Formulas for monodromy

Before introducing the main results of this paper, we first present an application. Let \(f(x_1, \ldots , x_n) \in \mathbb {C}[x_1, \ldots , x_n]\) be a complex polynomial with no constant term, and consider the induced map \(f: \mathbb {C}^n \rightarrow \mathbb {C}\). One would like to understand the singularity of \(f^{-1}(0) \subseteq \mathbb {C}^n\) at the origin (we assume this is an isolated singularity). A classical invariant used to distinguish the singularities arising from different polynomials is the action of monodromy on the cohomology of the associated Milnor fiber. In simple terms, this means associating to f a finite, square complex matrix \(M_f\) that is well defined up to change of basis. We assume that f is ‘general’ in the sense that it is nondegenerate with respect to its Newton polyhedron \(\Gamma _+(f)\), and also that it is convenient (see Sect. 6.3). The most famous result was proved by Varchenko in 1976 [61], who gave an explicit combinatorial formula for the eigenvalues (with multiplicity) of \(M_f\), involving alternating signs of volumes of polytopes associated with \(\Gamma _+(f)\). Unfortunately, since \(M_f\) is rarely diagonalizable, it is not, in general, determined by its eigenvalues. We are left with the problem of giving combinatorial formulas for the numbers \(J_{k,\alpha }\) of Jordan blocks of \(M_f\) of size k with eigenvalue \(\alpha \). Using tropical geometry, together with weighted Ehrhart theory, which involves the enumerative combinatorics of lattice points in dilates of polytopes, we settle this question:

We refer the reader to Corollary 6.22 for the specific formulas. By a ‘nonnegative’ formula, we mean that each \(J_{k,\alpha }\) is expressed as a sum of nonnegative integers. By summing over k, we obtain a nonnegative formula for the multiplicities of the eigenvalues of \(M_f\), which is equivalent to Varchenko’s formula (Remark 6.19). Using different techniques, special cases of formulas for the \(J_{k,\alpha }\) have previously been proved by Matsui and Takeuchi [40] and van Doorn and Steenbrink [60], and an algorithm to compute the numbers \(J_{k,\alpha }\) involving simplicial resolutions of toric varieties was proved by Matsui and Takeuchi in [40], extending work of Danilov [13] and Tanabé [58]. We note that algorithms in a more general setting are given by Schulze in [49].

The theorem above is a corollary of more general results that we will outline in the remainder of the introduction. Given a family of varieties \(f:X\rightarrow \mathbb {D}^*\) over a complex disk, Denef and Loeser [17] associate an invariant \(\psi _X\) called the motivic nearby fiber, which contains information about the variation of Hodge structures associated with the degeneration. In particular, if we fix a nonzero fiber \(X_{{{\mathrm{gen}}}} := f^{-1}(t)\) for some sufficiently small \(t \in \mathbb {D}^*\), then \(\psi _X\) encodes information on the induced monodromy map \(T: H^m_c(X_{{{\mathrm{gen}}}}) \rightarrow H^m_c(X_{{{\mathrm{gen}}}})\), which is a linear operator on the complex cohomology with compact supports of \(X_{{{\mathrm{gen}}}}\) and is quasi-unipotent, i.e., \(T = T_sT_u\), where \(T_s\) and \(T_u\) commute, \(T_s\) is semi-simple and has finite order, and \(T_u\) is unipotent. In [30], Eric Katz and the author showed that one can use tropical geometry to concretely compute a specialization of \(\psi _X\) that is invariant under base change, and hence encodes information about \(T_u\) but not \(T_s\). Using new combinatorics developed in [29], in the case of families of schön complex hypersurfaces of tori, explicit combinatorial formulas were then deduced for the refined limit mixed Hodge numbers, which, in particular, determine the Jordan block structure of the action of \(T_u\) on the graded pieces (with respect to the Deligne weight filtration) of \(H^m_c(X_{{{\mathrm{gen}}}})\). Here schön is a ‘generic’ condition introduced by Tevelev [59], generalizing the notion of nondegeneracy of a hypersurface of a complex torus [33]. Equivalent formulas were also given for the action of \(T_u\) on the intersection cohomology groups of a family of schön hypersurfaces of a projective toric variety.

The goal of this note is to extend the above results to the full generality of the motivic nearby fiber \(\psi _X\) and monodromy operator T. Our first main result is to show that the motivic nearby fiber may be computed using tropical geometry (see Sect. 3). A key point is that since there is a range of available software implementing the main algorithms in tropical geometry [26, 27], our main formula Theorem 3.2 can be computed in practice.

Weighted Ehrhart theory was introduced by the author in [52] both in order to extend and reprove many results in Ehrhart theory, the study of enumeration of lattice points in polytopes, and in order to give explicit computations of motivic integrals on toric stacks [53]. Roughly speaking, one associates with every lattice point v a ‘weight’ w(v) in \(\mathbb {Q}/\mathbb {Z}\) and then attempts to enumerate lattice points in polytopes keeping track of the associated weights. Extending work in [29], in Sect. 4.4, we associate new Ehrhart-theoretic invariants to a pair \((P,\nu )\), where P is a lattice polytope and \(\nu \) is the convex graph of an integral height function on P. In particular, we introduce the weighted refined limit mixed \(h^*\) -polynomial \(h^*(P,\nu ;u,v,w) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\). Roughly speaking, our second main result states that if \(X^\circ \) is a family of schön complex hypersurfaces of tori, then the associated equivariant refined limit mixed Hodge numbers are precisely the coefficients of \(h^*(P,\nu ;u,v,w)\) (Theorem 5.7, Corollary 5.12), where \((P,\nu )\) is the Newton polytope and associated convex graph associated with \(X^\circ \). In particular, this gives combinatorial formulas for the Jordan block structure of the action of T on the graded pieces (with respect to the Deligne weight filtration) of \(H^m_c(X^\circ _{{{\mathrm{gen}}}})\). Equivalent formulas are also given for the equivariant refined limit mixed Hodge numbers associated with the intersection cohomology groups of a family of schön hypersurfaces of a projective toric variety (Theorem 5.9, Corollary 5.12). In Example 5.13, we express these invariants in a special case in terms of dimensions of orbifold cohomology [11, 12] of a toric stack [4]. It would be interesting to have an explanation of this fact involving ‘mirror symmetry’.

As a special case of the above result, in Theorem 5.11, we give a formula for the equivariant Hodge–Deligne polynomial of a schön complex hypersurface of a torus invariant under the action of an element of the torus of finite order. An algorithm to determine this polynomial had previously been given by Matsui and Takeuchi in [39], generalizing an algorithm of Danilov and Khovanskiĭ in the nonequivariant setting [14]. Moreover, the above formula reduces to a formula of Borisov and Mavlyutov [5] in the nonequivariant setting, which itself is a simplification of a formula of Batyrev and Borisov [3].

We next present some applications to the monodromy of complex polynomials. Let \(f(x_1, \ldots , x_n) \in \mathbb {C}[x_1, \ldots , x_n]\) be a complex polynomial. It is well known that there exists a minimal finite subset \(B_f \subseteq \mathbb {C}\), such that \(f: \mathbb {C}^n \rightarrow \mathbb {C}\) is a locally trivial fibration away from \(B_f\). Then monodromy defines an action of the fundamental group \(\pi _1(\mathbb {C}\smallsetminus B_f)\) on the cohomology of a fixed generic fiber of f. On the one hand, we are interested in the monodromy action induced by moving anti-clockwise around a small loop around an element of \(b \in B_f\). After translating f by a constant, we may assume that \(b = 0\), and then the resulting action is called monodromy at 0. On the other hand, choose \(R > 0\) sufficiently large such that \(B_f\) is strictly contained in \(\{ z \in \mathbb {C}\mid |z| = R \}\). Then the monodromy action on cohomology induced by moving clockwise around such a loop is called the monodromy at infinity of f, and a fundamental result of Dimca and Némethi [18] states that monodromy at infinity essentially determines the monodromy action of \(\pi _1(\mathbb {C}\smallsetminus B_f)\). We assume that f is convenient in the sense that its Newton polytope has nonzero intersection with each ray through a coordinate vector [34]. Then under certain schönness (also called ‘nondegeneracy’) conditions, we give explicit combinatorial formulas that essentially completely describe both monodromy at 0 (Example 6.7) and monodromy at infinity (Example 6.8). In particular, we deduce ‘nonnegative’ combinatorial formulas for both the equivariant limit mixed Hodge numbers associated with monodromy at infinity (Corollary 6.13) and the Jordan block structure of monodromy at infinity (Corollary 6.14). Algorithms to compute the latter invariants were given by Matsui and Takeuchi in [39]. Also, these formulas specialize to known formulas for the spectrum at infinity of f [39, Theorem 5.11] and zeta function at infinity of f [35]. We note that some of the above results may be extended without the assumption that f is convenient (see Remark 6.1). We also present analogous ‘local’ results for the Milnor fiber of a polynomial f with an isolated singularity at the origin, including combinatorial formulas for both the corresponding equivariant limit mixed Hodge numbers, and, as outlined at the beginning of this introduction, the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber.

Finally, we mention some possible generalizations of the above results that we expect will follow by similar methods. Firstly, using the ‘Cayley trick’ of Danilov and Khovanskiĭ [14, Section 6], one may extend our results on hypersurfaces to the case of complete intersections. In particular, in the case of the monodromy at infinity and monodromy of the cohomology of the Milnor fiber, one may obtain explicit formulas that extend algorithms given by Esterov and Takeuchi in [23]. Secondly, in the case when f is not convenient, one may use the results above to obtain formulas extending algorithms of Takeuchi and Tibar in [57]. Lastly, analogous formulas to those in Sect. 6.3 for families of hypersurfaces of an affine toric variety (rather than simply affine space) may be obtained using the setup and results of Steenbrink in [56].

In this section we briefly recall the notion of the motivic nearby fiber and its associated geometry. We refer the reader to Bittner [7], Peters [43] and Peters and Steenbrink [44] for details. A similar exposition is given in the nonequivariant setting in [30, Section 3].

2.2 The motivic nearby fiber

Fix \(\mathbb {K}= \mathbb {C}(t)\). Then the motivic nearby fiber is a ring homomorphism

$$\begin{aligned}&\psi : K_0({{\mathrm{Var}}}_\mathbb {K}) \rightarrow K_0^{\widehat{\mu } }({{\mathrm{Var}}}_\mathbb {C}),\\&\quad [X] \mapsto \psi _X. \end{aligned}$$

A result of Bittner [8] implies that \(K_0({{\mathrm{Var}}}_\mathbb {K})\) is generated by the classes of smooth, proper varieties. In particular, the description below determines \(\psi \).

