homogeneous polynomial over \(\mathbb {K}\). Now,

*V*(

*F*) is a smooth cubic over \(\mathbb {K}\). Consequently, for \(t\ne 0\) sufficiently small, \(V(F)_{{{\mathrm{gen}}}}^\circ \) is a smooth cubic over \(\mathbb {C}\) with 9 points removed. By construction, the tropicalization of \(V(F)^\circ \subseteq (\mathbb {K}^*)^2\) is the same as that of \(V(G)^\circ \). Moreover, \((V(F)^\circ ,\mathbb {P}(\Sigma )_\mathcal {O})\) is tropical, where \(\Sigma \) is the standard polyhedral structure on the tropicalization of \(V(F)^\circ \). There is a single-bounded cell which is the origin. Since \({{\mathrm{in}}}_0 V(F)^\circ =V(G)^\circ \), we have \(\psi _{(V(F)^\circ ,\Sigma ,\{0\})} = [V(G)^\circ ]\). This is a nodal cubic minus the 9 points of intersection with the coordinate lines. Because a nodal cubic is isomorphic to a projective line with two points identified, we have \(e(V(G)^\circ ) = -8 \ne e(V(F)_{{{\mathrm{gen}}}}^\circ ) = -9\) violating the final statement in Corollary 1.4.

## 3 The motivic nearby fiber and limit mixed Hodge structures

As observed in the introduction, by composing the motivic nearby fiber (1) with a motivic invariant over \(\mathbb {C}\), we obtain a motivic invariant over \(\mathbb {K}\) to which we can apply Theorem 1.2. In this section, we introduce some known results from the theory of limit mixed Hodge structures. We recommend [49] and [50] as references. The theory was developed by many authors including Deligne et al. [19], Schmid [54], Steenbrink [57], and Saito [53].

*B*admits a mixed Hodge structure \((W_\bullet ,F^\bullet )\) [50] with associated graded pieces

*Hodge polynomial*of \(B_\bullet \) is defined by

### 3.1 Motivic invariants over \(\mathbb {C}\)

*m*th cohomology group with compact supports \(H^m_c (V)\) of a complex variety

*V*admits a canonical mixed Hodge structure with decreasing filtration \(F^\bullet \) called the

**Hodge filtration**and increasing filtration \(W_\bullet \) called the

**Deligne weight filtration**. The set of numbers \(\{ h^{p,q}(H_c^m(V)) \}_{p,q,m}\) is called the

**mixed Hodge numbers**of

*V*, and the corresponding Hodge polynomial

*E*(

*V*;

*u*,

*v*) is the

**Hodge–Deligne polynomial**of

*V*. The corresponding motivic invariant over \(\mathbb {C}\) is the

*Hodge–Deligne map*

*V*specializes to the \(\chi _y\)-

**characteristic**

*E*(

*V*;

*u*, 1) of

*V*. Its coefficients are alternating sums of the dimensions of the graded pieces of the Hodge filtration on the cohomology of

*V*with compact supports. The

**Euler characteristic**

*e*(

*V*) is obtained via the specialization \(e(V) = E(V;1,1)\).

###
*Example 3.1*

Recall that we write \(\mathbb {L}:= [\mathbb {A}^1]\) in the Grothendieck ring \(K_0({{\mathrm{Var}}}_k)\) for any field *k*. The complex affine line has \(h^{1,1}(H^2_c(\mathbb {A}^1)) = 1\) and all other mixed Hodge numbers equal to zero. Hence its Hodge–Deligne polynomial is \(E(\mathbb {A}^1) = uv\).

*n*-dimensional torus \((\mathbb {C}^*)^n\) has \([(\mathbb {C}^*)^n] = (\mathbb {L}- 1)^n\) in \(K_0({{\mathrm{Var}}}_\mathbb {C}), E((\mathbb {C}^*)^n) = (uv - 1)^n\), and its mixed Hodge numbers are all zero except

*k*.

### 3.2 Motivic invariants over \(\mathbb {K}\)

Recall from the introduction that we regard a variety *X* over \(\mathbb {K}\) as a family of complex varieties over the disk \(\mathbb {D}^*\), and we fix a nonzero fiber \(X_{{{\mathrm{gen}}}} := f^{-1}(t)\) for some \(t \in \mathbb {D}^*\). Then the cohomology groups \(H_c^m(X_{{{\mathrm{gen}}}})\) admit a weight filtration \(M_\bullet \) called the **monodromy weight filtration**. We will write \(H_c^m(X_{\infty })\) to denote \(H_c^m(X_{{{\mathrm{gen}}}})\) with the mixed Hodge structure \((F^\bullet ,M_\bullet )\). The corresponding mixed Hodge numbers are denoted \(h^{p,q}(H_c^m(X_{\infty }))\) and are called the **limit mixed Hodge numbers** of *X*. The corresponding Hodge polynomial is denoted \(E(X_\infty ;u,v)\) and is called the **limit Hodge–Deligne polynomial** of *X*. It is the composition of the motivic nearby fiber and the Hodge–Deligne map, i.e., \(E(X_\infty ;u,v) = E(\psi _X)\). It specializes to both the \(\chi _y\)-characteristic of \(X_{{{\mathrm{gen}}}}\) and the Euler characteristic of \(X_{{{\mathrm{gen}}}}\) (see Remark 3.5 below).

The monodromy weight filtration encodes the action of the logarithm of the monodromy operator on the \(W_\bullet \)-graded pieces of \(H_c^m(X_{{{\mathrm{gen}}}})\). We explain this statement in detail below.

The cohomology groups \(H^m_c(X_t)\) of the fibers are isomorphic as vector spaces but have a Hodge structure which varies. Because \(H^m_c(X_t)\) forms a locally trivial fiber bundle, parallel transport gives a monodromy transformation, \(T: H_c^m (X_{{{\mathrm{gen}}}}) \rightarrow H_c^m (X_{{{\mathrm{gen}}}})\). It turns out that *T* is quasi-unipotent, that is, some multiple of *T* is unipotent. After replacing *T* by some power which corresponds to pulling back the family by a map \(\mathbb {D}^*\rightarrow \mathbb {D}^*\) ramified over the puncture, we may take its logarithm, \(N=\log T\), and obtain a nilpotent operator. Moreover, the monodromy map *T* preserves the weight filtration \(W_\bullet \), and \(N: H_c^m(X_{\infty }) \rightarrow H_c^m(X_{\infty })\) is a morphism of mixed Hodge structures of type \((-1,-1)\).

*r*,

*N*restricts to a nilpotent operator

*N*(

*r*) on the graded piece \(Gr_r^W H_c^m (X_{{{\mathrm{gen}}}})\). Let \(F(r)^\bullet \) and \(M(r)_\bullet \) denote the filtrations on \(Gr_r^W H_c^m (X_{{{\mathrm{gen}}}})\) induced by \(F^\bullet \) and \(M_\bullet \), respectively. Then \(N(r)^{r + 1} = 0\) and \(M(r)_\bullet \) is the filtration obtained from

*N*(

*r*) which determines and is determined by the Jordan block decomposition of

*N*(

*r*). Indeed, we may inductively define a unique increasing filtration

*k*,

- (1)
\(N(r)( M(r)_k ) \subseteq M(r)_{k - 2}\),

- (2)
the induced map \(N(r)^k: Gr^{M(r)}_{r + k} Gr_r^W H_c^m (X_{{{\mathrm{gen}}}}) \rightarrow Gr^{M(r)}_{r - k} Gr_r^W H_c^m (X_{{{\mathrm{gen}}}})\) is an isomorphism.

