Formulas for monodromy

Given a family $X$ of complex varieties degenerating over a punctured disc, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of $X$. Our first main result is that the motivic nearby fiber of $X$ can be computed by first stratifying $X$ into locally closed subvarieties that are non-degenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when $X$ is a family of non-degenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.


Introduction
Before introducing the main results of this paper, we first present an application. Let f (x 1 , . . . , x n ) ∈ C[x 1 , . . . , x n ] be a complex polynomial with no constant term, and consider the induced map f : C n → C. One would like to understand the singularity of f −1 (0) ⊆ 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 non-degenerate with respect to its Newton polyhedron Γ + (f ), and also that it is convenient (see Section 6.3). The most famous result was proved by Varchenko in 1976 [62], who gave an explicit combinatorial formula for the eigenvalues (with multiplicity) of M f , involving alternating signs of volumes of polytopes associated to Γ + (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,α of Jordan blocks of M f of size k with eigenvalue α. Using tropical geometry, together with weighted Ehrhart theory, which involves the enumerative combinatorics of lattice points in dilates of polytopes, we settle this question: Theorem (Corollary 6.22). Let f ∈ C[x 1 , . . . , x n ] be a complex polynomial such that f −1 (0) admits an isolated singularity at the origin. Assume further that f is convenient and nondegenerate with respect to its Newton polyhedron Γ + (f ). Then there are explicit, non-negative combinatorial formulas for the numbers J k,α , involving the weighted Ehrhart theory of polytopes associated to Γ + (f ).
We refer the reader to Corollary 6.22 for the specific formulas. By a 'non-negative' formula, we mean that each J k,α is expressed as a sum of non-negative integers. By summing over k, we obtain a non-negative 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,α have previously been proved by Matsui and Takeuchi [41] and van Doorn and Steenbrink [21], and an algorithm to compute the numbers J k,α involving simplicial resolutions of toric varieties was proved by Matsui and Takeuchi in [41], extending work of Danilov [14] and Tanabé [60]. We note that algorithms in a more general setting are given by Schulze in [50].
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 → D * over a complex disc, Denef and Loeser [18] associate an invariant ψ X called the motivic nearby fiber, which contains information about the variation of Hodge structures associated to the degeneration. In particular, if we fix a non-zero fiber X gen := f −1 (t) for some sufficiently small t ∈ D * , then ψ X encodes information on the induced monodromy map T : H m c (X gen ) → H m c (X gen ), which is a linear operator on the complex cohomology with compact supports of X gen and is quasiunipotent i.e. T = T s T u , where T s and T u commute, T s is semi-simple and has finite order, and T u is unipotent. In [32], Eric Katz and the author showed that one can use tropical geometry to concretely compute a specialization of ψ X that is invariant under base change, and hence encodes information about T u but not T s . Using new combinatorics developed in [31], 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 gen ). Here schön is a 'generic' condition introduced by Tevelev [61], generalizing the notion of non-degeneracy of a hypersurface of a complex torus [35]. 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 ψ X and monodromy operator T . Our first main result is to show that the motivic nearby fiber may be computed using tropical geometry (see Section 3). A key point is that since there is a range of available software implementing the main algorithms in tropical geometry [28,29], our main formula Theorem 3.2 can be computed in practice.
Weighted Ehrhart theory was introduced by the author in [54] 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 [55]. Roughly speaking, one associates to every lattice point v a 'weight' w(v) in Q/Z, and then attempts to enumerate lattice points in polytopes keeping track of the associated weights. Extending work in [31], in Section 4.4, we associate new Ehrhart-theoretic invariants to a pair (P, ν), where P is a lattice polytope and ν is the convex graph of an integral height function on P . In particular, we introduce the weighted refined limit mixed h * -polynomial h * (P, ν; u, v, w) ∈ Z[Q/Z] [u, v, w]. Roughly speaking, our second main result states that if X • 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, ν; u, v, w) (Theorem 5.7, Corollary 5.12), where (P, ν) is the Newton polytope and associated convex graph associated to X • . 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 • gen ). Equivalent formulas are also given for the equivariant refined limit mixed Hodge numbers associated to 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 [12,13] 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 [40], generalizing an algorithm of Danilov and Khovanskiȋ in the non-equivariant setting [15]. Moreover, the above formula reduces to a formula of Borisov and Mavlyutov [9] in the non-equivariant 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 , . . . , x n ) ∈ C[x 1 , . . . , x n ] be a complex polynomial. It is well-known that there exists a minimal finite subset B f ⊆ C, such that f : C n → C is a locally trivial fibration away from B f . Then monodromy defines an action of the fundamental group π 1 (C 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 ∈ 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 ∈ C | |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 [19] states that monodromy at infinity essentially determines the monodromy action of π 1 (C B f ). We assume that f is convenient in the sense that its Newton polytope has non-zero intersection with each ray through a coordinate vector [36]. Then under certain schönness (also called 'non-degeneracy') 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 'non-negative' combinatorial formulas for both the equivariant limit mixed Hodge numbers associated to 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 [40]. Also, these formulas specialize to known formulas for the spectrum at infinity of f [40,Theorem 5.11] and zeta function at infinity of f [37]. 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 the 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 Khovanskiȋ [15,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 [25]. 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 [59]. Lastly, analogous formulas to those in Section 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 [58].