We follow the description of the motivic nearby fiber in [56]. A variety X over \(\mathbb {K}\) is naturally interpreted as a complex variety with a morphism \(f:X\rightarrow \mathbb {C}^* \smallsetminus \{ b_1, \ldots , b_r \}\), for some points \(b_1,\ldots , b_r\). Assume that X is smooth and extend X to a variety \(X'\) with flat morphism \(f: X' \rightarrow \mathbb {A}^1 \smallsetminus \{ b_1, \ldots , b_r \}\). After resolving singularities, we may assume that \(X'\) is smooth and that the central fiber \(f^{-1}(0) = \sum _i m_i D_i\) is a simple normal crossings divisor with irreducible components \(D_1,\ldots ,D_r\), with multiplicities \(m_1,\ldots , m_r\), respectively. For each nonempty subset \(I \subseteq \{ 1, \ldots , r \}\), let \(D_I^\circ = \cap _{i \in I} D_i \smallsetminus \cup _{j \notin I} D_j\) and \(m_I= \gcd (m_i \mid i \in I)\). Restrict f to a small complex disk \(\mathbb {D}\) about the origin and let m be a common multiple of \(m_1,\ldots , m_r\). Pullback \(X'\) via the map \(\mathbb {D}\rightarrow \mathbb {D}\), \(t \mapsto t^m\) and normalize to obtain a commutative diagram

The action of \(\mu _m\) on \(\mathbb {D}\), with \(\zeta \in \mu _m\) acting via \(t \mapsto \zeta t\), extends to an action on \(\widetilde{X'}\), and hence a good \(\widehat{\mu }\)-action on the central fiber. Let \(\widetilde{D_I^\circ }\) be proper transform of \(D_I^\circ \) via \(\rho \). Then \(\rho : \widetilde{D_I^\circ } \rightarrow D_I^\circ \) is a \(\mu _{m_I}\)-covering. The motivic nearby fiber \(\psi _X \in K_0^{\widehat{\mu } }({{\mathrm{Var}}}_\mathbb {C})\) of X is given by

$$\begin{aligned} \psi _X = \sum _{\emptyset \ne I \subseteq \{ 1, \ldots , r \}} [\widetilde{D}_I^\circ \circlearrowleft \widehat{\mu }] (1 - {\mathbb {L}})^{|I| - 1}. \end{aligned}$$

(2)

2.3 Motivic invariants

The importance of the motivic nearby fiber comes from its specializations. In this section, we will explain the following commutative diagram, which will be crucial in the rest of the paper, where the first vertical map is the motivic nearby fiber:

Throughout the paper, we will identify the group \(\mathbb {Q}/\mathbb {Z}\) with the group \({\mathbb {S}}^1_\mathbb {Q}\) of rational points on the circle \(\{ z \in \mathbb {C}\mid |z| = 1 \}\), sending \([k] \in \mathbb {Q}/\mathbb {Z}\) to \(\alpha = e^{2 \pi \sqrt{-1}k} \in {\mathbb {S}}^1_\mathbb {Q}\). We will consider the group algebra \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\).

Remark 2.3

Consider the \(\mathbb {Z}\)-algebra involution on \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) defined by setting \(\overline{[k]} = [-k]\), corresponding to complex conjugation on the circle. This extends coefficient-wise to an involution on a polynomial ring with coefficients in \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) that we will refer to as conjugation.

Remark 2.4

Consider the natural \(\mathbb {Z}\)-algebra homomorphism \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}] \rightarrow \mathbb {Z}, [k] \mapsto 1\). One may apply this homomorphism below to get invariants with coefficients in \(\mathbb {Z}\) rather than \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\). The corresponding diagram of invariants above over \(\mathbb {Z}\) is explained in detail in [30, Section 3].

Consider a complex vector space B that admits a mixed Hodge structure [44] with corresponding vector space decomposition

$$\begin{aligned} B \cong \bigoplus _{p,q} H^{p,q}(B). \end{aligned}$$

Suppose the \(\mu _n\) acts linearly on B, preserving the mixed Hodge structure. With the convention above, for \(\alpha \in \mathbb {Q}/\mathbb {Z}\), we write \(h^{p,q}(B)_\alpha \) for the dimension of the \(\alpha \)-eigenspace of \(H^{p,q}(B)\), and write \(h^{p,q}(B,\widehat{\mu }) := \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{p,q}(B)_\alpha \alpha \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\). For a sequence of such representations \(B_\bullet = \{ B_m \mid m \ge 0 \}\), set \(e^{p,q}(B_\bullet ,\widehat{\mu } ) = \sum _{m} (-1)^m h^{p,q}(B_m, \widehat{\mu }) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\). Then the equivariant Hodge polynomial of \(B_\bullet \) is defined by

$$\begin{aligned} E(B_\bullet ) = E(B_\bullet ; u,v) = \sum _{p,q} e^{p,q}(B_\bullet ,\widehat{\mu }) u^p v^q \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v]. \end{aligned}$$

In [15], Deligne proved that the \(m^{\text {th}}\) 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. Given a good action of \(\widehat{\mu }\) on V, we may set \(B_m = H^{p,q}(H^m_c (V))\) above. The corresponding equivariant Hodge polynomial is denoted \(E_{\widehat{\mu }}(V;u,v) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v]\), and is called the equivariant Hodge–Deligne polynomial of V. The invariants \(h^{p,q}(H^m_c (V))_\alpha \) are called the equivariant mixed Hodge numbers of V. We have a ring homomorphism called the equivariant Hodge–Deligne map:

$$\begin{aligned} E_{\widehat{\mu }}: K_0^{\widehat{\mu }}({{\mathrm{Var}}}_k) \rightarrow \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v], [V] \mapsto E_{\widehat{\mu }}(V;u,v). \end{aligned}$$

As in Remark 2.4, we will also consider the corresponding invariants over \(\mathbb {Z}\). In this case, we have the usual mixed Hodge numbers \(h^{p,q}(H^m_c (V)) = \sum _{\alpha } h^{p,q}(H^m_c (V))_\alpha \) and corresponding Hodge–Deligne polynomial \(E(V;u,v) \in \mathbb {Z}[u,v]\).

Remark 2.5

More generally, for any finite group G acting algebraically on a complex variety V, we obtain a linear action of G on \(H^{p,q}(H^m_c (V))\), and one can form the equivariant Hodge–Deligne polynomial

$$\begin{aligned} E_G(V;u,v) = \sum _{p,q} \sum _m (-1)^m H^{p,q}(H^m_c (V)) u^p v^q \in R(G)[u,v], \end{aligned}$$

where R(G) is the complex representation ring of G. This induces a ring homomorphism \(\widetilde{K}_0^{G}({{\mathrm{Var}}}_\mathbb {C}) \rightarrow R(G)[u,v]\). This invariant was introduced and studied by the author in [54], generalizing the notion of weight polynomial due to Dimca and Lehrer [19], and the notion of equivariant \(\chi _y\)-genus due to Cappell, Maxim and Shaneson [10]. In our case, \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) may be viewed as the direct limit associated with the restriction maps \(R(\mu _{n}) \rightarrow R(\mu _{nd})\) induced by the epimorphisms \(\mu _{nd} \rightarrow \mu _n, \zeta \mapsto \zeta ^d\).

Recall that a variety X over \(\mathbb {K}\) is naturally interpreted as a complex variety with a morphism \(f:X\rightarrow \mathbb {C}^* \smallsetminus \{ b_1, \ldots , b_r \}\), for some points \(b_1,\ldots , b_r\). Restricting X to a family over a sufficiently small punctured complex disk centered at the origin, and fixing a fiber \(X_{{{\mathrm{gen}}}} := f^{-1}(t)\) and counter-clockwise orientation around the disk, there exists a quasi-unipotent monodromy map \(T = T_sT_u: H^m_c(X_{{{\mathrm{gen}}}}; \mathbb {C}) \rightarrow H^m_c(X_{{{\mathrm{gen}}}}; \mathbb {C})\), where \(T_s\) is semi-simple and \(T_u\) is unipotent. Then the cohomology groups \(H_c^m(X_{{{\mathrm{gen}}}})\) admit a weight filtration \(M_\bullet \) called the monodromy weight filtration, and \(T_s\) acts preserving the filtrations \((F^\bullet ,M_\bullet , W_\bullet )\). We will write \(H_c^m(X_{\infty })\) to denote \(H_c^m(X_{{{\mathrm{gen}}}})\) with the filtrations \((F^\bullet ,M_\bullet ,W_\bullet )\). The filtrations \((F^\bullet ,M_\bullet )\) induce a mixed Hodge structure on \(H_c^m (X_{\infty })\), and the nilpotent operator \(N = \log T_u\) is a morphism of mixed Hodge structures of type \((-1,-1)\). The corresponding invariants \(h^{p,q}(H_c^m (X_{\infty }))_\alpha \) are called the equivariant limit mixed Hodge numbers, and the corresponding equivariant Hodge polynomial \(E(X_\infty , \widehat{\mu };u,v) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v]\) is called equivariant limit Hodge–Deligne polynomial. A deep result of Denef and Loeser states that this is a specialization of the motivic nearby fiber in the sense that \(E(X_\infty ,\widehat{\mu };u,v) = E_{\widehat{\mu }}(\psi _X)\). As in Remark 2.4, we may consider the corresponding invariants over \(\mathbb {Z}\), i.e., the limit mixed Hodge numbers and limit Hodge–Deligne polynomial.

Before proceeding, we recall the following standard linear algebra construction.

Definition 2.6

Let A be a nilpotent linear operator on a finite-dimensional vector space V such that \(A^{r + 1} = 0\). Then the A -weight filtration centered at r is the increasing filtration \(\{ V_\bullet \}\) of V by subspaces

$$\begin{aligned} 0 \subseteq V_0 \subseteq V_1 \subseteq \cdots \subseteq V_{2r} = V, \end{aligned}$$

uniquely determined by

1. (1)

   \(A( V_k ) \subseteq V_{k - 2}\),

    
2. (2)

   the induced map \(A^k: Gr_{r + k} V \rightarrow Gr_{r - k} V\) is an isomorphism,

    

for any nonnegative integer k. Here we set \(V_k = 0\) for \(k < 0\). The filtration encodes the Jordan block structure of A. Explicitly, for \(1 \le k \le r + 1\), the number of Jordan blocks of size k equals

$$\begin{aligned} \dim Gr_{r + 1 - k} V - \dim Gr_{r - 1 - k} V. \end{aligned}$$

The monodromy weight filtration \(M_\bullet \) has the following important geometric property: for every nonnegative integer r, the induced filtration \(M(r)_\bullet \) on \(Gr_r^W H_c^m (X_{\infty })\) coincides with the N(r)-weight filtration centered at r, where N(r) is the nilpotent operator induced by \(N = \log T_u\). It follows from Definition 2.6 that the monodromy weight filtration, together with the action of \(T_s\), encodes the Jordan block structure of the monodromy operator T acting on \(Gr_r^W H_c^m (X_{\infty })\). Moreover, \((F^\bullet ,M_\bullet )\) induces a mixed Hodge structure on \(Gr_r^W H_c^m (X_{\infty })\) that is invariant under \(T_s\), and N(r) is a morphism of mixed Hodge structures of type \((-1,-1)\). The associated equivariant mixed Hodge numbers

$$\begin{aligned} h^{p,q}(Gr_r^W H_c^m (X_{\infty }))_\alpha = ( Gr_F^p Gr_{p + q}^M Gr_r^W H_c^m (X_{\infty }))_\alpha \end{aligned}$$

are called the equivariant refined limit mixed Hodge numbers, and the associated equivariant Hodge polynomial is the coefficient of \(w^r\) in a polynomial \(E(X_\infty , \widehat{\mu };u,v,w) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\) called the equivariant refined limit Hodge–Deligne polynomial. As in Remark 2.4, we have corresponding invariants over \(\mathbb {Z}\) that were introduced in [30, Section 3], to where we refer the reader for further details. We have a ring homomorphism

$$\begin{aligned}&\overline{E}_{\widehat{\mu }}: K_0({{\mathrm{Var}}}_\mathbb {K}) \rightarrow \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w],\\&\quad \quad [X] \mapsto E(X_\infty ,\widehat{\mu };u,v,w). \end{aligned}$$

It follows from the definitions that we recover the equivariant limit mixed Hodge numbers and the usual mixed Hodge numbers of \(X_{{{\mathrm{gen}}}}\) via the specializations:

$$\begin{aligned} h^{p,q}(H_c^m(X_\infty ))_\alpha= & {} \sum _r h^{p,q,r}(H_c^m(X_{\infty }))_\alpha , \nonumber \\ h^{p,r-p}(H_c^m(X_{{{\mathrm{gen}}}}))= & {} \sum _q \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{p,q,r}(H_c^m(X_{\infty }))_\alpha . \end{aligned}$$

(3)

Correspondingly, we have \(E(X_\infty , \widehat{\mu };u,v,1) = E(X_\infty , \widehat{\mu };u,v)\), and \(E(X_\infty , \widehat{\mu };uw^{-1},1,w)\) restricts to the Hodge–Deligne polynomial \(E(X_{{{\mathrm{gen}}}}; u,w) \in \mathbb {Z}[u,w]\). The further specializations \(E(X_{{{\mathrm{gen}}}}; u,1)\) and \(E(X_{{{\mathrm{gen}}}}; 1,1)\) are the \(\chi _y\)-characteristic and Euler characteristic of \(X_{{{\mathrm{gen}}}}\), respectively.