*N*(

*r*) is a morphism of mixed Hodge structures of type \((-1,-1)\). We will write \(Gr_r^W H_c^m (X_\infty )\) when referring to \(Gr_r^W H_c^m (X_{{{\mathrm{gen}}}})\) with the limit mixed Hodge structure. We denote the corresponding mixed Hodge numbers by \(h^{p,q,r}(H_c^m(X_{\infty }))\) and call them the

**refined limit mixed Hodge numbers**of

*X*. For each

*r*, we encode the corresponding Hodge polynomial as the coefficient of \(w^r\) in a polynomial \(E(X_\infty ;u,v,w)\) that we will call the

**refined limit Hodge–Deligne polynomial**.

###
*Remark 3.2*

*X*over \(\mathbb {K}\) give rise to morphisms of complex varieties \(X_{{{\mathrm{gen}}}}\) that respect the filtrations \((F^\bullet ,W_\bullet ,M_\bullet )\). In particular, if \(U \subseteq X\) is an open inclusion and \(V = X \backslash U\), then the corresponding long exact sequence of cohomology with compact supports for the triple \((X_{{{\mathrm{gen}}}},U_{{{\mathrm{gen}}}},V_{{{\mathrm{gen}}}})\) consists of morphisms that preserve the Hodge filtration and both the Deligne and monodromy weight filtrations (c.f. proof of [50, Lemma 14.61], see also [27] for the classical approach). In particular, it follows from Remark 3.4 that the refined limit Hodge–Deligne polynomial is a motivic invariant over \(\mathbb {K}\). That is, we may consider the

*refined Hodge–Deligne map*

### 3.3 Properties of the refined limit Hodge–Deligne polynomial

###
*Remark 3.3*

*r*, we discard the refinement by the weight filtration and obtain the following relation with the limit mixed Hodge numbers,

*q*, we discard the monodromy filtration refinement and obtain mixed Hodge numbers of \(X_{{{\mathrm{gen}}}}\),

*q*and

*r*gives the following relation between the limit mixed Hodge numbers and the mixed Hodge numbers of \(X_{{{\mathrm{gen}}}}\),

###
*Remark 3.4*

###
*Remark 3.5*

###
*Remark 3.6*

With the notation above, since *N*(*r*) is a morphism of mixed Hodge structures of type \((-1,-1)\), the isomorphisms (2) imply that each vertical strip of the Hodge diamond of \(Gr_r^W H_c^m (X_\infty )\) is a symmetric, unimodal sequence of nonnegative integers. That is, for \(0 \le k \le r\), the sequence \(\{ h^{k + i,i,r}(H_c^m(X_{\infty })) \mid 0 \le i \le r -k \}\) is symmetric and unimodal.

###
*Remark 3.7*

*p*and

*q*, i.e., \(h^{p,q,r}(H_c^m(X_{\infty })) = h^{q,p,r}(H_c^m(X_{\infty }))\). It follows from Remark 3.6 that they satisfy the additional symmetry:

###
*Example 3.8*

*V*is a complex variety and \(X = V \times _\mathbb {C}\mathbb {K}\), then

*X*may be regarded as a trivial family over \(\mathbb {D}^*\). In this case,

*N*is identically zero, \(M_\bullet \) coincides with the Deligne filtration \(W_\bullet \), and \(E(X_\infty ;u,v) = E(V;u,v)\). Moreover,

###
*Example 3.9*

*X*is smooth and proper, then \(Gr_r^W H^m (X_{{{\mathrm{gen}}}}) = 0\) unless \(m = r\). In this case, the monodromy weight filtration encodes the Jordan block decomposition of \(N: H^m(X_{\infty }) \rightarrow H^m(X_{\infty })\). Moreover,

###
*Remark 3.10*

*p*,

*q*), either the source or target of the induced map \(N: H^{p,q}(H_c^m(X_\infty )) \rightarrow H^{p-1,q-1}(H_c^m(X_\infty ))\) is zero.

## 4 Subdivisions of lattice polytopes

In this section, we gather together the relevant facts that we will need about the combinatorics of subdivisions of polytopes. Details and proofs of all statements can be found in [39]. We say that the empty polytope has dimension \(-1\).

A **polyhedral subdivision** of a polytope \(P\subset \mathbb {R}^n\) is a subdivision of *P* into a finite number of polytopes such that the intersection of any two polytopes is a (possibly empty) face of both. A **lattice polyhedral subdivision** of a lattice polytope *P* is a polyhedral subdivision of *P* into lattice polytopes. A natural class of polyhedral subdivisions is the regular subdivisions. They are induced by a height function \(\omega :P\cap \mathbb {Z}^n\rightarrow \mathbb {R}\). The cells of the subdivision are the projections of the bounded faces of the convex hull of \({\text {UH}}=\{ (u, \lambda ) \mid \lambda \ge \omega (u) \} \in \mathbb {R}^n \times \mathbb {R}\). A subdivision is said to be **regular** if it is induced by some height function. For more details, see [22, 29].

We first recall some definitions concerning the combinatorics of Eulerian posets. Consider a finite poset *B* containing a minimal element \(\widehat{0}\) and a maximal element \(\widehat{1}\). For any pair \(z \le x\) in *B*, we can consider the interval \([z,x] = \{ y \in B \mid z \le y \le x \}\). Assume that *B* is *graded* in the sense that for every \(x \in B\), every maximal chain in the interval \([\widehat{0},x]\) has the same length \(\rho (x)\). We call \(\rho : B \rightarrow \mathbb {N}\) the *rank function* of *B*, and call \(\rho (\widehat{1})\) the *rank* of *B*. Then *B* is **Eulerian** if every interval [*z*, *x*] with \(z < x\) has as many elements of odd rank as even rank.

###
*Example 4.1*

The poset of faces of a polytope *P* (including the empty face) is an Eulerian poset under inclusion. Then \(\rho (Q) = \dim Q + 1\), for any face *Q* of *P*. Let \(\mathcal {S}\) be a polyhedral subdivision of *P*, and let *F* be a (possibly empty) cell of \(\mathcal {S}\). As a poset, the **link**
\({{\mathrm{lk}}}_\mathcal {S}(F)\) of *F* in \(\mathcal {S}\) consists of all cells \(F'\) of \(\mathcal {S}\) that contain *F* under inclusion, and we have that the interval \([F,F']\) is an Eulerian poset.

###
*Example 4.2*

If *B* is a poset, then \(B^*\) is the poset with the same elements as *B* and all orderings reversed. In particular, *B* is Eulerian if and only if \(B^*\) is Eulerian.