1.1. Organization of the paper. This paper is structured as follows. In Section 2, we review the necessary geometric background. In Section 3, we show how to compute the motivic nearby fiber using tropical geometry. In Section 4, we review the necessary combinatorial background before introducing our main combinatorial invariants. In Section 5, we prove our combinatorial formulas for geometric invariants associated to degenerations of hypersurfaces. Finally, in Section 6, we apply our results to deduce formulas for invariants associated with the monodromy of complex polynomials.
Notation and conventions. All cohomology has complex coefficients. If N is a lattice, then N R = N ⊗ R R. We identify the group Q/Z with the group S 1 Q of rational points on the circle A variety X over K is naturally interpreted as a complex variety with a morphism f : X → D * , for some sufficiently small punctured complex disc D * around the origin, and X gen := f −1 (t) denotes a fixed fiber for some choice of t ∈ D * .
Acknowledgements. The author would like to thank Anthony Henderson for some useful comments.

Geometry of degenerations
In this section we briefly recall the notion of the motivic nearby fiber and its associated geometry. We refer the reader to [7], [44] and [45] for details. A similar exposition is given in the non-equivariant setting in [32, Section 3].
2.1. The equivariant Grothendieck ring. We follow the treatment in [7] below. Fix a field k. The Grothendieck ring K 0 (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 whenever U is an open subvariety of V . Multiplication is given by tensor product of varieties. Suppose that G is a finite group. An (algebraic) action of G on a variety over k is good if every orbit lies in an open affine subvariety. Then we may similarly define K G 0 (Var k ) to be the free abelian group generated by isomorphism classes [V ] of varieties V over k together with a good action of G, modulo the relation where V → X is a vector bundle of rank n + 1 with a good G-action that is linear over a good G-action on X. Here P(V ) denotes the projectivization of V with induced G-action, and P n admits a trivial G-action. A group homomorphism G → H induces a ring homomorphism K H 0 (Var k ) → K G 0 (Var k ) by restriction. In particular, K G 0 (Var k ) is a module over K 0 (Var k ), and forgetting the action of G gives a ring homomorphism K G 0 (Var k ) → K 0 (Var k ). We will follow the convention that L := [A 1 ] is the class of the affine line with trivial G-action. A motivic invariant is a ring homomorphism K G 0 (Var k ) → R, for some ring R. Remark 2.1. Suppose that a finite group G acts linearly on a k-vector space V of dimension n + 1 i.e. V is a representation of G. Then taking X = pt in the condition above, we see that [P(V )] = [P n ] = L n + L n−1 + · · · + 1 ∈ K G 0 (Var k ). Now assume that k = C and identify µ n ∼ = Z/nZ with the group of n th roots of unity in C generated by exp(2π √ −1i/n). The epimorphisms µ nd → µ n , ζ → ζ d give rise to a projective limit µ = lim ← − µ n . If g is an element of a group G of finite order, then we have a well-defined group homomorphism µ → µ n → G sending exp(2π √ −1i/n) → g whenever g n = 1. In this way, group actions of elements of finite order give rise to group actions of µ.