Remark 2.7

With the notation above, since N(r) is a morphism of mixed Hodge structures of type \((-1,-1)\), the isomorphisms (2) in Definition 2.6 imply that for \(0 \le j \le r\), the sequence \(\{ h^{i+ j,i,r}(H_c^m(X_{\infty }))_\alpha \mid 0 \le i \le r -j \}\) is symmetric and unimodal.

Remark 2.8

Since \(H^{p,q}(H_c^m(X_\infty ))\) and \(H^{q,p}(H_c^m(X_\infty ))\) are conjugate, and by Remark 2.7, the refined limit mixed Hodge numbers satisfy the following symmetries:

$$\begin{aligned} h^{p,q,r}(H_c^m(X_{\infty }))_\alpha= & {} h^{r-q,r-p,r}(H_c^m(X_{\infty }))_\alpha = h^{q,p,r}(H_c^m(X_{\infty }))_{\alpha ^{-1}}\\= & {} h^{r-p,r-q,r}(H_c^m(X_{\infty }))_{\alpha ^{-1}} . \end{aligned}$$

Hence the equivariant refined limit Hodge–Deligne polynomial satisfies the following symmetries with respect to the conjugation action on \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\) described in Remark 2.3:

$$\begin{aligned} E(X_{\infty }, \widehat{\mu };u,v,w)= & {} \overline{E(X_{\infty }, \widehat{\mu };v,u,w)},\\ E(X_{\infty }, \widehat{\mu };u,v,w)= & {} \overline{E(X_{\infty }, \widehat{\mu };u^{-1},v^{-1},uvw)}. \end{aligned}$$

Example 2.9

If 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 \(E(X_\infty ,\widehat{\mu };u,v,w) = E(V;uw,vw)\).

Example 2.10

If 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, together with the action of \(T_s\), encodes the Jordan block decomposition of T. In this case,

$$\begin{aligned} E(X_\infty ,\widehat{\mu };u,v,w) = \sum _{p,q,m} (-1)^m h^{p,q}(H^m(X_{\infty }))_\alpha u^p v^q w^m \alpha . \end{aligned}$$

In this section we prove our formula for the motivic nearby fiber. The nonequivariant version of this result is [30, Theorem 1.2]. We continue with the notation of the previous section. In particular, \(\mathbb {K}= \mathbb {C}(t)\).

A variety X over \(\mathbb {K}\) is very affine if it admits a closed embedding \(X \subseteq (\mathbb {K}^*)^n\) for some n. Tevelev introduced the notion of a schön, very affine variety. A closed subvariety \(X^\circ \subseteq (\mathbb {K}^*)^n\) is schön if \({{\mathrm{in}}}_w X^\circ \subseteq (\mathbb {C}^*)^n\) is a smooth (reduced) subvariety for every \(w \in \mathbb {R}^n\) [25, Prop 3.9]. A very affine variety X over \(\mathbb {K}\) is schön if it admits a schön closed embedding \(X \subseteq (\mathbb {K}^*)^n\) for some n (cf. [37, Lemma 2.11]). Note that a complex variety V may be viewed via base change as a variety over \(\mathbb {K}\), and hence we may consider the same notions. The definition of schön for a hypersurface of a complex torus agrees with the notion of nondegeneracy with respect to the Newton polytope [33].

Proof

We first recall the relevant geometric setting, as described in [25, Sections 1,2]. We refer the reader to [24] for the appropriate background on toric geometry.

Let \(N = \mathbb {Z}^n\), and for each cell \(\gamma \in \Sigma \), let \(C_\gamma \) denote the cone generated by \(\gamma \times \{ 1\}\) in \(N_\mathbb {R}\times \mathbb {R}\). The cones \(C_\gamma \) form a fan \(\widetilde{\Sigma }\) in \(N_\mathbb {R}\times \mathbb {R}\), and the recession fan \(\Delta \) is defined to be the intersection of \(\widetilde{\Sigma }\) with \(N_\mathbb {R}\times \{ 0\}\). Projection \({{\mathrm{pr}}}: N_\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) onto the second coordinate induces a morphism of fans \(\widetilde{\Sigma } \rightarrow \mathbb {R}_{\ge 0}\). Fix a positive integer m such that \(m\Sigma \) has a lattice polyhedral structure, and let \(\widetilde{\mathbb {P}}\) and \(\mathbb {P}\) denote the complex toric varieties corresponding to the fan \(\widetilde{\Sigma }\) with respect to the lattices \(N \times m\mathbb {Z}\) and \(N \times \mathbb {Z}\), respectively. We have an induced commutative diagram corresponding to pulling back and normalizing \(\mathbb {P}\):

where the vertical arrows are flat. After removing the closed torus invariant divisors corresponding to nonzero cones in \(\Delta \), the nonzero fibers of \(\widetilde{\mathbb {P}}\) and \(\mathbb {P}\) are tori and the central fibers have decompositions into disjoint unions of tori:

$$\begin{aligned} \widetilde{\mathbb {P}}^\circ = \bigcup _{\begin{array}{c} \gamma \in \Sigma \\ \gamma {\text {bounded}} \end{array}} \widetilde{U}_\gamma , \; \; \; \mathbb {P}^\circ = \bigcup _{\begin{array}{c} \gamma \in \Sigma \\ \gamma {\text {bounded}} \end{array}} U_\gamma . \end{aligned}$$

Let \(\pi _\gamma : \widetilde{U}_\gamma \rightarrow U_\gamma \) denote the morphism of tori induced by \(\pi \). This has the following explicit description. Let \(L_\gamma '\) denote the linear span of \(C_\gamma \), and let \(L_\gamma = L_\gamma ' \cap (N_\mathbb {R}\times \{ 0 \}) \subseteq N_\mathbb {R}\). That is, \(L_\gamma \) is the translate of the affine span of \(\gamma \) to the origin in \(N_\mathbb {R}\). Fix \(m_\gamma \in \mathbb {Z}_{>0}\) such that \({{\mathrm{pr}}}( (N \times \mathbb {Z}) \cap L_\gamma ') = m_\gamma \mathbb {Z}\). Let \(N_\gamma = N / N \cap L_\gamma \) with distinguished rational point \([\gamma ] \in (N_\gamma )_\mathbb {Q}\). Then projection along \(L_\gamma '\) induces an isomorphism \(N_\mathbb {Q}\times \mathbb {Q}/ (N_\mathbb {Q}\times \mathbb {Q}) \cap L_\gamma ' \rightarrow (N_\gamma )_\mathbb {Q}\) that sends \((0,1) = ([-\gamma ],0) + ([\gamma ],1) \mapsto [-\gamma ]\). This isomorphism induces the vertical maps in the following commutative diagrams:

Applying the functors \({{\mathrm{Hom}}}(\underline{,} \mathbb {Z})\) and then \({{\mathrm{Hom}}}(\underline{}, \mathbb {C}^*)\) induces

$$\begin{aligned} 1 \rightarrow \mu _{m_\gamma } \rightarrow \widetilde{U}_\gamma \xrightarrow {\pi _\gamma } U_\gamma \rightarrow 1. \end{aligned}$$

Letting \(M = {{\mathrm{Hom}}}(N, \mathbb {Z})\) and \(M_\gamma = {{\mathrm{Hom}}}(N_\gamma ,\mathbb {Z})\), we identify \(\widetilde{U}_\gamma = {{\mathrm{Hom}}}(M_\gamma , \mathbb {C}^*)\). Then the above sequence says that the rational point \([-\gamma ] \in (N_\gamma )_\mathbb {Q}\) corresponds to an element \(\exp (2\pi \sqrt{-1} [-\gamma ]) \in \widetilde{U}_\gamma \) of order \(m_\gamma \) such that \(U_\gamma \) is the quotient of \(\widetilde{U}_\gamma \) by the action of multiplication by \(\exp (2\pi \sqrt{-1} [-\gamma ])\). In what follows, we let \(T = {{\mathrm{Hom}}}(M, \mathbb {C}^*)\) and consider the multiplication map \(T \times \widetilde{U}_\gamma \rightarrow \widetilde{U}_\gamma \). We also fix the distinguished element \(p \in \widetilde{U}_\gamma = {{\mathrm{Hom}}}(M_\gamma , \mathbb {C}^*)\) corresponding to the constant map 1.