The *g*-**polynomial** of an Eulerian poset is defined recursively and was introduced by Stanley [55, Corollary 6.7].

###
**Definition 4.3**

*B*be an Eulerian poset of rank

*n*. If \(n = 0\), then \(g(B;t) = 1\). If \(n > 0\), then

*g*(

*B*;

*t*) is the unique polynomial of degree strictly less the

*n*/ 2 satisfying

The following theorem of Stanley giving an inversion formula for *g* is used in Sect. 6:

###
**Theorem 4.4**

*B*is Eulerian and has positive rank, then

We will be interested in the following example [55, Example 7.2].

###
*Example 4.5*

*P*. If

*F*is a (possibly empty) cell of \(\mathcal {S}\), then the

*h*-

**polynomial**of \({{\mathrm{lk}}}_\mathcal {S}(F)\) is defined by

*P*be a non-empty lattice polytope in a lattice

*M*of rank

*n*. For \(m\in \mathbb {Z}_{> 0}\), consider the function \(f_P(m)=\#(mP\cap M)\). By Ehrhart’s theorem [10, Section 3.3], \(f_P(m)\) is a polynomial of degree \(\dim P\), called the

**Ehrhart polynomial**of

*P*. It follows that we can write

*P*is a polynomial of degree at most \(\dim P\) called the \(h^*\)-

**polynomial**of

*P*(see, e.g., [10, Section 3.3]). Note that if

*P*is empty, then we set \(f_P(m) \equiv 0\) and \(h^*(P;u) = 1\). We have \(h^*(P;1) = (\dim P)!{{\mathrm{vol}}}(P)\) where \({{\mathrm{vol}}}(P)\) is the Euclidean volume of

*P*.

These invariants play the central role in the theory of valuations on polytopes. The definition given below is a priori weaker than the usual definition of valuations but is equivalent as a consequence of Lemma 4.7.

###
**Definition 4.6**

*M*and let

*G*be a group. A

*G*-valued

**valuation**on \(\mathcal {P}_M\) is a map \(\varphi :\mathcal {P}_{M}\rightarrow G\) satisfying

- (1)If \(\mathcal {S}\) is a regular lattice subdivision of
*P*with top-dimensional cells \(P_1,\dots ,P_m, \varphi \) satisfies the inclusion/exclusion relation$$\begin{aligned} \varphi (P) = \sum _{ \begin{array}{c} F \in \mathcal {S}\\ F \nsubseteq \partial P \end{array} } (-1)^{\dim P - \dim F} \varphi (F), \end{aligned}$$ - (2)
\(\varphi (\emptyset )=0\), and

- (3)
\(\varphi (P)=\varphi (UP+u)\) for \(P\in \mathcal {P}_{M}, U\in {{\mathrm{Aut}}}(M), u\in M\).

The lemma below is non-trivial since not every lattice polytope admits a lattice polyhedral subdivision into unimodular simplices, let alone a regular one. This lemma is an adaptation of [31, Prop 19.2] which is stated for general lattice subdivisions.

###
**Lemma 4.7**

Valuations are determined by their values on unimodular simplices: If \(\varphi _1,\varphi _2\) are valuations that are equal on unimodular simplices, then \(\varphi _1=\varphi _2\).

###
*Proof*

*M*. Let \(\mathcal {H}\) be the subgroup generated by the following:

- (1)For \(\mathcal {S}\) is a regular subdivision of
*P*,$$\begin{aligned} P - \sum _{ \begin{array}{c} F \in \mathcal {S}\\ F \nsubseteq \partial P \end{array} } (-1)^{\dim P - \dim F} \varphi (F), \end{aligned}$$ - (2)
\(\emptyset ,\)

- (3)
\(P-(UP+u)\) for \(P\in \mathcal {P}_{\mathbb {Z}^n}, U\in {{\mathrm{Aut}}}(M), u\in M\)

*M*. Let \(\mathcal {H}'\) be the subgroup of \(\mathcal {G}\) generated by convex lattice polytopes of

*M*whose affine span is not full-dimensional. By induction, we may suppose \(\mathcal {H}'/(\mathcal {H}'\cap \mathcal {H})\) is generated by unimodular simplices. Now, it suffices to show that \(\mathcal {G}/(\mathcal {H}+\mathcal {H}')\) is generated by a

*d*-dimensional unimodular simplex.

*d*-dimensional lattice simplex. If \({{\mathrm{vol}}}(P)=1\) then we are done. Suppose \({{\mathrm{vol}}}(P)=V\ge 2\), and let \(F_0,\dots ,F_d\) denote the facets of

*P*. By the proof of [31, Proposition 19.1], there is a point \(p\in M\) such that \({{\mathrm{vol}}}({{\mathrm{Conv}}}(F_i\cup \{p\}))<V\). Let \(\omega :{\text {Vert}}(P)\cup \{p\}\rightarrow \mathbb {R}\) be the height function that is 0 on the vertices of

*P*and 1 on

*p*. The graph of the height function lies in \(M_\mathbb {R}\times \mathbb {R}\), and because the points in the graph are affinely independent, their convex hull is a simplex. The projections of the convex hull by \(\pi :M\times \mathbb {R}\rightarrow M\) are \({{\mathrm{Conv}}}(P\cup \{p\})\). Moreover, the projections of the upper faces or the lower faces each give regular subdivisions \(\mathcal {S}_{{\text {upper}}}, \mathcal {S}_{{\text {lower}}}\) of \({{\mathrm{Conv}}}(P\cup \{p\})\). The top-dimensional lower faces of the convex hull are

*P*and some faces \({{\mathrm{Conv}}}(F_i\cup \{p\})\) for \(i\in I\) for some subset \(I\subset \{0,\dots ,n\}\). The top-dimensional upper faces of the convex hull are \({{\mathrm{Conv}}}(F_i\cup \{p\})\) for \(i\not \in I\). The subdivision relation in \(\mathcal {G}/(\mathcal {H}+\mathcal {H}')\) gives

*P*as a formal sum of simplices of smaller volume. \(\square \)

###
*Example 4.8*

*m*, we have a valuation

**local**\(h^*\)-

**polynomial**\(l^*(P;u)\) of

*P*was introduced by Stanley in [55, Example 7.13], generalizing the definition of Betke and McMullen in the case of a simplex [13], and was independently introduced by Borisov and Mavlyutov in [15],

###
**Definition 4.9**

*P*. Then the

**limit mixed**\(h^*\)-

**polynomial**of \((P,\mathcal {S})\) is

**local limit mixed**\(h^*\)-

**polynomial**of \((P,\mathcal {S})\) is

**refined limit mixed**\(h^*\)-

**polynomial**of \((P,\mathcal {S})\) is

*P*, with cells of \(\mathcal {S}\) given by the faces of

*P*, then we write \(h^*(P;u,v) = h^*(P,\mathcal {S};u,v)\) and call the polynomial the

**mixed**\(h^*\)-

**polynomial**. If

*P*is empty, then \(h^*(P,\mathcal {S};u,v,w) = h^*(P,\mathcal {S};u,v) = l^*(P, \mathcal {S};u,v) = 1\).