A good µ-action on a complex variety is an action of µ induced by a good µ n -action for some n > 0. The equivariant Grothendieck ring K µ 0 (Var C ) is defined similarly as above, and coincides with the direct limit of the restriction maps K µn gives a good µ-action on (C * ) n . This action extends to P n , which admits a natural stratification into tori each invariant under µ, and it follows from Remark 2.1 and induction on dimension that [(C * ) n µ] = ( L − 1) n ∈ K µ 0 (Var C ).

2.2.
The motivic nearby fiber. Fix K = C(t). Then the motivic nearby fiber is a ring homomorphism A result of Bittner [8] implies that K 0 (Var K ) is generated by the classes of smooth, proper varieties. In particular, the description below determines ψ.
We follow the description of the motivic nearby fiber in [58]. A variety X over K is naturally interpreted as a complex variety with a morphism f : X → C * {b 1 , . . . , b r }, for some points b 1 , . . . , b r . Assume that X is smooth and extend X to a variety X ′ with flat morphism f : X ′ → A 1 {b 1 , . . . , b r }. After resolving singularities, we may assume that X ′ is smooth and that the central fiber f −1 (0) = i m i D i is a simple normal crossings divisor with irreducible components D 1 , . . . , D r , with multiplicities m 1 , . . . , m r respectively. For each non-empty subset I ⊆ {1, . . . , r}, let D • I = ∩ i∈I D i ∪ j / ∈I D j and m I = gcd(m i | i ∈ I). Restrict f to a small complex disc D about the origin and let m be a common multiple of m 1 , . . . , m r . Pullback X ′ via the map D → D, t → t m and normalize to obtain a commutative diagram The action of µ m on D, with ζ ∈ µ m acting via t → ζt, extends to an action on X ′ , and hence a good µ-action on the central fiber. Let D • I be proper transform of D • I via ρ. Then ρ :

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 Q/Z with the group S 1 Q of rational points on the circle {z ∈ C | |z| = 1}, sending [k] ∈ Q/Z to α = e 2π √ −1k ∈ S 1 Q . We will consider the group algebra Z[Q/Z]. , corresponding to complex conjugation on the circle. This extends coefficient-wise to an involution on a polynomial ring with coefficients in Z[Q/Z], that we will refer to as conjugation.
Remark 2.4. Consider the natural Z-algebra homorphism Z[Q/Z] → Z, [k] → 1. One may apply this homomorphism below to get invariants with coefficients in Z rather than Z[Q/Z]. The corresponding diagram of invariants above over Z is explained in detail in [32,Section 3].
Consider a complex vector space B that admits a mixed Hodge structure [45] with corresponding vector space decomposition Suppose the µ n acts linearly on B, preserving the mixed Hodge structure. With the convention above, for α ∈ Q/Z, we write h p,q (B) α for the dimension of the α-eigenspace of In [16], Deligne proved that the 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 • called the Hodge filtration and increasing filtration W • called the Deligne weight filtration. Given a good action of µ on V , we may set B m = H p,q (H m c (V )) above. The corresponding equivariant Hodge polynomial is denoted E µ (V ; u, v) ∈ Z[Q/Z] [u, v], and is called the equivariant Hodge-Deligne polynomial of V . The invariants h p,q (H m c (V )) α are called the equivariant mixed Hodge numbers of V . We have a ring homomorphism called the equivariant Hodge-Deligne map: . As in Remark 2.4, we will also consider the corresponding invariants over Z. In this case, we have the usual mixed Hodge numbers h p,q (H m c (V )) = α h p,q (H m c (V )) α and corresponding 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 where R(G) is the complex representation ring of G. This induces a ring homomorphism K G 0 (Var C ) → R(G) [u, v]. This invariant was introduced and studied by the author in [56], generalizing the notion of weight polynomial due to Dimca and Lehrer [20], and the notion of equivariant χ y -genus due to Cappell, Maxim and Shaneson [11]. In our case, Z[Q/Z] may be viewed as the direct limit associated to the restriction maps R(µ n ) → R(µ nd ) induced by the epimorphisms µ nd → µ n , ζ → ζ d .