Recall that \(X^\circ \) may viewed as a family of complex varieties over \(\mathbb {C}^* \smallsetminus \{ b_1, \ldots , b_r \}\), for some points \(b_1,\ldots , b_r\). Let \(\mathbb {D}\) be a sufficiently small complex disk centered at the origin, and restrict \(X^\circ \) to a family over the punctured disk \(\mathbb {D}^*\), and \(\widetilde{\mathbb {P}}\) and \(\mathbb {P}\) to families over \(\mathbb {D}\). Taking the closure \({\mathcal {X}}\) of \(X^\circ \) in \(\mathbb {P}\) induces a commutative diagram:

where for each bounded cell \(\gamma \in \Sigma \), we have a corresponding \(\mu _{m_\gamma }\)-covering \(\widetilde{V}^\circ _\gamma := \widetilde{{\mathcal {X}}} \cap \widetilde{U}_\gamma \rightarrow V^\circ _\gamma := {\mathcal {X}} \cap U_\gamma \) corresponding to the action of multiplication by \(\exp (2\pi \sqrt{-1} [-\gamma ])\). As above, consider the induced multiplication map \(m: T \times \widetilde{V}^\circ _\gamma \rightarrow \widetilde{U}_\gamma \) and consider the fiber \(m^{-1}(p)\). By [25, p. 8], projection onto T identifies \(m^{-1}(p)\) with \({{\mathrm{in}}}_\gamma X^\circ \), while projection onto \(\widetilde{V}^\circ _\gamma \) identifies \(m^{-1}(p)\) with \(T_\gamma \times \widetilde{V}^\circ _\gamma \), where \(T_\gamma \) is the subtorus that fixes \(U_\gamma \) pointwise. Fix a rational point \(w \in N_\mathbb {Q}\) in the relative interior of \(\gamma \), and consider the corresponding point \(\exp (2\pi \sqrt{-1} w) \in T\) of finite order. Then multiplication by \((\exp ( 2\pi \sqrt{-1} w), \exp (2\pi \sqrt{-1} [-\gamma ])\) preserves \(m^{-1}(p)\), and, using Example 2.2, we conclude that

$$\begin{aligned}{}[{{\mathrm{in}}}_\gamma X^\circ \circlearrowleft \widehat{\mu }] = [ \widetilde{V}^\circ _\gamma \circlearrowleft \widehat{\mu }] (\mathbb {L}- 1)^{\dim \gamma }, \end{aligned}$$

(5)

where \(\widehat{\mu }\) acts via multiplication by \(\exp (2\pi \sqrt{-1} w)\) and \(\exp (2\pi \sqrt{-1} [-\gamma ])\), respectively. In particular, this shows that the left-hand side is independent of the choice of w in the relative interior of \(\gamma \).

We now complete the proof. We first verify that the right-hand side of the statement in the theorem is independent of the choice of polyhedral structure \(\Sigma \). Indeed, if \(\Sigma '\) is a polyhedral structure refining \(\Sigma \), then, using the independence of the choice of w above

$$\begin{aligned}&\sum _{\begin{array}{c} \gamma ' \in \Sigma ' \\ \gamma {\text {bounded}} \end{array}}\, (-1)^{\dim \gamma '} [{{\mathrm{in}}}_{\gamma '} X^\circ \circlearrowleft \widehat{\mu }] \\&\quad = \sum _{\gamma \in \Sigma } (-1)^{\dim \gamma } [{{\mathrm{in}}}_{\gamma } X^\circ \circlearrowleft \widehat{\mu }] \left( \sum _{\begin{array}{c} \gamma ' \in \Sigma ', \gamma ' {\text {bounded}}\\ {{\mathrm{Int}}}(\gamma ') \subseteq {{\mathrm{Int}}}(\gamma ) \end{array}} (-1)^{\dim \gamma - \dim \gamma '}\right) . \end{aligned}$$

Hence, it is enough to show that

$$\begin{aligned} \sum _{\begin{array}{c} \gamma ' \in \Sigma ', \gamma ' {\text {bounded}},\\ {{\mathrm{Int}}}(\gamma ') \subseteq {{\mathrm{Int}}}(\gamma ) \end{array}} (-1)^{\dim \gamma '} = \left\{ \begin{array}{cl} (-1)^{\dim \gamma } &{} \quad \text {if } \gamma \text { is bounded } \\ 0 &{} \quad \text {otherwise}. \end{array}\right. \end{aligned}$$

Topological and combinatorial proofs of the above fact are given in the proofs of [30, Lemma 2.2] and [28, Theorem 3.4], respectively.

After possibly replacing \(\widetilde{\Sigma }\) with a smooth fan refinement, and replacing \(\Sigma \) with \(\widetilde{\Sigma } \cap (N_\mathbb {R}\times \{ 1 \})\), we may assume that \(\mathbb {P}\) is smooth and the central fiber \({\mathcal {X}}_0\) is a simple normal crossings divisor. Let \(\{v_1, \ldots , v_s \}\) denote the vertices of \(\gamma \). Then \({\mathcal {X}}_0\) has irreducible components \(\{ D_1, \ldots , D_s \}\) with multiplicities \(\{ m_1, \ldots , m_s \}\), respectively, where \(m_i\) is the smallest positive integer m such that \(mv_i \in N\). For every nonempty subset \(I \subseteq \{ 1,\ldots s \}\), let \(\gamma _I\) be the bounded cell of \(\Sigma \) given by the convex hull of \(\{ v_i \mid i \in I \}\). Observe that every bounded cell appears in this way. As above, \(D_I^\circ = \cap _{i \in I} D_i \smallsetminus \cup _{j \notin I} D_j = V^\circ _\gamma \) admits an \(\mu _{m_\gamma }\)-covering \(\widetilde{V}^\circ _\gamma \rightarrow V^\circ _\gamma \) corresponding to the action of multiplication by \(\exp (2\pi \sqrt{-1} [-\gamma ])\). It follows from the fact that \(C_\gamma \) is a smooth cone in \(N \times \mathbb {Z}\) that \(m_\gamma = \gcd (m_i \mid i \in I )\). The result now follows from (5) above together with the definition of the motivic nearby fiber given in (2). \(\square \)

We will see interesting examples of the above theorem in Sect. 5 and Sect. 6. For the moment, we have the following simple example.

In this section, we present the relevant combinatorics of this paper. Weighted Ehrhart theory was introduced by the author in [52] both in order to extend and reprove many results in Ehrhart theory, the study of enumeration of lattice points in polytopes, and in order to give explicit computations of motivic integrals on toric stacks [53]. We present a generalization of this theory below using an extension of Stanley’s theory of subdivisions [51] presented in [29]. For brevity, we will only present the relevant results and refer the reader to [29] and [52] for details of proofs.

4.3 Weighted Ehrhart theory

Fix a lattice polytope P with respect to a lattice M. After possibly replacing M with a sublattice, we may assume that \(\dim P = \dim M_\mathbb {R}\). Under an isomorphism \(M \cong \mathbb {Z}^n\), the normalized volume \({{\mathrm{Vol}}}(P) \in \mathbb {Z}_{> 0}\) of P with respect to M is n! times the Euclidean volume of P. Let \(\nu : P \rightarrow \mathbb {R}\) be the convex graph of an integral height function on P, with corresponding regular lattice polyhedral subdivision \(\mathcal {S}(\nu )\) of P.

Remark 4.11

The setup above is slightly different than the ‘weight 0’ setup in [52], although the same arguments hold. We explain the relationship below. In [52], we consider a stacky fan \((N,\Sigma ,\{ b_i \})\), where \(\Sigma \) is a full-dimensional simplicial fan with respect to a lattice N, and \(\{ b_i \}\) is a choice of a nonzero lattice point on each ray of \(\Sigma \). We let \(\psi \) be the piecewise \(\mathbb {Q}\)-linear function with respect to \(\Sigma \) satisfying \(\psi (b_i) = 1\), and let \(Q = \{ v \in N_\mathbb {R}\mid \psi (v) \le 1 \}\). If Q is a convex polytope, then we may set \(M = N\), \(P = Q\) and \(\nu = \psi \) above to recover this setup. Here \(\Sigma \) induces a triangulation \(\mathcal {S}\) of P refining \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). We also note that the definition of the weighted \(h^*\)-polynomial below is slightly different (although equivalent) to that in [52] (see Example 5.13).

Let C be the cone over \(P \times \{ 1\}\) in \(M_\mathbb {R}\times \mathbb {R}\), and extend \(\nu \) by linearity to a piecewise \(\mathbb {Q}\)-linear function on C. For a nonnegative integer m, we will often identify mP with \(C \cap (M_\mathbb {R}\times \{ m \})\) and \(M_\mathbb {R}\) with \(M_\mathbb {R}\times \{ m \}\). We define a weight function

$$\begin{aligned}&w: C \cap (M \times \mathbb {Z}) \rightarrow \mathbb {Q}/\mathbb {Z},\\&w(v) = [\nu (v)]. \end{aligned}$$

Note that adding a global affine function with respect to M to \(\nu \) does not change w. Recall from Remark 2.3 that the group algebra \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) admits a conjugation action such that \(\overline{[k]} = [-k]\), and a ‘forgetful’ \(\mathbb {Z}\)-algebra homomorphism \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}] \rightarrow \mathbb {Z}, [k] \mapsto 1\). The weighted Ehrhart polynomial is defined by

$$\begin{aligned} f(P,\nu ;m) = \sum _{v \in mP \cap M } w(v) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}], \end{aligned}$$

for every nonnegative integer m. As in Remark 2.4, the corresponding polynomial with coefficients in \(\mathbb {Z}\) is the usual Ehrhart polynomial of P [20, 21], defined by \(f(P;m) = \#(mP \cap M )\) for every nonnegative integer m. By [52, Theorem 3.7],  \(f(P,\nu ;m)\) is a polynomial in m of degree \(\dim P\), and for every positive integer m,

$$\begin{aligned} (-1)^{\dim P} f(P,\nu ;-m) = \sum _{v \in {{\mathrm{Int}}}(mP) \cap M} \overline{w(v)} \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}]. \end{aligned}$$

The latter result is known as weighted Ehrhart reciprocity, since it reduces to the classical Ehrhart reciprocity: \(f(P;-m) = \#({{\mathrm{Int}}}(mP) \cap M )\) for every positive integer m [22]. In fact, it is shown in [52, Remark 3.10] that weighted Ehrhart reciprocity generalizes Ehrhart reciprocity for rational polytopes. Let \(\mathcal {S}\) be a lattice polyhedral decomposition refining \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). For each maximal cell \(F \in \mathcal {S}\), let \(m_F\) be the minimal positive integer such that \(m_F \nu |_{F}\) is an affine function with respect to M. Then one verifies that the leading coefficient of \(f(P,\nu ;m)\) as a polynomial in m is given by:

$$\begin{aligned} \frac{1}{(\dim P)!} \sum _{ \begin{array}{c} F \in \mathcal {S}\\ \dim F = \dim P \end{array} } \frac{{{\mathrm{Vol}}}(F)}{m_F} \sum _{i = 0}^{m_F - 1} [i/m_F] \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}], \end{aligned}$$

(6)

where \({{\mathrm{Vol}}}(F)\) is the normalized volume of F and \({{\mathrm{Vol}}}(F)/m_F\) is a positive integer. Define the weighted \(h^*\) -polynomial \( h^*(P,\nu ;u) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]\) by

$$\begin{aligned} \sum _{m \ge 0} f(P,\nu ;m) u^m = \frac{ h^*(P,\nu ;u) }{(1 - u)^{\dim P + 1}}. \end{aligned}$$

When P is the empty polytope, we set \(h^*(P,\nu ;u) = 1\). As in Remark 2.4, the corresponding polynomial with coefficients in \(\mathbb {Z}\) is the usual \(h^*\) -polynomial \(h^*(P;u) \in \mathbb {Z}[u]\) of P introduced by Stanley in [50]. We define the local weighted \(h^*\) -polynomial \(l^*(P,\nu ;u) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]\) by

$$\begin{aligned} l^*(P,\nu ;u) = \sum _{Q \subseteq P} (-1)^{\dim P - \dim Q}h^*(Q,\nu |_Q;u)g([Q,P]^*;u). \end{aligned}$$

Here the sum runs over all faces of P including the empty face. When P is the empty polytope, we set \(l^*(P,\nu ;u) = 1\). As in Remark 2.4, the corresponding polynomial with coefficients in \(\mathbb {Z}\) is the usual local \(h^*\) -polynomial \(l^*(P;u) \in \mathbb {Z}[u]\) of P, introduced by Stanley in [51, Example 7.13] and later independently by Borisov and Mavlyutov in [5]. By Theorem 4.6 and Remark 4.9, the weighted \(h^*\)-polynomial can be recovered as a ‘nonnegative’ combination of local weighted \(h^*\)-polynomials:

$$\begin{aligned} h^*(P,\nu ;u) = \sum _{Q \subseteq P} l^*(Q,\nu |_Q;u)g([Q,P];u). \end{aligned}$$

We present some properties and examples below. For the corresponding invariants over \(\mathbb {Z}\), proofs of the properties below can be found in [6] and [29].