The following theorem is proved in [39, Theorem 9.2].

###
**Theorem 4.10**

*P*. Then the refined limit mixed \(h^*\)-polynomial satisfies the following properties:

- (1)The refined limit mixed \(h^*\)-polynomial is invariant under the interchange of
*u*and*v*and satisfies the additional symmetry$$\begin{aligned} h^*(P,\mathcal {S},u,v,w) = h^*(P,\mathcal {S},u^{-1},v^{-1},uvw). \end{aligned}$$ - (2)The refined limit mixed \(h^*\)-polynomial specializes to the limit mixed \(h^*\)-polynomial$$\begin{aligned} h^*(P,\mathcal {S};u,v,1) = h^*(P,\mathcal {S};u,v). \end{aligned}$$
- (3)The refined limit mixed \(h^*\)-polynomial specializes to the mixed \(h^*\)-polynomial$$\begin{aligned} h^*(P,\mathcal {S};uw^{-1},1,w) = h^*(P;u,w). \end{aligned}$$
- (4)The refined limit mixed \(h^*\)-polynomial specializes to the \(h^*\)-polynomial$$\begin{aligned} h^*(P,\mathcal {S};u,1,1) = h^*(P;u). \end{aligned}$$
- (5)
The degree of \(h^*(P,\mathcal {S};u,v,w)\) as a polynomial in

*w*is at most \(\dim P + 1\). Moreover, the coefficient of \(w^{\dim P + 1}\) is the local limit mixed \(h^*\)-polynomial \(l^*(P, \mathcal {S}; u,v)\). - (6)The limit mixed \(h^*\)-polynomial can be written in terms of mixed \(h^*\)-polynomials,where \(\partial P\) denotes the boundary of$$\begin{aligned} h^*(P, \mathcal {S};u,v) = \sum _{ \begin{array}{c} F \in \mathcal {S}\\ F \nsubseteq \partial P \end{array} } (uv - 1)^{\dim P - \dim F} h^*(F;u,v), \end{aligned}$$
*P*.

*P*.

*P*with all maximal cones removed. The cones \(\gamma _Q\) in \(\Delta _P\) are in inclusion-reserving correspondence with the positive dimensional faces

*Q*of

*P*. Let \(\Delta _P'\) denote a simplicial fan refinement of \(\Delta _P\) which exists by the resolution of singularities algorithm for toric varieties [28, Sec.2.6]. That is, every cone \(\gamma '\) in \(\Delta _P'\) is generated by precisely \(\dim \gamma '\) rays and is contained in a cone of \(\Delta _P\). We let \(\sigma (\gamma ')\) denote the smallest cone in \(\Delta _P\) containing \(\gamma '\) and set

###
**Corollary 4.11**

- (1)
The degree of \(h^*(P,\mathcal {S};u,v,w)\) as a polynomial in

*w*is at most \(\dim P + 1\). - (2)The refined limit mixed \(h^*\)-polynomial specializes to the limit mixed \(h^*\)-polynomial, i.e.,$$\begin{aligned} h^*(P,\mathcal {S};u,v,1) = h^*(P,\mathcal {S};u,v). \end{aligned}$$
- (3)If \(\Delta _P'\) denotes a simplicial fan refinement of \(\Delta _P\) then for \(\Lambda \) defined in terms of the refined limit mixed \(h^*\)-polynomial as above, we have$$\begin{aligned} \Lambda (P,\mathcal {S},\Delta _P';u,v,w) = (uvw^2)^{\dim P + 1}\Lambda (P,\mathcal {S},\Delta _P';u^{-1},v^{-1},w^{-1}). \end{aligned}$$

###
**Corollary 4.12**

- (1)
All terms in \(h^*(P;u,w)\) have combined degree in

*u*and*w*at most \(\dim P + 1\). - (2)The mixed \(h^*\)-polynomial specializes to the \(h^*\)-polynomial, i.e.,$$\begin{aligned} h^*(P;u,1) = h^*(P;u). \end{aligned}$$
- (3)If \(\Delta _P'\) denotes a simplicial fan refinement of \(\Delta _P\) then for \(\Lambda \) defined in terms of the mixed \(h^*\)-polynomial as above, we have$$\begin{aligned} \Lambda (P,\mathcal {S},\Delta _P';u,w) = (uw)^{\dim P + 1}\Lambda (P,\mathcal {S},\Delta _P';u^{-1},w^{-1}). \end{aligned}$$

The following example is computed in [39, Example 9.10]:

###
*Example 4.13*

*F*is a cell of \(\mathcal {S}\), then let \(\sigma (F)\) denote the smallest face of

*P*containing

*F*. Then for \(q,r > 0\),

## 5 Refined limit mixed Hodge numbers of hypersurfaces

The goal of this section is to present a proof of Theorem 1.5 giving a combinatorial formula for the refined limit Hodge–Deligne polynomial of a schön hypersurface in \((\mathbb {K}^*)^n\) which is interpreted as a family of hypersurfaces. We first reprove a theorem of Danilov–Khovanskiĭ for the \(\chi _y\)-characteristic of a complex hypersurface in terms of the \(h^*\)-polynomial of its Newton polytope. Then we give combinatorial formulas of the following progressively finer cohomological invariants: the Hodge–Deligne polynomial of a generic fiber; the limit Hodge–Deligne polynomial, the limit Hodge–Deligne polynomial of a smooth compactification of the family of hypersurfaces, and then the refined limit Hodge–Deligne polynomial. We will make use of the fact that the cohomology of a hypersurface is tightly constrained by Poincaré duality and the weak Lefschetz theorem.

### 5.1 Tropical geometry for hypersurfaces

Let \(X^\circ = \{ \sum _{u \in M} \alpha _u x^u = 0 \} \subset T \cong (\mathbb {K}^*)^n\) be a schön hypersurface. The **Newton polytope**
*P* of \(X^\circ \) is the convex hull of \(\{ u \in M \mid \alpha _u \ne 0 \}\). Note that *P* may be viewed as a full-dimensional lattice polytope in the translation *M* of the saturation of its integer affine span in \(\mathbb {Z}^n\) to the origin, and \(X^\circ \cong X' \times (\mathbb {K}^*)^k\), for some *k*, where \(X' \subseteq {{\mathrm{Spec}}}\mathbb {K}[M] \) is a schön hypersurface with Newton polytope *P*. Hence we may and will assume that \(\dim P = n\).