Recall that a variety X over K is naturally interpreted as a complex variety with a morphism f : X → C * {b 1 , . . . , b r }, for some points b 1 , . . . , b r . Restricting X to a family over a sufficiently small punctured complex disc centered at the origin, and fixing a fiber X gen := f −1 (t) and counter-clockwise orientation around the disc, there exists a quasiunipotent monodromy map T = T s T u : H m c (X gen ; C) → H m c (X gen ; C), where T s is semisimple and T u is unipotent. Then the cohomology groups H m c (X gen ) admit a weight filtration M • called the monodromy weight filtration, and T s acts preserving the filtrations (F • , M • , W • ). We will write H m c (X ∞ ) to denote H m c (X gen ) with the filtrations (F • , M • , W • ). The filtrations (F • , M • ) induce a mixed Hodge structure on H m c (X ∞ ), 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 m c (X ∞ )) α are called the equivariant limit mixed Hodge numbers, and the corresponding equivariant Hodge polynomial E(X ∞ , µ; u, v) ∈ Z[Q/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 ∞ , µ; u, v) = E µ (ψ X ). As in Remark 2.4, we may consider the corresponding invariants over 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 uniquely determined by for any non-negative integer k. Here we set V k = 0 for k < 0. The filtration encodes the Jordan block structure of A. Explicitly, for 1 ≤ k ≤ r + 1, the number of Jordan blocks of size k equals dim The monodromy weight filtration M • has the following important geometric property: for every non-negative integer r, the induced filtration M(r) • on Gr W r H m c (X ∞ ) 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 W r H m c (X ∞ ). Moreover, (F • , M • ) induces a mixed Hodge structure on Gr W r H m c (X ∞ ) 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 h p,q (Gr W r H m c (X ∞ )) α = (Gr p F Gr M p+q Gr W r H m c (X ∞ )) α 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 ∞ , µ; u, v, w) ∈ Z[Q/Z][u, v, w] called the equivariant refined limit Hodge-Deligne polynomial. As in Remark 2.4, we have corresponding invariants over Z, that were introduced in [32, Section 3], to where we refer the reader for further details. We have a ring homomorphism It follows from the definitions that we recover the equivariant limit mixed Hodge numbers and the usual mixed Hodge numbers of X gen via the specializations: Correspondingly, we have E(X ∞ , µ; u, v, 1) = E(X ∞ , µ; u, v), and E(X ∞ , µ; uw −1 , 1, w) restricts to the Hodge-Deligne polynomial E(X gen ; u, w) ∈ Z[u, w]. The further specializations E(X gen ; u, 1) and E(X gen ; 1, 1) are the χ y -characteristic and Euler characteristic of X 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 ≤ j ≤ r, the sequence {h i+j,i,r (H m c (X ∞ )) α | 0 ≤ i ≤ r − j} is symmetric and unimodal. Remark 2.8. Since H p,q (H m c (X ∞ )) and H q,p (H m c (X ∞ )) are conjugate, and by Remark 2.7, the refined limit mixed Hodge numbers satisfy the following symmetries: Hence the equivariant refined limit Hodge-Deligne polynomial satisfies the following symmetries with respect to the conjugation action on Z[Q/Z][u, v, w] described in Remark 2.3: Example 2.9. If V is a complex variety and X = V × C K, then X may be regarded as a trivial family over D * . In this case, N is identically zero, M • coincides with the Deligne filtration W • , and E(X ∞ , µ; u, v, w) = E(V ; uw, vw). Example 2.10. If X is smooth and proper, then Gr W r H m (X 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,

Tropical geometry
In this section we prove our formula for the motivic nearby fiber. The non-equivariant version of this result is [32,Theorem 1.2]. We continue with the notation of the previous section. In particular, K = C(t).
n ] of f with respect to w is defined as follows: Assume that w is rational and consider the element Then multiplication by exp(2π √ −1w) induces a finite order automorphism β w of (C * ) n and hence a good action of µ on (C * ) n . Note that if w = 0, then β w acts freely on (C * ) n . We compute Then the initial degeneration in w X • is the closed subscheme of (C * ) n defined by the ideal in w I : . By (4), in w I is invariant under β w , and hence we have an induced action on in w X • .
A variety X over K is very affine if it admits a closed embedding X ⊆ (K * ) n for some n. Tevelev introduced the notion of a schön, very affine variety. With the notation above, Prop 3.9]. This definition is independent of the choice of embedding [38,Lemma 2.3]. Note that a complex variety V may be viewed via base change as a variety over 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 non-degeneracy with respect to the Newton polytope [35].