Example 4.12

One may verify that (6) translates into the following:

$$\begin{aligned} h^*(P,\nu ;1) = \sum _{ \begin{array}{c} F \in \mathcal {S}\\ \dim F = \dim P \end{array} } \frac{{{\mathrm{Vol}}}(F)}{m_F} \sum _{i = 0}^{m_F - 1} [i/m_F] \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}], \end{aligned}$$

where \(m_F\) is the minimal positive integer such that \(m_F \nu |_{F}\) is an affine function with respect to M.

Example 4.13

Suppose that P is a simplex and \(\nu \) is \(\mathbb {Q}\)-affine, and let \(v_0, \ldots , v_n\) denote the primitive integer vectors on the rays of C corresponding to the vertices of P. Let \({{\mathrm{Box}}}= \{ v \in C \cap (M \times \mathbb {Z}) \mid v = \sum _{i = 0}^n a_i v_i, 0 \le a_i < 1 \}\). If \({{\mathrm{pr}}}: N_\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) denotes projection onto the second coordinate, then

$$\begin{aligned} h^*(P,\nu ;u)= & {} \sum _{v \in {{\mathrm{Box}}}} w(v) u^{{{\mathrm{pr}}}(v)} \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u],\\ l^*(P,\nu ;u)= & {} \sum _{v \in {{\mathrm{Box}}}\cap {{\mathrm{Int}}}(C)} w(v) u^{{{\mathrm{pr}}}(v)} \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]. \end{aligned}$$

Remark 4.14

One may verify that weighted Ehrhart reciprocity is equivalent to the following statements about the weighted \(h^*\)-polynomial and local weighted \(h^*\)-polynomial, respectively (cf. [29, Lemma 7.11]). Firstly, the weighted \(h^*\)-polynomial is acceptable in the sense of Stanley [51]. That is,

$$\begin{aligned} u^{\dim P + 1} \overline{h^*(P,\nu ;u^{-1})} = \sum _{Q \subseteq P} h^*(Q,\nu |_Q;u)(u - 1)^{\dim P - \dim Q}, \end{aligned}$$

where the above sum ranges over all (possibly empty) faces Q of P. Secondly, the local weighted \(h^*\)-polynomial is symmetric in the sense that

$$\begin{aligned} l^*(P,\nu ;u) = u^{\dim P + 1} \overline{l^*(P,\nu ;u^{-1})}. \end{aligned}$$

Example 4.15

If P is nonempty, then \(h^*(P,\nu ;u)\) and \(l^*(P,\nu ;u)\) have degree in u at most \(\dim P\), with the coefficient of \(u^{\dim P}\) in both cases equal to \(\sum _{ v \in {{\mathrm{Int}}}(P) \cap M } \overline{w(v)}\). By the symmetry in Remark 4.14 above, this implies that \(l^*(P,\nu ;u)\) has constant term 0 and linear coefficient \(\sum _{ v \in {{\mathrm{Int}}}(P) \cap M } w(v)\). The polynomial \(h^*(P,\nu ;u)\) has constant term 1 and linear coefficient \(\sum _{v \in P \cap M} w(v) - \dim P - 1\).

Remark 4.16

Let \(\mathcal {S}\) be a lattice polyhedral subdivision of P that refines \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). Then it follows from the definitions that for every positive integer m,

$$\begin{aligned} f(P,\nu ;m) = \sum _{ F \in \mathcal {S}} \sum _{v \in {{\mathrm{Int}}}(mF) \cap M} w(v). \end{aligned}$$

Using weighted Ehrhart reciprocity and following the proof of [29, Lemma 7.12], the above statement is equivalent to any of the three statements below:

$$\begin{aligned} h^*(P,\nu ;u)= & {} \sum _{ \begin{array}{c} F \in \mathcal {S}\\ \sigma (F) = P \end{array}} h^*(F,\nu |_F;u) (u - 1)^{\dim P - \dim F},\\ h^*(P,\nu ;u)= & {} \sum _{ F \in \mathcal {S}} l^*(F,\nu |_F;u) h({{\mathrm{lk}}}_\mathcal {S}(F);u),\\ l^*(P,\nu ;u)= & {} \sum _{ F \in \mathcal {S}} l^*(F,\nu |_F;u) l_P(\mathcal {S},F;u). \end{aligned}$$

Remark 4.17

One may always choose a lattice triangulation \(\mathcal {S}\) of P that refines \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). In fact, there exists a choice with the same vertices as \(\mathcal {S}(\nu )\) (see, for example, [42]). Then Remark 4.16 together with Example 4.13 gives explicit formulas that one may use to compute \(h^*(P,\nu ;u)\) and \(l^*(P,\nu ;u)\) in practice. In particular, using Remark 4.9, it follows that all integer coefficients in \(h^*(P,\nu ;u)\) and \(l^*(P,\nu ;u)\) are nonnegative.

Remark 4.18

The following lower bound theorem generalizes [29, Theorem 7.20] in the case when \(\nu \equiv 0\): if we write \(l^*(P,\nu ;u) = \sum _{k \in \mathbb {Q}/\mathbb {Z}} l^*(P,\nu ;u)_{[k]} [k]\) and \(l^*(P,\nu ;u)_{[0]} = l^*_{1}(P,\nu )_{[0]}u + \cdots + l_{\dim P}^*(P,\nu )_{[0]} u^{\dim P}\), then

$$\begin{aligned} l^*_{1}(P,\nu )_{[0]} = \# \{ v \in {{\mathrm{Int}}}(P) \cap \mathbb {Z}^d \mid w(v) = 0 \} \le l^*_{i}(P,\nu )_{[0]} \end{aligned}$$

for \(1 \le i \le \dim P\). This follows by the proof of [29, Theorem 7.20]. Explicitly, consider a regular, lattice polyhedral subdivision \(\mathcal {S}\) of P that refines \(\mathcal {S}(\nu )\) such that the lattice points of weight 0 in P are precisely the vertices of \(\mathcal {S}\). Then \(\nu \) takes integer values on the vertices of \(\mathcal {S}\), and every positive-dimensional cell in \(\mathcal {S}\) contains no interior lattice points of weight 0. By Remark 4.16 and Example 4.15,

$$\begin{aligned} l^*(P,\nu ;u)_{[0]} = l_P(\mathcal {S};u) + \beta (u)u^2, \end{aligned}$$

where \(l_P(\mathcal {S};u)\) has nonnegative, symmetric, unimodal coefficients and the polynomial \(\beta (u)\) has nonnegative integer coefficients. The result follows.

Example 4.19

The following example appeared in [52, Example 1.1]. Let \(M = {\mathbb {Z}}^{2}\) and let P be the lattice polytope with vertices (1, 0),(0, 2), \((-1,2)\), \((-2,1)\), \((-2,0)\) and \((0,-1)\). Let \(\nu : P \rightarrow \mathbb {R}\) be the piecewise \(\mathbb {Q}\)-linear function satisfying \(\nu (0) = 0\) and \(\nu (v) = 1\) for every vertex v of P. We write \(f(P,\nu ; m) = \sum _{[k] \in \mathbb {Q}/\mathbb {Z}} f_{[k]}(P,\nu ; m)[k]\) below. Then the invariants above may be computed to be:

$$\begin{aligned} h^*(P,\nu ; u)= & {} 1+ 4u + u^{2} + 2u(1 + u)[1/2] + u[2/3] + u^2[1/3],\\ h^*(P; u)= & {} 1+ 7u + 4u^{2},\\ l^*(P,\nu ; u)= & {} u(1 + u)(1 + 2[1/2]) + u[2/3] + u^2[1/3],\\ l^*(P; u)= & {} 4u(1 + u). \end{aligned}$$

In this section, we use Theorem 3.2 together with the combinatorics of Sect. 4 to prove a formula for the equivariant refined limit Hodge–Deligne polynomial of a schön hypersurface of a torus. We also give an equivalent formula for the equivariant refined limit Hodge–Deligne polynomial associated with the intersection cohomology of a canonical compactification of such a hypersurface. The corresponding results for invariants with coefficients in \(\mathbb {Z}\) appear in [30]. We will continue with the notation of Sects. 2 and 4.

Let M be a lattice and let \(N = {{\mathrm{Hom}}}(M,\mathbb {Z})\). Let \(X^\circ = \{ f = 0 \} \subseteq {{\mathrm{Spec}}}\mathbb {K}[M] \cong (\mathbb {K}^*)^n\) be a schön hypersurface, where \(f = \sum _{u \in M} \gamma _u(t) x^u\) for some \(\gamma _u(t) \in \mathbb {C}(t)\). The Newton polytope P of \(X^\circ \) is the convex hull of \(S = \{ u \in M \mid \gamma _u(t) \ne 0 \}\) in \(M_\mathbb {R}\). Note that P may be viewed as a full-dimensional lattice polytope in its affine span with induced lattice structure, and \(X^\circ \cong X' \times (\mathbb {K}^*)^k\), for some k, where \(X'\) is a schön hypersurface in a torus of dimension \(\dim P\) with Newton polytope P. Hence we may and will assume that \(\dim P = n\).

Consider the induced integral height function \(\omega _f: S \rightarrow \mathbb {Z}\) defined by \(\omega _f(u) = {{\mathrm{ord}}}_t \gamma _u(t)\). Recall from Sect. 4.2 that we may consider the convex graph \(\nu _f\) associated with \(\omega _f\), with corresponding regular lattice polyhedral subdivision \(\mathcal {S}(\nu _f)\).

Tropical geometry of hypersurfaces reduces to the study of Newton polytopes and polyhedral subdivisions [31, 46]. In particular, \({{\mathrm{Trop}}}(X^\circ )\) admits a polyhedral structure \(\Sigma \) with cells \(\gamma \) in inclusion-reversing correspondence with positive-dimensional cells F of \(\mathcal {S}(\nu _f)\), such that the bounded cells of \(\Sigma \) correspond to the cells of \(\mathcal {S}(\nu _f)\) not contained in the boundary of P. For example, if F is a maximal dimensional cell in \(\mathcal {S}(\nu _f)\), then \(\nu _f|_F\) is a \(\mathbb {Q}\)-affine function and the corresponding \(\mathbb {Q}\)-linear function is an rational point in \(N_\mathbb {Q}\), also denoted \(\nu _f|_F\). Then \(\gamma = -\nu _f|_F\) is a vertex in \(\Sigma \).

Recall the definition of the weighted refined limit mixed \(h^*\)-polynomial \(h^*(P,\nu ;u,v,w) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\) from Definition 4.20. Our goal is to prove the following formula for the equivariant refined limit Hodge–Deligne polynomial.