Tropical geometry of hypersurfaces reduces to the study of Newton polytopes and polyhedral subdivisions [29, 51]. Recall that the field \(\mathbb {K}\) has a natural valuation by considering the vanishing order of a function on \(\mathbb {D}^*\) at the origin. With the notation above, the function \(P \cap \mathbb {Z}^n \rightarrow \mathbb {Z}, u \mapsto {{\mathrm{ord}}}(\alpha _u)\) induces a regular, lattice subdivision \(\mathcal {S}\) of *P*. Explicitly, the cells of \(\mathcal {S}\) are the projections of the bounded faces of the convex hull of \({\text {UH}}=\{ (u, \lambda ) \mid \alpha _u \ne 0, \lambda \ge {{\mathrm{ord}}}(\alpha _u) \}\) in \(\mathbb {R}^n \times \mathbb {R}\), and the bounded faces of \({\text {UH}}\) are the graph of a function \(\omega :P\rightarrow \mathbb {R}\). Restricting to \(P\cap \mathbb {Z}^n\), we get a height function. There is a dual complex associated with the height function that generalizes the normal fan. The cells of this complex are in inclusion-reversing bijective correspondence with the cells of \(\mathcal {S}\). See [32, 9.11] for details.

###
*Remark 5.1*

Since the initial degeneration \({{\mathrm{in}}}_wX^{\circ }\) of a hypersurface is given by the corresponding initial form of its defining polynomial, for a generic choice of coefficients (in a certain analytic topology), a hypersurface with a given height function is schön, i.e., all initial degenerations are smooth. Hence every pair \((P,\mathcal {S})\), where \(\mathcal {S}\) is a regular, lattice polyhedral subdivision of a lattice polytope *P* arises from the construction above for some schön hypersurface. See [33, Section 8.1] for a more detailed discussion of genericity and schönness.

*P*with the maximal cones removed. Recall from Sect. 2 that we may define a toric scheme \(\mathbb {P}(\Sigma )_\mathcal {O}\) over \(\mathcal {O}\) from \(\Sigma \) with generic fiber equal to the toric variety \(\mathbb {P}(\Delta )_\mathbb {K}\). Let \(\mathcal {X}\) denote the closure of \(X^\circ \) in \(\mathbb {P}(\Sigma )_\mathcal {O}\), and let \(X_{\Delta }\) and \(X_0\) denote the generic fiber and central fiber of \(\mathcal {X}\), respectively. Then we can write the stratifications of \(X_{\Delta }\) and \(X_0\) in dual language with respect to the Newton polytope and subdivision as the following:

*P*in its corresponding complex torus, which we denote as \(T_{{{\mathrm{gen}}}}\). For every cell

*F*of \(\mathcal {S}\) with \(\dim F > 0\), the corresponding complex variety \(X^\circ _{F}\) is a complex schön hypersurface with Newton polytope

*F*, and, if

*w*lies in the relative interior of the cell in \(\Sigma \) corresponding to

*F*, then

###
**Corollary 5.2**

*F*.

### 5.2 The \(\chi _y\)-characteristic of a complex hypersurface

We apply Corollary 5.2 to give a new proof of a formula of Danilov–Khovanskiĭ [20, Section 4] for the \(\chi _y\)-characteristic of schön hypersurfaces in \((\mathbb {C}^*)^n\).

###
*Remark 5.3*

The fact that the Hodge–Deligne polynomial of a schön hypersurface of a complex torus is determined by its Newton polytope can be seen directly. One considers the closure *V* of a schön hypersurface \(V^\circ \) given by a Laurent polynomial with Newton polytope *P* in a toric resolution of the complex toric variety determined by *P*. It is a smooth variety. Since Hodge numbers are locally constant through families of smooth proper varieties, the Hodge–Deligne polynomial of *V* is independent of the choice of polynomial. The result can then be deduced from the motivic nature of the Hodge–Deligne polynomial.

*P*. We may suppose \(\dim P = \dim V^\circ + 1\). Recall from Remark 5.1 and Remark 5.3 that the Hodge–Deligne polynomial of \(V^\circ \) only depends on

*P*and that given any

*P*, there exists a corresponding schön hypersurface \(V^\circ \). Hence we may define

*P*is empty, then we let \(V_P^\circ \) be the empty set. To identify the \(\chi _y\)-characteristic, we build a valuation out of it (see Definition 4.6).

###
**Lemma 5.4**

###
*Proof*

*P*. By Remark 5.1, there exists a schön hypersurface \(X^\circ \subset (\mathbb {K}^*)^{\dim P}\) with corresponding Newton polytope

*P*and polyhedral subdivision \(\mathcal {S}\). This hypersurface satisfies \(E(X^\circ _{{{\mathrm{gen}}}};u,1)=E(V_P^\circ ;u,1)\). By Corollary 5.2 and (9), we obtain

With the notation of Sect. 4, we obtain a new proof of Danilov and Khovanskiĭ’s theorem.

###
**Theorem 5.5**

*P*be a non-empty lattice polytope and let \(V_P^\circ \) be a complex schön hypersurface with Newton polytope

*P*. Then we have the following formula for the \(\chi _y\)-characteristic of \(V_P^\circ \):

*P*.

###
*Proof*

*l*of these hyperplanes as coordinate hyperplanes and the last one as some generic hyperplane, we get the motivic relation \([V_{\Delta _l}]=[(\mathbb {C}^*)^{l-1}]-[V_{\Delta _{l-1}}]\). Because \(V^\circ \mapsto E(V^\circ ;u,1)\) is motivic and \(E((\mathbb {C}^*)^{l-1};u,1)=(u-1)^{l-1}\), we have

By specializing the above theorem to \(u=1\) and using the fact that \(h^*(P;1) = (\dim P)!{{\mathrm{vol}}}(P)\) where \({{\mathrm{vol}}}(P)\) is the Euclidean volume of *P*, we get the following well-known result of Kouchnirenko [40]:

###
**Corollary 5.6**

*P*be a non-empty lattice polytope and let \(V_P^\circ \) be a schön hypersurface with Newton polytope

*P*. Then we have the following formula for the topological Euler characteristic of \(V_P^\circ \):

### 5.3 A Danilov–Khovanskiĭ type algorithm

In [20], Danilov and Khovanskiĭ use their formula for the \(\chi _y\)-characteristic in Theorem 5.5 in connection with the weak Lefschetz theorem and Poincaré duality to give an algorithm to compute the Hodge–Deligne polynomial of a complex schön hypersurface. We use an analogous approach to providing an algorithm to compute the refined limit Hodge–Deligne polynomial of a schön hypersurface from the limit Hodge–Deligne polynomial. We continue with the notation from earlier in this section.

We consider the cohomology with compact supports of the complex variety \(X^{\circ }_{{{\mathrm{gen}}}} \subseteq T_{{{\mathrm{gen}}}}\), and set \(n = \dim T_{{{\mathrm{gen}}}}\). The following weak Lefschetz result implies that the only interesting cohomology is in middle dimension.

###
**Proposition 5.7**

[20, Proposition3.9] The Gysin map \(H^k_c (X^{\circ }_{{{\mathrm{gen}}}}) \rightarrow H^{k + 2}_c (T_{{{\mathrm{gen}}}})\) is an isomorphism for \(k > n - 1\), and a surjection for \(k = n - 1\). Since \(X^{\circ }_{{{\mathrm{gen}}}}\) is affine, \(H^k_c (X_{{{\mathrm{gen}}}}^\circ ) = 0\) for \(k < n - 1\).

**primitive cohomology**of \(X^\circ _{{{\mathrm{gen}}}}\) to be

*N*preserves the primitive cohomology of \(X^\circ _{{{\mathrm{gen}}}}\).