The tropical variety Trop(X • ) can be given a rational polyhedral structure Σ such that initial degeneration at w ∈ Trop(X • ) only depends on the cell containing w in its relative interior (this follows from [38, Theorem 1.5]). Hence for every cell γ of Σ, we may define in γ X • := in w X • for any w ∈ R n in the relative interior of γ. Moreover, by fixing such a w ∈ Q n , we obtain a good action of µ on in γ X • via multiplication by exp(2π √ −1w).
for some m i ∈ Z >0 and e ∈ Z. Let e 1 , . . . , e n denote the standard basis vectors of R n . Then X • is schön and Trop(X • ) ⊆ R n is the affine translation by (e/m 1 , . . . , e/m n ) of the (n − 1)-dimensional fan with cones spanned by subsets of size at most n−1 of {e 1 , . . . , e n , −e 1 /m 1 −· · ·−e n /m n }. Let w = (e/m 1 , . . . , e/m n ). Then µ acts on in w X . We deduce that the action of µ coincides with the action of monodromy induced by moving anti-clockwise around the unit circle (more generally, see Example 5.6).
Recall that our goal is to compute the motivic nearby fiber defined in Section 2.2: A result of Luxton and Qu [38,Theorem 6.11] that was conjectured by Tevelev in [61], implies that every variety over K admits a stratification into locally closed, very affine, schön subvarieties. By the additivity property (1) of K 0 (Var K ), it follows that we may reduce the computation of ψ to the case when X • ⊆ (K * ) n is a schön subvariety. In this case, we have the following equivariant generalization of [32, Theorem 1.2].
Theorem 3.2. Let X • ⊆ (K * ) n be a schön closed subvariety and let Σ be a rational polyhedral structure on Trop(X • ). Then the motivic nearby fiber ψ X • ∈ K µ 0 (Var C ) is given by where the action of µ on in γ X • ⊆ (C * ) n is induced by multiplication by exp(2π Proof. We first recall the relevant geometric setting, as described in [27,Sections 1,2]. We refer the reader to [26] for the appropriate background on toric geometry.
Let N = Z n , and for each cell γ ∈ Σ, let C γ denote the cone generated by γ × {1} in N R × R. The cones C γ form a fan Σ in N R × R, and the recession fan ∆ is defined to be the intersection of Σ with N R × {0}. Projection pr : N R × R → R onto the second coordinate induces a morphism of fans Σ → R ≥0 . Fix a positive integer m such that mΣ has a lattice polyhedral structure, and let P and P denote the complex toric varieties corresponding to the fan Σ with respect to the lattices N × mZ and N × Z respectively. We have an induced commutative diagram corresponding to pulling back and normalizing P: where the vertical arrows are flat. After removing the closed torus invariant divisors corresponding to non-zero cones in ∆, the non-zero fibers of P and P are tori and the central fibers have decompositions into disjoint unions of tori: Let π γ : U γ → U γ denote the morphism of tori induced by π. This has the following explicit description. Let L ′ γ denote the linear span of C γ , and let . This isomorphism induces the vertical maps in the following commutative diagrams: Applying the functors Hom( , Z) and then Hom( , C * ) induces Letting M = Hom(N, Z) and M γ = Hom(N γ , Z), we identify U γ = Hom(M γ , C * ). Then the above sequence says that the rational point In what follows, we let T = Hom(M, C * ) and consider the multiplication map T × U γ → U γ . We also fix the distinguished element p ∈ U γ = Hom(M γ , C * ) corresponding to the constant map 1.
Recall that X • may viewed as a family of complex varieties over C * {b 1 , . . . , b r }, for some points b 1 , . . . , b r . Let D be a sufficiently small complex disc centered at the origin, and restrict X • to a family over the punctured disc D * , and P and P to families over D. Taking the closure X of X • in P induces a commutative diagram: where for each bounded cell γ ∈ Σ, we have a corresponding µ mγ -covering V • γ := X ∩ U γ → V • γ := X ∩ U γ corresponding to the action of multiplication by exp(2π √ −1[−γ]). As above, consider the induced multiplication map m : T × V • γ → U γ and consider the fiber m −1 (p). By [27, p. 8 where T γ is the subtorus that fixes U γ pointwise. Fix a rational point w ∈ N Q in the relative interior of γ, and consider the corresponding point exp(2π √ −1w) ∈ T of finite order. Then multiplication by (exp(2π , and, using Example 2.2, we conclude that where µ acts via multiplication by exp(2π √ −1w) and exp(2π √ −1[−γ]) respectively. In particular, this shows that the left hand side is independent of the choice of w in the relative interior of γ.