We will also prove an equivalent statement involving intersection cohomology. We will use middle perversity throughout. As explained in detail in [30, Section 6], given a projective variety X over \(\mathbb {K}\), for each of the geometric invariants described in Section 2.3, one may consider the corresponding invariant with respect to intersection cohomology rather than cohomology, e.g., the equivariant refined limit Hodge–Deligne polynomial \(E_{{{\mathrm{int}}}}(X_\infty , \widehat{\mu };u,v,w)\) associated with the intersection cohomology of X. The key benefit of using intersection cohomology is the fact that the intersection cohomology groups of a complex projective variety admit a pure (rather than just mixed) Hodge structure.

We will need the following sum-over-strata formula. Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function \((P,\nu )\) and \(\dim P = n\). Let \(\Delta _P\) denote the normal fan to P with cones \(\sigma \) in inclusion-reserving bijection with the faces \(Q_{\sigma }\) of P. Let X denote the closure of \(X^\circ \) in the projective toric variety \(\mathbb {P}(\Delta _P)_\mathbb {K}\) over \(\mathbb {K}\) corresponding to \(\Delta _P\). Then X admits a stratification \(X = \bigcup _{\sigma \in \Delta _P} X_\sigma ^\circ \), where \(X_\sigma ^\circ \) is a schön hypersurface with Newton polytope \(Q_{\sigma }\) and convex graph of an integral height function \(\nu |_{Q_\sigma }\). The theorem below is proved in the nonequivariant setting in [30, Theorem 6.1], but the proof goes through unchanged in our situation.

Our goal is to prove the following theorem.

The following lemma is a formal consequence of Theorem 5.8. Details are provided in [30, Lemma 6.2].

We now complete the proofs of Theorems 5.7 and 5.9, which we have established are both equivalent to equivalent statements (9) and (10).

Using Remark 5.1 and the above proof, we immediately deduce the following geometric description of the combinatorial invariants introduced in Sect. 4. As in Remark 2.4, the corresponding result for invariants with coefficients in \(\mathbb {Z}\) can be found in [30, Corollary 5.11,Corollary 6.3].

In this section, we apply Theorem 5.7 to deduce explicit combinatorial equations for some important invariants associated with the monodromy of complex polynomials. Recall that we identify the group \(\mathbb {Q}/\mathbb {Z}\) with the group \({\mathbb {S}}^1_\mathbb {Q}\) of rational points on the circle \(\{ z \in \mathbb {C}\mid |z| = 1 \}\), sending \([k] \in \mathbb {Q}/\mathbb {Z}\) to \(\alpha = e^{2 \pi \sqrt{-1}k} \in {\mathbb {S}}^1_\mathbb {Q}\). We will continue with the notation of Sects. 2, 4 and 5, and set \(M = \mathbb {Z}^n\).

6.1 Degenerations of hypersurfaces in affine space

We first consider applications of the results of Sect. 5 for families of hypersurfaces of affine space. Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function \((P,\nu )\) and \(\dim P = n\). Assume that \(P \subseteq \mathbb {R}_{\ge 0}^n\), and for each (possibly empty) subset S of \(\{1,\ldots ,n\}\), let \(\mathbb {R}^S = \{ (v_1,\dots ,v_n) \mid v_i = 0 \text { if } i \notin S \}\), \(\mathbb {Z}^S = \mathbb {R}^S \cap \mathbb {Z}^n\) and \(P^S = P \cap \mathbb {R}^S\). We assume that P is convenient in the sense that \(\dim P^S\) equals the cardinality |S| of S for every subset S of \(\{1,\ldots ,n\}\). Equivalently, P contains the origin and has nonzero intersection with each ray through a coordinate vector. We call a face (including the empty face) of P that does not contain the origin, a face at infinity, and write \(P_\infty \) for the union of faces at infinity of P. Then the faces of P are precisely the faces at infinity together with \(\{ P^S \mid S \subseteq \{ 1, \ldots , n\} \}\). Let X denote the closure of \(X^\circ \) in \(\mathbb {K}^n\). Then \(X_{{{\mathrm{gen}}}} \subseteq \mathbb {C}^n\) is a smooth hypersurface, and as in Sect. 5, one has a weak Lefschetz result (see, for example, [14, Corollary 3.8]) stating that \(H^{2(n - 1)}_c(X_{{{\mathrm{gen}}}}) = \mathbb {C}\) with trivial monodromy action and the only other nonzero cohomology is in middle dimension \(H^{n - 1}_c(X_{{{\mathrm{gen}}}})\). By Corollary 5.3, the motivic nearby fiber of X is given by

$$\begin{aligned} \psi _{X} = \sum _{S \subseteq \{ 1, \ldots , n \} } \sum _{\begin{array}{c} F \in \mathcal {S}(\nu ) \\ \ \sigma (F) = P^S \end{array}} [V_{F}^\circ \circlearrowleft \widehat{\mu }](1 - \mathbb {L})^{|S| - \dim F}, \end{aligned}$$

(11)

where \(\sigma (F)\) is the smallest face of P containing F. After rearranging terms, Theorem 5.7 implies that

$$\begin{aligned} uvw^2E(X_\infty , \widehat{\mu };u,v,w) = (uvw^2)^n + (-1)^{n - 1} \sum _{S \subseteq \{ 1, \ldots , n \} } (-1)^{n - |S|} h^*(P^S, \nu |_{P^S}; u,v,w). \end{aligned}$$

(12)

Specializing by setting \(w = 1\) gives a formula for the equivariant limit Hodge–Deligne polynomial. We may specialize further by setting \(v = 1\) and then \(u = 1\). In particular, by Example 4.12, we have the following formula for the eigenvalues (with multiplicity) of the action of monodromy on the cohomology of \(X_{{{\mathrm{gen}}}}\):

$$\begin{aligned} \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} \dim H_{c}^{n - 1}(X_\infty )_\alpha \alpha = \sum _{S \subseteq \{ 1, \ldots , n \} } \sum _{ \begin{array}{c} F \in \mathcal {S}(\nu )|_{P^S} \\ \dim F = \dim P^S \end{array} } (-1)^{n - |S|} \frac{{{\mathrm{Vol}}}(F)}{m_F} \sum _{i = 0}^{m_F - 1} [i/m_F], \end{aligned}$$

(13)

where \(m_F\) is the minimal positive integer such that \(m_F \nu |_{F}\) is an affine function with respect to \(\mathbb {Z}^S\). Here \({{\mathrm{Vol}}}(F)\) is the normalized volume of F, and \({{\mathrm{Vol}}}(F)/m_F\) is a positive integer.

Remark 6.1

With the setup above, do not assume that P is convenient, and instead only assume that P contains the origin. Then we may not apply the same weak Lefschetz argument. However, we still obtain the following formulas for the motivic nearby fiber and equivariant refined limit Hodge–Deligne polynomial:

$$\begin{aligned}&\psi _{X} = \sum _{ S \subseteq \{ 1, \ldots , n \} } (-1)^{|S| - \dim P^S} \sum _{\begin{array}{c} F \in \mathcal {S}(\nu ) \\ \ \sigma (F) = P^S \end{array}} [V_{F}^\circ \circlearrowleft \widehat{\mu }](1 - \mathbb {L})^{|S| - \dim F}.\\&uvw^2E(X_\infty , \widehat{\mu };u,v,w) = (uvw^2)^n \\&\quad + \sum _{ S \subseteq \{ 1, \ldots , n \} } (-1)^{\dim P^S - 1} (uvw^2 - 1)^{|S| - \dim P^S} h^*(P^S, \nu |_{P^S}; u,v,w). \end{aligned}$$

Specializing at \(u = v = w = 1\) gives:

$$\begin{aligned} E(X_\infty , \widehat{\mu };1,1,1)&= \sum _m \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} (-1)^m \dim H_{c}^{m}(X_\infty )_\alpha \alpha \\&= \sum _{ \begin{array}{c} \emptyset \ne S \subseteq \{ 1, \ldots , n \} \\ \dim P^S = |S| \end{array} } (-1)^{|S| - 1} \sum _{ \begin{array}{c} F \in \mathcal {S}(\nu )|_{P^S} \\ \dim F = \dim P^S \end{array} } \frac{{{\mathrm{Vol}}}(F)}{m_F} \sum _{i = 0}^{m_F - 1} [i/m_F], \end{aligned}$$

where \(m_F\) is the minimal positive integer such that \(m_F \nu |_{F}\) is an affine function with respect to \(\mathbb {Z}^S\).

Assume further that the restriction of \(\nu \) to \(P_\infty \) is constant Substitute in Definition 4.20 and observe that \(g([P^S, P^{S'}]; t) = 1\) for any \(S \subseteq S'\) by Example 4.5. By assumption, for every face Q at infinity, \(l^*(Q, \nu |_Q;u,v) = v^{\dim Q + 1} l^*(Q;uv^{-1})\). In this case, we may rewrite (12) as:

$$\begin{aligned} uvw^2E(X_\infty , \widehat{\mu };u,v,w) = (uvw^2)^n + (-1)^{n - 1} \big [ F(uw,vw) +w^{n + 1} l^*(P,\nu ;u,v) \big ], \end{aligned}$$

where

$$\begin{aligned} F(u,v):= & {} \sum _{ Q \subseteq P_\infty } v^{\dim Q + 1} l^*(Q;uv^{-1}) G(Q;uv),\\ G(Q;t):= & {} \sum _{ \begin{array}{c} S \subseteq \{1,\ldots ,n\} \\ Q \subseteq P^S \end{array} } (-1)^{n - |S|}g([Q,P^S]; t). \end{aligned}$$

Using Theorem 4.6 and Example 4.8, we have:

$$\begin{aligned} G(Q;t)&= \sum _{Q \subseteq Q' \subseteq P_\infty } (-1)^{n - 1 - \dim Q'} g([Q,Q']; t)g([Q',P]^*; t) \\&= \left[ \sum _{ \begin{array}{c} S \subseteq \{1,\ldots ,n\} \\ Q \subseteq P^S \end{array} } (-1)^{n - |S|} h({{\mathrm{lk}}}_{\mathcal {S}(\nu )|_{P^S}}(Q);t) \right] - l_P(\mathcal {S}(\nu ),Q;uv). \end{aligned}$$

Example 6.2

With the notation above, let Q be a face at infinity and consider G(Q; t). By definition, G(Q; t) has degree strictly less than \((\dim P - \dim Q)/2\). If \({{\mathrm{Int}}}(Q) \subseteq \mathbb {R}_{> 0}^n\), then \(G(Q;t) = g([Q,P];t)\) has constant term 1. Otherwise, G(Q; t) has no constant term. Using Example 4.4, when \(Q = \emptyset \), we compute that the linear coefficient of G(Q; t) is the number of vertices of P contained in \(\mathbb {R}_{> 0}^n\).

One then computes that F(u, v) has the form

$$\begin{aligned} F(u,v)= & {} \left[ \sum _{ \begin{array}{c} Q \subseteq P_\infty \\ \dim Q \le 1 \\ {{\mathrm{Int}}}(Q) \subseteq \mathbb {R}_{> 0}^n \end{array} } \#( {{\mathrm{Int}}}(Q) \cap \mathbb {Z}^n)\right] uv \\&+ \left[ \sum _{ \begin{array}{c} Q \subseteq P_\infty \\ \dim Q =2 \\ {{\mathrm{Int}}}(Q) \subseteq \mathbb {R}_{> 0}^n \end{array} } \#( {{\mathrm{Int}}}(Q) \cap \mathbb {Z}^n)\right] uv(u + v) + \beta (u,v), \end{aligned}$$

where every term in \(\beta (u,v)\) has combined degree in u and v at least 4.