It follows that the refined limit Hodge–Deligne polynomial \(E(X_\infty ^\circ ;u,v,w)\) determines and is determined by the refined limit Hodge numbers of the primitive cohomology of \(X_\infty ^\circ \). In particular, we have the following lemma:

###
**Lemma 5.8**

Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and polyhedral subdivision \((P,\mathcal {S})\). Then, as a polynomial in *w*, \(uvw^2E(X_\infty ^\circ ;u,v,w)\) has the same coefficient as \((uvw^2 - 1)^{\dim P + 1}\) in all degrees strictly greater than \(\dim P + 1\).

###
*Proof*

Since \(X^\circ _{{{\mathrm{gen}}}}\) is a smooth complex variety, the graded pieces of the Deligne weight filtration \(Gr_r^W H_c^m (X^\circ _{{{\mathrm{gen}}}})\) are zero for \(r > m\) by, e.g., [50, Thm 5.39]. In particular, the contributions from the primitive cohomology of \(X^\circ _{{{\mathrm{gen}}}}\) to \(E(X_\infty ^\circ ;u,v,w)\) all have degree at most \(\dim P - 1\) in *w*. The result then follows from the above discussion and Example 3.11. \(\square \)

The above lemma may be viewed as a generalization of the corresponding statement for the Hodge–Deligne polynomial, due to Danilov and Khovanskiĭ, which follows by the exact same argument as above.

###
**Lemma 5.9**

[20, Sec. 3.11] Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope *P*. Then the coefficient of \(u^pw^q\) in \(uwE(X_{{{\mathrm{gen}}}}^\circ ;u,w)\) equals the coefficient of \(u^pw^q\) in \((uw - 1)^{\dim P + 1}\) for \(p + q > \dim P + 1\).

*P*with all maximal cones removed, with cones \(\gamma _Q\) in inclusion-reserving correspondence with the positive dimensional faces

*Q*of

*P*. As in Sect. 4, let \(\Delta _P'\) denote a simplicial fan refinement of \(\Delta _P\), and let \(\sigma (\gamma ')\) denote the smallest cone in \(\Delta _P\) containing a cone \(\gamma '\) in \(\Delta _P'\). Then we have an induced proper, birational map of toric varieties over \(\mathbb {K}, \pi : \mathbb {P}(\Delta _P')_\mathbb {K}\rightarrow \mathbb {P}(\Delta _P)_\mathbb {K}\), which, by standard toric geometry, is locally a projection in the sense that if \(\mathbb {P}(\Delta _P')_\mathbb {K}= \bigcup _{\gamma ' \in \Delta _P'} U_{\gamma '}\) and \(\mathbb {P}(\Delta _P)_\mathbb {K}= \bigcup _{ Q \subseteq P , \dim Q > 0 } U_{\gamma _{Q}}\) are unions of the toric varieties into torus orbits, then \(\pi |_{U_{\gamma '}}\) is given by

*w*. Firstly, Lemma 5.8 implies that we know \(E(X_{\infty }^\circ ; u,v,w)\) in all degrees strictly greater than \(\dim P - 1\). Secondly, by induction on dimension and (11), we know \(E(X_{P,\infty }'; u,v,w)\) in all degrees strictly greater than \(\dim P - 1\), and by (12), we know \(E(X_{P,\infty }'; u,v,w)\) and hence \(E(X_{\infty }^\circ ; u,v,w)\) in all degrees strictly less than \(\dim P - 1\). Finally, \(E(X_{\infty }^\circ ; u,v,1)\) now determines \(E(X_{\infty }^\circ ; u,v,w)\) in degree \(\dim P - 1\).

###
*Remark 5.10*

The same argument gives the Danilov and Khovanskiĭ algorithm to determine the Hodge–Deligne polynomial \(E(X_{\infty }^\circ ; uw^{-1},1,w) = E(X_{{{\mathrm{gen}}}}^\circ ; u,w)\) from the \(\chi _y\)-characteristic \(E(X_{\infty }^\circ ; u,1,1) = E(X_{{{\mathrm{gen}}}}^\circ ; u,1)\). Explicitly, Lemma 5.9 implies that we know the coefficient of \(u^pw^q\) in \(E(X_{{{\mathrm{gen}}}}^\circ ;u,w)\) for \(p + q > \dim P - 1\). Secondly, by induction on dimension and (11), we know the coefficient of \(u^pw^q\) in \(E(X_{P,{{\mathrm{gen}}}}'; u,w)\) for \(p + q > \dim P\), and by (12), we know the coefficient of \(u^pw^q\) in \(E(X_{P,{{\mathrm{gen}}}}'; u,w)\) and hence \(E(X_{{{\mathrm{gen}}}}^\circ ; u,w)\) in all degrees strictly less than \(\dim P - 1\). Finally, \(E(X_{{{\mathrm{gen}}}}^\circ ; u,1)\) now determines the coefficient of \(u^pw^q\) in \(E(X_{{{\mathrm{gen}}}}^\circ ; u,w)\) when \(p + q = \dim P - 1\).

### 5.4 A formula for the refined limit Hodge–Deligne polynomial

We now complete the proof of Theorem 1.5 and deduce an explicit description of the refined limit mixed Hodge numbers of a schön hypersurface. We will see that the proof reduces to some combinatorial results which are proved in [39]. We also state some immediate consequences of the theorem.

*P*. We claim that

Using our description of the cohomology of \(X_{{{\mathrm{gen}}}}^\circ \) in Sects. 5.3 and 5.1 together with the above formula, we immediately deduce the following corollary. The second two statements below follow from (2) and (5) in Theorem 4.10, respectively. We refer the reader to Example 4.13 and Theorem 4.10 for explicit combinatorial descriptions of the invariants below in the cases when \(n = 2,3\).

###
**Corollary 5.11**

As in Corollary 2.4, the motivic nature of the invariants above means that we obtain formulas for invariants of partial compactifications of schön hypersurfaces. We now state this explicitly for possible future reference.

*Q*of

*P*. Let \(\widetilde{\Delta }_P'\) denote a fan refinement (not necessarily simplicial) of a subfan \(\widetilde{\Delta }_P\) of \(\Delta _P\). We let \(\sigma (\gamma ')\) denote the smallest cone in \(\widetilde{\Delta }_P\) containing \(\gamma '\). Let \(\widetilde{X}_P'\) denote the closure of \(X^\circ \) in the toric variety \(\mathbb {P}(\widetilde{\Delta }_P')_\mathbb {K}\) over \(\mathbb {K}\). Let \(\widetilde{X}_P'\) and \(\widetilde{X}_P\) denote the closure of \(X^\circ \) in the toric varieties \(\mathbb {P}(\widetilde{\Delta }_P')_\mathbb {K}\) and \(\mathbb {P}(\Delta _P)_\mathbb {K}\), respectively. Then \(\widetilde{X}_P'\) and \(\widetilde{X}_P\) have toroidal singularities, and \(X_P\) has a stratification into schön subvarieties

*Q*of

*P*since \(X_{Q}^{\circ } = \emptyset \) when \(\dim Q = 0\).