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 Σ. Indeed, if Σ ′ is a polyhedral structure refining Σ, then, using the independence of the choice of w above Hence, it is enough to show that Topological and combinatorial proofs of the above fact are given in the proofs of [ After possibly replacing Σ with a smooth fan refinement, and replacing Σ with Σ ∩ (N R × {1}), we may assume that P is smooth and the central fiber X 0 is a simple normal crossings divisor. Let {v 1 , . . . , v s } denote the vertices of γ. Then X 0 has irreducible components {D 1 , . . . , D s } with multiplicities {m 1 , . . . , m s } respectively, where m i is the smallest positive integer m such that mv i ∈ N. For every non-empty subset I ⊆ {1, . . . s}, let γ I be the bounded cell of Σ given by the convex hull of {v i | i ∈ I}. Observe that every bounded cell appears in this way. As above, . It follows from the fact that C γ is a smooth cone in N ×Z that m γ = gcd(m i | i ∈ I). The result now follows from (5) above together with the definition of the motivic nearby fiber given in (2).
It is a corollary of the proof above that in the statement of Theorem 3.2, one may consider the action on in γ X • ⊆ (C * ) n to be induced by multiplication by exp(2π √ −1w) for any fixed choice of w ∈ Q n in the affine span of γ.
We will see interesting examples of the above theorem in Section 5 and Section 6. For the moment, we have the following simple example.

The combinatorics of height functions
In this section, we present the relevant combinatorics of this paper. Weighted Ehrhart theory was introduced by the author in [54] 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 [55]. We present a generalization of this theory below using an extension of Stanley's theory of subdivisions [52] presented in [31]. For brevity, we will only present the relevant results and refer the reader to [31] and [54] for details of proofs. 4.1. Eulerian posets. We first recall some basic notions on Eulerian posets. Consider a finite poset B containing a minimal element 0 and a maximal element 1. For any pair z ≤ x in B, we can consider the interval [z, x] = {y ∈ B | z ≤ y ≤ x}. Assume that B is graded in the sense that for every x ∈ B, every maximal chain in the interval [ 0, x] has the same length ρ(x). We call ρ : B → N the rank function of B, and call ρ( 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 4.4. By considering the leading terms above, we see that g(B; t) has constant term 1, and linear term #{x ∈ B | ρ(x) = 1} − n. We will need the following theorem of Stanley.

Polyhedral subdivisions.
A polyhedral subdivision of a polytope P ⊆ 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 polyhedral subdivision is a rational (respectively lattice) subdivision if each polytope has rational (respectively integral) vertices. A natural class of lattice polyhedral subdivisions are the regular lattice subdivisions. They are induced by a function ω : S → Z, called an integral height function , for some finite subset S ⊆ P ∩ Z n containing the vertices of P . The cells of the subdivision are the projections of the bounded faces of the convex hull UH of {(u, λ) | u ∈ S, λ ≥ ω(u)} ⊆ R n × R. The bounded faces of UH form the graph of a function ν : P → R, called the convex graph of ω. Note that ν| S is not necessarily the same as ω| S . Let S(ν) denote the regular lattice polyhedral subdivision induced by ω. Then ν is convex and piecewise Q-affine with respect to S(ν), and takes integer values on the vertices of S(ν). Conversely, if ν : P → R is a convex function that is piecewise Q-affine with respect to a lattice polyhedral subdivision S, such that ν takes integer values on the vertices of S, then the restriction of ν to the vertices of S is an integral height function such that ν is the corresponding convex graph and S is a refinement of S(ν). For more details, including the general definition of a regular polyhedral subdivision, see [53,33].
By abuse of notation, we write S for the poset of faces (including the empty face) of a polyhedral subdivision S of P . Let [∅, P ] denote the face poset of P . In what follows, we will consider the function σ : S → [∅, P ], where σ(∅) = ∅, and for a non-empty cell F ∈ S, σ(F ) is the smallest face of P containing F . As a poset, the link lk S (F ) of F in S consists of all cells F ′ of S that contain F under inclusion.