Observe that every term in F(uw, vw) has combined degree in u and v equal to its degree in w. Hence, the only contribution above corresponding to a nontrivial action of monodromy is \((-w)^{n+ 1} l^*(P,\nu ;u,v)\). We conclude that we have the following corollary.

Corollary 6.3

Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function \((P,\nu )\) and \(\dim P = n\). Let \(X \subseteq \mathbb {K}^n\) denote the closure of \(X^\circ \) in \(\mathbb {K}^n\). Assume that \(P \subseteq \mathbb {R}_{\ge 0}^n\) is convenient, and the restriction of \(\nu \) to every face at infinity is constant. Then the action of monodromy is trivial on the graded pieces \(Gr^W_{r} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\) of the Deligne weight filtration for \(r \ne n - 1\), and we have the following explicit combinatorial formula for the equivariant limit mixed Hodge numbers of \(Gr^W_{ n - 1} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\):

$$\begin{aligned} uv \sum _{p,q} \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{p,q}(Gr^W_{n - 1} H^{n - 1}_c (X_{\infty }))_{\alpha } \alpha u^p v^q = l^*(P,\nu ; u,v). \end{aligned}$$

In particular, one may use Definition 2.6 to give a formula for the Jordan block structure of the action of monodromy on \(Gr^W_{ n - 1} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\).

Example 6.4

With the notation of Corollary 6.3, let \(n = 2\) and consider \(X \subseteq \mathbb {K}^2\). Using Example 4.24 and Example 6.2, we compute

$$\begin{aligned} E(X_\infty , \widehat{\mu };u,v,w) {=} uvw^2 - \big [ b +w[ h^*_{0,0,1}(P,\nu )(1 + uv) + h^*_{0,1,1}(P,\nu )v + \overline{h^*_{0,1,1}(P,\nu )}u] \big ], \end{aligned}$$

where \(b = \#(\partial P \cap \mathbb {Z}^{2}_{> 0})\), \(\partial P\) denotes the boundary of P, and

$$\begin{aligned} h^*_{0,0,1}(P,\nu ) = \sum _{\begin{array}{c} F \in \mathcal {S}(\nu ) \\ \ \dim F \le 1 \\ \ \sigma (F) = P \end{array}} \sum _{v \in {{\mathrm{Int}}}(F) \cap \mathbb {Z}^2} w(v), \; \; h^*_{0,1,1}(P,\nu ) = \sum _{\begin{array}{c} F \in \mathcal {S}(\nu ) \\ \ \dim F = 2 \\ \ \sigma (F) = P \end{array}} \sum _{v \in {{\mathrm{Int}}}(F) \cap \mathbb {Z}^2} w(v). \end{aligned}$$

In particular, the action of monodromy on \(Gr^W_{0} H^{1}_c (X_{{{\mathrm{gen}}}})\) is trivial and \(\dim Gr^W_{0} H^{1}_c (X_{{{\mathrm{gen}}}}) = b\). The equivariant limit mixed Hodge numbers of \(Gr^W_{1} H^{1}_c (X_{{{\mathrm{gen}}}})\) are given by

$$\begin{aligned}&\sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{0,0}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha \alpha = \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{1,1}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha \alpha = h^*_{0,0,1}(P,\nu ),\\&\sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{0,1}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha \alpha = \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{1,0}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha \alpha ^{-1} = h^*_{0,1,1}(P,\nu ). \end{aligned}$$

The Jordan normal form of the action of monodromy on \(Gr^W_{1} H^{1}_c (X_{{{\mathrm{gen}}}})\) has

$$\begin{aligned} h^{0,1}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha + h^{1,0}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha \end{aligned}$$

Jordan blocks of size 1 with eigenvalue \(\alpha \), and \(h^{0,0}(Gr^W_{1} H^{1}_c (X_{\infty }))_\alpha \) Jordan blocks of size 2 with eigenvalue \(\alpha \).

Remark 6.5

Specializing by setting \(w = 1\) above gives a formula for the equivariant limit mixed Hodge numbers:

$$\begin{aligned} uv \sum _{p,q} \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{p,q}( H^{n - 1}_c (X_{\infty }))_{\alpha } \alpha u^p v^q = F(u,v) + l^*(P,\nu ; u,v). \end{aligned}$$

Substituting in the definitions and simplifying, the right-hand side may be rewritten as follows:

$$\begin{aligned}&\sum _{\begin{array}{c} F \in \mathcal {S}(\nu ) \\ F \nsubseteq P_\infty \end{array} } v^{\dim F + 1}l^*(F, \nu |_F; uv^{-1}) l_P(\mathcal {S}(\nu ),F;uv)\\&\quad + \sum _{Q \subseteq P_\infty } v^{\dim Q + 1} l^*(Q;uv^{-1}) \sum _{ \begin{array}{c} S \subseteq \{1,\ldots ,n\} \\ Q \subseteq P^S \end{array} } (-1)^{n - |S|} h({{\mathrm{lk}}}_{\mathcal {S}(\nu )|_{P^S}}(Q);uv). \end{aligned}$$

We present the following two examples of the above situation. Below we let \(f \in \mathbb {C}[x_1, \ldots , x_n]\) be a complex polynomial with Newton polytope \({{\mathrm{NP}}}(f)\). For every face Q of \({{\mathrm{NP}}}(f)\), we let \(\Delta _Q\) denote the convex hull of Q and the origin. We set \(P = \Delta _{{{\mathrm{NP}}}(f)}\).

We assume that f is convenient in the sense that \({{\mathrm{NP}}}(f)\) has nonzero intersection with each ray through a coordinate vector [34]. Recall that for a complex hypersurface of a torus, schön is also called nondegenerate. Let \(\Gamma _+(f) = {{\mathrm{NP}}}(f) + \mathbb {R}_{\ge 0}^n\) denote the Newton polyhedron of f. Observe that every bounded face of \(\Gamma _+(f)\) is a face of NP(f). We let \(\Gamma _f\) denote the union of the bounded faces of \(\Gamma _+(f)\).

Remark 6.6

For any proper face Q of \({{\mathrm{NP}}}(f)\) not containing the origin, consider a \(\mathbb {Q}\)-affine function \(\nu \) on \(\Delta _Q\) with value \(m \pm 1\) at the origin, and value m on Q, for some integer m. Consider \(l^*(Q,\nu ;u) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]\) as a sum of integer-valued polynomials indexed by \(\alpha \in \mathbb {Q}/\mathbb {Z}\). Then the coefficient of \(\alpha = 1\) is zero. To see this, one may for example use Remark 4.16 to reduce to the case when Q is a simplex, and then apply Example 4.13.

Example 6.7

(Monodromy at 0) Assume that \(f \in \mathbb {C}[x_1, \ldots , x_n]\) has no constant term, f is convenient and \(\{ f = 0 \} \subseteq (\mathbb {C}^*)^n\) defines a schön hypersurface. Then for a general choice of \(\lambda \in \mathbb {C}^*\), \(X^\circ = \{ f = \lambda t \} \subseteq (\mathbb {K}^*)^n\) defines a schön, convenient hypersurface with Newton polytope P, and \(\nu _0 := \nu \) is the piecewise \(\mathbb {Q}\)-affine function on P with value 1 at the origin and value identically 0 on NP(f). The cells of \(\mathcal {S}(\nu _0)\) are given by the union of \(\{ Q \mid Q \subseteq {{\mathrm{NP}}}(f) \}\) and \(\{ \Delta _Q \mid Q \subseteq \Gamma _f \}\). By (11), we have the following equation for the motivic nearby fiber:

$$\begin{aligned} \psi _{X} = \sum _{Q \subseteq \Gamma _f } [V_{\Delta _Q}^\circ \circlearrowleft \widehat{\mu }](1 - \mathbb {L})^{\dim \sigma (\Delta _Q) - \dim \Delta _Q} + \sum _{\begin{array}{c} Q \subseteq {{\mathrm{NP}}}(f) \\ \ Q \nsubseteq P_\infty \end{array}} [V_{Q}^\circ ](1 - \mathbb {L})^{\dim \sigma (Q) - \dim Q}. \end{aligned}$$

(14)

By Corollary 6.3, the action of monodromy is trivial on the graded pieces \(Gr^W_{r} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\) of the Deligne weight filtration for \(r \ne n - 1\), and the equivariant limit mixed Hodge numbers of \(Gr^W_{ n - 1} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\) are given by:

$$\begin{aligned} uv \sum _{p,q} \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} h^{p,q}(Gr^W_{n - 1} H^{n - 1}_c (X_{\infty }))_{\alpha } \alpha u^p v^q = l^*(P,\nu _0; u,v). \end{aligned}$$

By Remarks 6.5 and  6.6, we have the following formulas for the equivariant limit mixed Hodge numbers of \(H^{n - 1}_c (X_{{{\mathrm{gen}}}})\):

$$\begin{aligned}&uv \sum _{p,q} \sum _{ \begin{array}{c} \alpha \in \mathbb {Q}/\mathbb {Z}\\ \alpha \ne 1 \end{array} } h^{p,q}( H^{n - 1}_c (X_{\infty }))_{\alpha } \alpha u^p v^q \\&\quad = \sum _{Q \subseteq \Gamma _f } v^{\dim \Delta _Q + 1}l^*( \Delta _Q , \nu _0|_{ \Delta _Q }; uv^{-1}) l_P(\mathcal {S}(\nu _0),\Delta _Q;uv),\\&uv \sum _{p,q} h^{p,q}( H^{n - 1}_c (X_{\infty }))_{1} u^p v^q\\&\quad = \sum _{\begin{array}{c} Q \subseteq {{\mathrm{NP}}}(f) \\ Q \nsubseteq P_\infty \end{array}} v^{\dim Q + 1}l^*( Q , \nu _0|_{ Q }; uv^{-1}) l_P(\mathcal {S}(\nu _0), Q;uv),\\&\qquad + \sum _{Q \subseteq P_\infty } v^{\dim Q + 1} l^*(Q;uv^{-1})\\&\qquad \times \sum _{ \begin{array}{c} S \subseteq \{1,\ldots ,n\} \\ Q \subseteq P^S \end{array} } (-1)^{n - |S|} h({{\mathrm{lk}}}_{\mathcal {S}(\nu _0)|_{P^S}}(Q);uv). \end{aligned}$$

By (13), we have a formula for the eigenvalues (with multiplicity) of the action of monodromy on the cohomology of \(X_{{{\mathrm{gen}}}}\). Explicitly, \(\sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} \dim H_{c}^{n - 1}(X_\infty )_\alpha \alpha \) is equal to

$$\begin{aligned}&\sum _{ \begin{array}{c} Q \subseteq \Gamma _f \\ \dim \sigma (\Delta _Q) = \dim \Delta _Q \end{array}} (-1)^{n - 1 - \dim Q} {{\mathrm{Vol}}}(Q) \sum _{i = 0}^{m(Q) - 1} [i/m(Q)] \nonumber \\&\quad + \sum _{\begin{array}{c} Q \subseteq {{\mathrm{NP}}}(f) \\ \ Q \nsubseteq P_\infty \\ \ \dim \sigma (Q) = \dim Q \end{array}} (-1)^{n - \dim Q} {{\mathrm{Vol}}}(Q), \end{aligned}$$

(15)

where \({{\mathrm{Vol}}}(Q)\) is the normalized volume of Q and m(Q) is the lattice distance of Q from the origin, i.e., m(Q) is the minimal positive integer such that \(m(Q) \nu _0|_{\Delta _Q}\) is an affine function with respect to the lattice given by intersecting M with the affine span of \(\Delta _Q\). When Q is empty, \({{\mathrm{Vol}}}(Q) = m(Q) = 1\) and \(\dim Q = -1\).