###
*Remark 5.12*

*P*containing

*F*. If we further assume that \(\Delta _P\) is smooth, then the above expression for the motivic nearby fiber appeared in [38, Section 6].

Finally, we present the following application of Theorem 1.5.

###
*Example 5.13*

Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and polyhedral subdivision \((P,\mathcal {S})\). Let *X* denote the closure of \(X^\circ \) in the projective toric variety over \(\mathbb {K}\) corresponding to the normal fan of *P*. We assume that *P* is **reflexive** in the sense of Batyrev [5, Section 4.1]. That is, we assume that *P* contains the origin in its relative interior, and the associated dual polytope \(P^*\) is also a lattice polytope. In this case, there is an inclusion-reversing correspondence between faces *Q* of *P* and faces \(Q^*\) of \(P^*\). Moreover, \(X_{{{\mathrm{gen}}}}\) is a projective Calabi–Yau variety with at worst canonical singularities. Similarly, let \(X^*\) denote a family of projective Calabi–Yau varieties corresponding to the pair \((P^*,\mathcal {S}^*)\), for some polyhedral subdivision \(\mathcal {S}^*\) of \(P^*\).

**stringy invariant**\(E_{st}(V;u,w)\) of a complex variety

*V*with at worst log-terminal singularities in [6], such that if

*V*admits a crepant resolution \(V'\) then \(E_{st}(V;u,w) = E(V';u,w)\). In [15, Theorem 7.2], Borisov and Mavlyutov proved a result equivalent to the following formula for the projective complex variety \(V = X_{{{\mathrm{gen}}}}\):

*X*over \(\mathbb {K}\). Using the methods of [15], together with Theorem 1.5, yields the formula

*X*and \(X^*\) are either trivial degenerations or maximally degenerate, in the sense that all nonzero limit mixed Hodge numbers are of type (

*p*,

*p*), in which case one recovers a statement equivalent to the Batyrev–Borisov result above.

## 6 Intersection cohomology of schön subvarieties

In this section, we give a sum-over strata formula for the refined limit Hodge–Deligne polynomial of the intersection cohomology of the closure of a schön subvariety in certain projective toric varieties over a punctured curve *C*. This formula is analogous to a special case of the motivic formula obeyed by the usual Hodge–Deligne polynomial. It differs in that it only works for stratifications induced by the ambient toric variety and that it requires a weighting of terms by the *g*-polynomial to account for singularities along strata. By considering the case of schön hypersurfaces, we will give an alternative proof of Theorem 1.5 for families of schön hypersurfaces over a punctured curve.

We will let \(\mathbb {K}=\mathbb {C}(t)\) instead of the function field of germs of analytic functions on a punctured disK. We view varieties defined over \(\mathbb {K}\) as algebraic families of varieties over a curve *C* that has a distinguished puncture 0. All monodromy will be computed around this puncture.

### 6.1 Sum-over-strata formulas in intersection cohomology

In the proof [8] of their formula for the cohomology of a schön hypersurface of a toric variety, Batyrev and Borisov observe that the intersection cohomology of schön hypersurfaces in the projective toric variety associated with their Newton polytope obeys a sum-over-strata formula analogous to that of the cohomology of projective toric varieties. Cappell, Maxim, and Shaneson [18] who study what they call the ‘stratified multiplicative property of intersection cohomology’ prove a natural generalization of that observation. They study an intersection cohomology Euler characteristic (such as topological Euler characteristic, \(\chi _y\)-characteristic, or Hodge–Deligne polynomial), extend its definition to open strata and study how it behaves under a stratified fibration \(f:X\rightarrow Y\). One can generalize these results to give a sum-over-strata formula for the refined limit Hodge–Deligne polynomial of a family of schön subvarieties over a punctured disc.

We first establish our framework. We will use middle-perversity throughout. All stratification will be complex algebraic stratifications. The compactly supported intersection cohomology of a quasi-projective variety over \(\mathbb {C}\) has a mixed Hodge structure [53] and therefore one can define a Hodge–Deligne polynomial. For a quasi-projective variety \(X^\circ \) over a curve *C*, the intersection cohomology with compact supports of the family forms a mixed Hodge module by the work of Saito [52]. In an arbitrarly small disk around the puncture, we can suppose that this is an admissible variation of mixed Hodge structures. Therefore, we have a Hodge, weight, and monodromy weight filtration on the compactly supported intersection cohomology of a generic fiber, and as in Sect. 3.2, we can form the refined limit Hodge–Deligne polynomial and refined limit mixed Hodge numbers. The advantage of using intersection cohomology is that projective varieties carry a *pure* Hodge structure and that toric strata can be treated as if they are smoothly embedded once one uses a combinatorial correction term coming from the *g*-polynomial. Let \(X^\circ \) be a schön subvariety of \((\mathbb {K}^*)^n\). Let \(\Delta _P\) be the normal fan of a lattice polytope *P* such that the recession fan of \({{\mathrm{Trop}}}(X^\circ )\) is supported on \(\Delta _P\), i.e., the support of the recession fan is a union of cones in \(\Delta _P\). Such a polytope always exists by arguments using the Hilbert scheme [60]. In this case, we say \(\mathbb {P}(\Delta _P)\) is **adapted** to \(X^\circ \). If \(X^\circ \) is a schön hypersurface, it suffices to take *P* to be the Newton polytope of \(X^\circ \). We will let *X* be the closure of \(X^\circ \) in \(\mathbb {P}(\Delta _P)_\mathbb {K}\). We say that *X* is a **schön**, projective variety. Note that for \(X^\circ =(\mathbb {K}^*)^n\), we have \(X=\mathbb {P}(\Delta _P)_\mathbb {K}\). We may also study the case where \(X^\circ \) is the schön subvariety of \((\mathbb {C}^*)^n\). In this case, we say \(\mathbb {P}(\Delta _P)_\mathbb {K}\) is adapted if it is adapted to \(X^\circ \times _{\mathbb {C}}\mathbb {K}\).

We begin with the analog of the sum-over-strata formula analogous to the motivic formula for compactly supported cohomology. Please note that our convention for the *g*-polynomial differs from that of [8].

###
**Theorem 6.1**

###
*Proof*

The analogous formula for the Hodge–Deligne polynomial for varieties over \(\mathbb {C}\) is deduced from the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber [9] in [8, Corollary 3.17]. There it is stated for hypersurfaces, but the arguments also hold for the closure of schön subvarieties in adapted projective toric varieties. Work in a similar direction has been done by Cappell et al. [18] who study a ‘stratified multiplicative property’ which is proved for the Hodge–Deligne polynomial in intersection cohomology. Again, the result is an application of the decomposition theorem. One can derive the sum-over-strata formula from the stratified multiplicative property as follows: one takes a projective toric resolution of singularities \(f:\mathbb {P}({\Delta }_{\widetilde{P}})\rightarrow \mathbb {P}(\Delta _{P})\); there is an induced resolution of singularities of the closures of the schön subvariety, \(f:\widetilde{X}\rightarrow X\) to which one applies the stratified multiplicative property; and one then deduces the sum-over-strata formula from the analogous formula on \(\widetilde{X}\) where the intersection cohomology Hodge–Deligne polynomial reduces to the usual Hodge–Deligne polynomial which is known to be motivic.