We will need the following two closely related polynomials. In the case F = ∅, the local h-polynomial defined below was introduced by Stanley in [52,Corollary 7.7]. In the case of a triangulation of a simplex, it was introduced by Athanasiadis [1,2], and the general case of a polyhedral subdivision of a polytope was first considered by Nill and Schepers [42], where it was called the 'relative local h-polynomial'. A more general definition of the local h-polynomial appears in [31, Definition 4.1].
Definition 4.7. Let S be a polyhedral subdivision of a polytope P . If F is a (possibly empty) cell of S, then the h-polynomial of lk S (F ) is defined by Example 4.8. If S is the trivial subdivision of P , with cells of S given by the faces of P , then one can verify that h(lk S (Q); t) = g([Q, P ]; t) and l P (S, Q; t) = 0 whenever Q is a (possibly empty) proper face of P .
We state some properties of these polynomials below. The non-negativity of the gpolynomial and h-polynomial and the local h-polynomial when F = ∅ is due to Stanley [52, Theorem 5.2,Theorem 7.9]. Remark 4.9. [31, Theorem 6.1] Let S be a rational polyhedral subdivision of a lattice polytope P . For any cell F ∈ S and face Q of P , the polynomials g([Q, P ] * ; t), g([Q, P ]; t), h(lk S (F ); t) and l P (S, F ; t) have non-negative coefficients. We have the symmetry If S is a regular subdivision, then the coefficients of l P (S, F ; t) are symmetric and unimodal.
In order that one may compute examples, we list some of the coefficients below (see [ is symmetric with constant term 0 and linear term 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 R . Under an isomorphism M ∼ = Z n , the normalized volume Vol(P ) ∈ Z >0 of P with respect to M is n! times the Euclidean volume of P . Let ν : P → R be the convex graph of an integral height function on P , with corresponding regular lattice polyhedral subdivision S(ν) of P .
Remark 4.11. The setup above is slightly different than the 'weight 0' setup in [54], although the same arguments hold. We explain the relationship below. In [54], we consider a stacky fan (N, Σ, {b i }), where Σ is a full-dimensional simplicial fan with respect to a lattice N, and {b i } is a choice of a non-zero lattice point on each ray of Σ. We let ψ be the piecewise Q-linear function with respect to Σ satisfying ψ(b i ) = 1, and let Q = {v ∈ N R | ψ(v) ≤ 1}. If Q is a convex polytope, then we may set M = N, P = Q and ν = ψ above to recover this setup. Here Σ induces a triangulation S of P refining S(ν) such that ν takes integer values on the vertices of S. We also note that the definition of the weighted h * -polynomial below is slightly different (although equivalent) to that in [54] (see Example 5.13).
Let C be the cone over P ×{1} in M R ×R, and extend ν by linearity to a piecewise Q-linear function on C. For a non-negative integer m, we will often identify mP with C ∩(M R ×{m}) and M R with M R × {m}. We define a weight function for every non-negative integer m. As in Remark 2.4, the corresponding polynomial with coefficients in Z is the usual Ehrhart polynomial of P [22,23], defined by f (P ; m) = #(mP ∩ M) for every non-negative integer m. By [54,Theorem 3.7], f (P, ν; m) is a polynomial in m of degree dim P , and for every positive integer m, The latter result is known as weighted Ehrhart reciprocity, since it reduces to the classical Ehrhart reciprocity: f (P ; −m) = #(Int(mP ) ∩ M) for every positive integer m [24]. In fact, it is shown in [54, Remark 3.10] that weighted Ehrhart reciprocity generalizes Ehrhart reciprocity for rational polytopes. Let S be a lattice polyhedral decomposition refining S(ν) such that ν takes integer values on the vertices of S. For each maximal cell F ∈ S, let m F be the minimal positive integer such that m F ν| F is an affine function with respect to M. Then one verifies that the leading coefficient of f (P, ν; m) as a polynomial in m is given by: where Vol(F ) is the normalized volume of F and Vol(F )/m F is a positive integer. Define the weighted h * -polynomial h * (P, ν; u When P is the empty polytope, we set h * (P, ν; u) = 1. As in Remark 2.4, the corresponding polynomial with coefficients in Z is the usual h * -polynomial h * (P ; u) ∈ Z[u] of P introduced by Stanley in [51]. We define the local weighted h * -polynomial l * (P, ν; u Here the sum runs over all faces of P including the empty face. When P is the empty polytope, we set l * (P, ν; u) = 1. As in Remark 2.4, the corresponding polynomial with coefficients in Z is the usual local h * -polynomial l * (P ; u) ∈ Z[u] of P , introduced by Stanley in [52,Example 7.13], and later independently by Borisov and Mavlyutov in [9]. By Theorem 4.6 and Remark 4.9, the weighted h * -polynomial can be recovered as a 'nonnegative' combination of local weighted h * -polynomials: We present some properties and examples below. For the corresponding invariants over Z, proofs of the properties below can be found in [6] and [31].