Example 6.8

(Monodromy at infinity) Assume that \(f \in \mathbb {C}[x_1, \ldots , x_n]\) is convenient. Assume that f is schön at infinity, meaning that \(\{ f|_Q = 0 \} \subseteq (\mathbb {C}^*)^n\) defines a smooth hypersurface whenever Q is a face of P at infinity. Then for a general choice of \(\lambda \in \mathbb {C}^*\), \(X^\circ = \{ ft = \lambda \} \subseteq (\mathbb {K}^*)^n\) defines a schön, convenient hypersurface with Newton polytope P, and \(\nu _\infty := \nu \) is the piecewise \(\mathbb {Q}\)-affine function on P with value 0 at the origin and value identically 1 on the faces at infinity of P. The cells of \(\mathcal {S}(\nu _\infty )\) are given by the union of \(\{ Q \mid Q \subseteq P_\infty \}\) and \(\{ \Delta _Q \mid Q \subseteq P_\infty \}\). By (11), we have the following equation for the motivic nearby fiber:

$$\begin{aligned} \psi _{X} = \sum _{Q \subseteq P_\infty } [V_{\Delta _Q}^\circ \circlearrowleft \widehat{\mu }](1 - \mathbb {L})^{\dim \sigma (\Delta _Q) - \dim \Delta _Q}. \end{aligned}$$

The motivic nearby fiber at infinity of a convenient polynomial f was introduced independently and from different perspectives in [39] and [45], and an explicit formula is given in [39, Theorem 5.3], which agrees with our formula for the motivic nearby fiber above. In particular, the motivic nearby fiber at infinity is equal to the usual motivic nearby fiber of X.

Let \(\Delta ^\infty \) be the simplex \(P \cap \{ x_1 + \cdots + x_n = \epsilon \}\) for fixed \(\epsilon > 0\) sufficiently small. Let \(\mathcal {S}_\infty \) denote the regular, rational polyhedral subdivision obtained by intersecting \(\mathcal {S}(\nu _\infty )\) with \(\Delta ^\infty \). That is, the cells of \(\mathcal {S}_\infty \) are \(\{ Q_\infty := \Delta _Q \cap \Delta ^\infty \mid Q \subseteq P_\infty \}\). By Remark 6.5 and Remark 6.6, we have the following formulas for the equivariant limit mixed Hodge numbers of \(H^{n - 1}_c (X_{{{\mathrm{gen}}}})\):

$$\begin{aligned}&uv \sum _{p,q} \sum _{ \begin{array}{c} \alpha \in \mathbb {Q}/\mathbb {Z}\\ \alpha \ne 1 \end{array} } h^{p,q}( H^{n - 1}_c (X_{\infty }))_{\alpha } \alpha u^p v^q \\&\quad = \sum _{Q \subseteq P_\infty } v^{\dim \Delta _Q + 1}l^*(\Delta _Q , \nu _\infty |_{\Delta _Q }; uv^{-1}) l_{\Delta ^\infty }(\mathcal {S}_\infty ,Q_\infty ;uv),\\&uv \sum _{p,q} h^{p,q}( H^{n - 1}_c (X_{\infty }))_{1} u^p v^q = \sum _{Q \subseteq P_\infty } v^{\dim Q + 1} l^*(Q;uv^{-1}) l_{\Delta ^\infty }(\mathcal {S}_\infty ,Q_\infty ;uv). \end{aligned}$$

An algorithm to compute the equivariant limit mixed Hodge numbers above is given in [39, Section 5].

Remark 6.9

Recall from Remark 4.9 that \( l_{\Delta ^\infty }(\mathcal {S}_\infty ,Q_\infty ;t) = t^{n - 1 - \dim Q} l_{\Delta ^\infty }(\mathcal {S}_\infty ,Q_\infty ;t^{-1})\) has nonnegative, symmetric, unimodal coefficients. Also, by Remark 4.14, the coefficients of the local weighted \(h^*\)-polynomial are nonnegative, and we have symmetries \(l^*(Q;u) = u^{\dim Q + 1} l^*(Q;u^{-1})\) and \(l^*(\Delta _Q ,\nu _\infty |_{\Delta _Q };u) = u^{\dim \Delta _Q + 1} \overline{l^*(\Delta _Q ,\nu _\infty |_{\Delta _Q };u^{-1})}\).

Recall also from (12) that we have the following equivalent formula for the equivariant refined limit Hodge–Deligne polynomial:

$$\begin{aligned} uvw^2E(X_\infty , \widehat{\mu };u,v,w) = (uvw^2)^n + (-1)^{n - 1} \sum _{S \subseteq \{ 1, \ldots , n \} } (-1)^{n - |S|} h^*(P^S, \nu _\infty |_{P^S}; u,v,w). \end{aligned}$$

After specializing at \(v = w = 1\), one obtains a formula for the spectrum at infinity of f [47] that is equivalent to [39, Theorem 5.11]. Recall from (13) that specialization at \(u = v = w = 1\) yields a formula for the eigenvalues (with multiplicity) of the action of monodromy:

$$\begin{aligned} \sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} \dim H_{c}^{n - 1}(X_\infty )_\alpha \alpha = \sum _{ \begin{array}{c} Q \subseteq P_\infty \\ \dim \sigma (\Delta _Q) = \dim \Delta _Q \end{array}} (-1)^{n - 1 - \dim Q} {{\mathrm{Vol}}}(Q) \sum _{i = 0}^{m(Q) - 1} [i/m(Q)], \end{aligned}$$

where \({{\mathrm{Vol}}}(Q)\) is the normalized volume of Q and m(Q) is the lattice distance of Q from the origin, i.e., m(Q) is the minimal positive integer such that \(m(Q) \nu _\infty |_{\Delta _Q}\) is an affine function with respect to the lattice given by intersecting M with the affine span of \(\Delta _Q\). When Q is empty, \({{\mathrm{Vol}}}(Q) = m(Q) = 1\) and \(\dim Q = -1\). The above is equivalent to a formula of Libgober and Sperber [35] for the zeta function at infinity of f.

Example 6.10

Following on with Example 3.1, Example 3.4 and Example 5.2, let \(f = \sum _{i = 1}^n x_i^{m_i} \in \mathbb {C}[x_1,\ldots ,x_n]\), for some \(m_i \in \mathbb {Z}_{>0}\). Let \(f_1, \ldots , f_n\) denote the standard basis vectors of \(M_\mathbb {R}= \mathbb {R}^n\). Then P is the convex hull of the origin and \(\{ m_i f_i \mid 1 \le i \le n \}\). In this case, \(f: \mathbb {C}^n \rightarrow \mathbb {C}\) is a locally trivial fibration away from the origin, and it follows that the actions of monodromy at 0 and monodromy at infinity on cohomology are inverse to each other. In both cases, \(\nu \) is \(\mathbb {Q}\)-affine and \(\mathcal {S}(\nu )\) is trivial. For \((i_1, \ldots i_n) \in [1,m_1 - 1] \times \cdots \times [1,m_n - 1]\), let

$$\begin{aligned} \nu _\infty (i_1,\ldots ,i_n) := i_1/m_1 + \cdots + i_n/m_n. \end{aligned}$$

The action of monodromy is semi-simple, and, using Example 4.23, we compute the equivariant refined limit Hodge–Deligne polynomial \(uvw^2E(X_\infty , \widehat{\mu };u,v,w)\) for monodromy at infinity:

$$\begin{aligned}&(uvw^2)^n + (-1)^{n - 1} \\&\quad \sum _{(i_1, \ldots i_n) \in [1,m_1 - 1] \times \cdots \times [1,m_n - 1]} [ \nu _\infty (i_1,\ldots ,i_n)] (uw)^{\lceil \nu _\infty (i_1,\ldots ,i_n) \rceil } (vw)^{\lceil n - \nu _\infty (i_1,\ldots ,i_n) \rceil }. \end{aligned}$$

For the corresponding invariant for monodromy at 0, one simply reverses the roles of u and v.

Example 6.11

Let \(f = a_0 x^4 + a_1 x^5 + a_2 x^4y^2 + a_3 xy^5 + a_4 y^5 + a_5 xy^2 + a_6 x^2y \in \mathbb {C}[x,y]\) for a general choice of \(a_0, \ldots , a_6 \in \mathbb {C}^*\). Then the pairs \((P, \nu )\) with their corresponding subdivisions \(\mathcal {S}(\nu )\) in Example 6.7 and Example 6.8 are shown below. We have labeled all interior lattice points with nonzero weight. We may calculate the invariants above using Example 6.4. In particular, using the notation of Example 6.4, we have \(\dim Gr^W_{0} H^{1}_c (X_{{{\mathrm{gen}}}}) = b = 4\). Also, for monodromy at 0, we have

$$\begin{aligned} h^*_{0,0,1}(P,\nu _0) = 2, \; h^*_{0,1,1}(P,\nu _0) = 7 + [1/3]. \end{aligned}$$

In this case, the action of monodromy on \(Gr^W_{1} H^{1}_c (X_{{{\mathrm{gen}}}})\) has 16 Jordan blocks of size 1, 14 with eigenvalue 1 and 1 with eigenvalue \(\exp (2\pi \sqrt{-1}/3)\) and \(\exp (4\pi \sqrt{-1}/3)\), respectively, as well as 2 Jordan blocks of size 2 with eigenvalue 1. For monodromy at infinity, we have

$$\begin{aligned} h^*_{0,0,1}(P,\nu _\infty )= & {} [1/2],\\ h^*_{0,1,1}(P,\nu _\infty )= & {} [7/10] + [9/10] + [1/3] + [1/2] + 2[2/3] + 3[5/6]. \end{aligned}$$

In this case, the action of monodromy on \(Gr^W_{1} H^{1}_c (X_{{{\mathrm{gen}}}})\) has 18 Jordan blocks of size 1, 3 with eigenvalues \(\exp (2\pi \sqrt{-1}/6)\), \(\exp (2\pi \sqrt{-1}/3)\), \(\exp (4\pi \sqrt{-1}/3)\) and \(\exp (10\pi \sqrt{-1}/6)\), respectively, 2 with eigenvalue \(-1\) and 1 with eigenvalues \(\exp (\pi \sqrt{-1}/5)\), \(\exp (3\pi \sqrt{-1}/5)\), \(\exp (7\pi \sqrt{-1}/5)\) and \(\exp (9\pi \sqrt{-1}/5)\), respectively, as well as a single Jordan block of size 2 with eigenvalue \(-1\).