To justify the formula for the refined limit Hodge–Deligne polynomial, we use the approach of [18]. Their results are stated for stratifications with simply connected strata, but they note that they only need the property that the local systems involved in the decomposition theorem have trivial monodromy along strata. This property is verified for schön hypersurfaces in [8, Cor 3.17]. The same proof holds for schön subvarieties. To obtain the result for the limit mixed Hodge structure coming from a family, it suffices to show that the isomorphism in the decomposition theorem respects the monodromy weight filtration.

*C*. Here, we will be concerned with the monodromy around 0. Write \(p:X\rightarrow C\). Take a toric resolution of singularities \(f:\mathbb {P}(\Delta _{\widetilde{P}})_C\rightarrow \mathbb {P}(\Delta _P)_C\), and let \(\widetilde{X}\) be the closure of \(X^\circ \) in \(\mathbb {P}(\Delta _{\widetilde{P}})\). By applying the decomposition theorem to the resolution of singularities \(f:\widetilde{X}\rightarrow X\), we have a non-canonical isomorphism,

*f*by \(X=\coprod _l S_l, 0\le l\le \dim X, \alpha _l:S_l\hookrightarrow X\) and the local system are given by \(L_{i,l}=\alpha _l^*\mathcal {H}^{-l}({}^p\mathcal {H}^i(f_*\mathbb {Q}_X[n]))\). In this case, the stratification coincides with that induced by the ambient toric variety and the local systems \(L_{i,l}\) are equal to those that occur in the decomposition theorem applied to \(f:\mathbb {P}(\Delta _{\widetilde{P}})_C\rightarrow \mathbb {P}(\Delta _P)\) and are therefore constant. Consequently, the cohomology sheaves of all terms in (13) give local families in a punctured disk around 0. Therefore, we may write down a monodromy operator and form the weight-monodromy filtration which is compatible with the isomorphism. The sheaf \(\mathbb {Q}_{\widetilde{X}}\) has the structure of a Hodge module, and hence by Saito’s theory, the derived pushforwards \(f_*\mathbb {Q}_{\widetilde{X}}\) and \((p\circ f)_*\mathbb {Q}_{\widetilde{X}}\) have the structure of mixed Hodge modules. Likewise, the relevant cohomology sheaves carry the structure of a mixed Hodge module [52], so their pushforwards do as well. See [35, Section 8.3.3] for an exposition. Consequently, the formula (13) is an isomorphism of mixed Hodge modules over

*C*. Therefore, over a small punctured disk about 0, we get an isomorphism of admissible variations of mixed Hodge structures. \(\square \)

Note that the sum in the above theorem only needs to be over cones of \(\Delta _P\) in the support of the recession fan of \({{\mathrm{Trop}}}(X^\circ )\) because for other cones \(\sigma , X^\circ _\sigma \) is empty and does not contribute.

### 6.2 Intersection cohomology of families of schön hypersurfaces of toric varieties

We will compute the refined limit Hodge–Deligne polynomial of schön, projective hypersurfaces using intersection cohomology. Our proof is inspired by that of [8] where one sums over strata in the stratification induced by the ambient toric variety and then constrains the intersection cohomology by Poincaré duality and the weak Lefschetz theorem.

*P*. We will need the following observations about the intersection cohomology of

*V*:

- (a)
The intersection cohomology of

*V*obeys Poincaré duality [30]. - (b)
The Hodge structure on \({{\mathrm{IH}}}^*(V)\) is pure [52].

- (c)
By the weak Lefschetz theorem, the Gysin map \({{\mathrm{IH}}}^{k}(V)\rightarrow {{\mathrm{IH}}}^{k+2}(\mathbb {P}(\Delta )_\mathbb {C})\) is a surjective map if \(k\ge \dim V\) and is an isomorphism if \(k>\dim V\). Moreover, it is a morphism of Hodge structures [30].

*C*that has a distinguished puncture such that \(\mathbb {P}(\Delta _P)_\mathbb {C}\) is adapted for each \(X_t\). By naturality, the monodromy around the puncture commutes with Poincaré duality and the Gysin map. We write \(X = X_{\mathbb {K}}\) for \(X_t\) considered as a subvariety of \(\mathbb {P}(\Delta _P)\) over \(\mathbb {K}\).

*X*. Because for each face

*Q*of

*P*, we have \(\mathbb {P}(\Delta _P)^\circ _Q=(\mathbb {K}^*)^{\dim Q}\) and \(E((\mathbb {P}(\Delta _P)^\circ _Q)_\infty ;u,v,w)=(uvw^2-1)^{\dim Q}\), we have from Theorem 6.1:

*f*is the

*toric h-polynomial*of

*P*which is well known to give the dimensions of the topological intersection cohomology of the toric variety \(\mathbb {P}\) (see, e.g., [16, 26]).

*X*that we know must exist by the weak hyperplane theorem and Poincaré duality. Because all the relevant cohomology is of type (

*p*,

*p*), the action of monodromy must be trivial by Remark 3.10. Then from the above expression for \(E_{{{\mathrm{int}}}}(\mathbb {P}_\infty ;u,v,w)\) and Definition 4.3, one may deduce that

*F*,

*W*) is pure and concentrated in

*W*-degree \(\dim P-1\).

*X*denote the closure of \(X^\circ \) in \(\mathbb {P}(\Delta _P)_\mathbb {K}\). Then Theorem 1.5, which we proved in Sect. 5.4, states that

###
*Proof*

*Q*of

*P*, (15) implies

*Q*to obtain

We now give a new proof of (16) in the following equivalent form:

###
**Corollary 6.3**

*X*denote the closure of \(X^\circ \) in \(\mathbb {P}(\Delta _P)_\mathbb {K}\). Then the refined limit Hodge–Deligne polynomial associated with the intersection cohomology of

*X*is given by

###
*Proof*

In Sect. 5.4, we proved that if (15) holds when specialized to \(\mathbb {Z}[u,w]\) then (15) holds when specialized to \(\mathbb {Z}[u,v]\). Hence, we are left with the vertical arrows of the diagram. Because primitive cohomology is concentrated in *W*-degree equal to \(\dim P-1\), it is clear that if (17) holds for \(\mathbb {Z}[u]\), then it holds for \(\mathbb {Z}[u,w]\). Similarly, if (17) holds for \(\mathbb {Z}[u,v]\), then it holds for \(\mathbb {Z}[u,v,w]\). \(\square \)

## Declarations

### Acknowledgements

We would like to thank Mark Andrea de Cataldo, Laurentiu Maxim, and Gregory Pearlstein for valuable discussions. Particular thanks should go to Patrick Brosnan who suggested the relevant mixed Hodge theory framework and to Benjamin Nill who introduced the authors to Stanley’s subdivision theory and advocated its importance in establishing the relevant combinatorial theory.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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