Example 4.12. One may verify that (6) translates into the following: where m F is the minimal positive integer such that m F ν| F is an affine function with respect to M.
Example 4.13. Suppose that P is a simplex and ν is Q-affine, and let v 0 , . . . , v n denote the primitive integer vectors on the rays of C corresponding to the vertices of P . Let 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. [31,Lemma 7.11]). Firstly, the weighted h * -polynomial is acceptable in the sense of Stanley [52]. That is, 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 l * (P, ν; u) = u dim P +1 l * (P, ν; u −1 ). Example 4.15. If P is non-empty, then h * (P, ν; u) and l * (P, ν; u) have degree in u at most dim P , with the coefficient of u dim P in both cases equal to v∈Int(P )∩M w(v). By the symmetry in Remark 4.14 above, this implies that l * (P, ν; u) has constant term 0 and linear coefficient v∈Int(P )∩M w(v). The polynomial h * (P, ν; u) has constant term 1 and linear Using weighted Ehrhart reciprocity and following the proof of [31, Lemma 7.12], the above statement is equivalent to any of the three statements below: Remark 4.17. One may always choose a lattice triangulation S of P that refines S(ν) such that ν takes integer values on the vertices of S. In fact, there exists a choice with the same vertices as S(ν) (see, for example, [43]). Then Remark 4.16 together with Example 4. 13 give explicit formulas that one may use to compute h * (P, ν; u) and l * (P, ν; u) in practice. In particular, using Remark 4.9, it follows that all integer coefficients in h * (P, ν; u) and l * (P, ν; u) are non-negative.
This follows by the proof of [31,Theorem 7.20]. Explicitly, consider a regular, lattice polyhedral subdivision S of P that refines S(ν) such that the lattice points of weight 0 in P are precisely the vertices of S. Then ν takes integer values on the vertices of S, and every positive-dimensional cell in S contains no interior lattice points of weight 0. By Remark 4.16 and Example 4.15, where l P (S; u) has non-negative, symmetric, unimodal coefficients and the polynomial β(u) has non-negative integer coefficients. The result follows.

Mixed invariants.
We introduce our main combinatorial invariants below. As in Remark 2.4, the corresponding invariants with coefficients in Z first appeared in [31]. A geometric description of these invariants is provided in Corollary 5.12.
If P is empty, then h * (P, ν; u, v, w) = h * (P, ν; u, v) = l * (P, ν; u, v) = 1. We write h * (P, ν; u, v, w) = 1 + uvw 2 · 0≤p,q,r≤dim P −1 h * p,q,r (P, ν)u p v q w r , where h * p,q,r (P, ν) We summarize the main properties of these invariants in the following theorem. They may all be deduced from the definitions and the properties in the previous sections, and hence we omit the proof. We refer the reader to [31, Theorem 9.2, Theorem 9.3,Theorem 9.4] for a similar statement for the corresponding invariants with coefficients in Z.
(9) The coefficients h * p,q,r (P, ν) [k] are non-negative integers. For 0 ≤ j ≤ r and [k] ∈ Q/Z, the sequence {h i+j,i,r (P, ν) [k] | 0 ≤ i ≤ r − j} is symmetric and unimodal. Moreover, Remark 4.22. As in [31, Section 9], after fixing r, one may view the coefficients h * p,q,r (P, ν) in a diamond, for example, by placing h * p,q,r (P, ν) at point (q − p, p + q) in Z 2 . The resulting diamond of coefficients is symmetric with respect to its symmetry about the horizontal axis and symmetric up to conjugation with respect to its symmetry about the vertical axis. Fixing  As in Example 4.13, suppose that P is a simplex and ν is Q-affine. For every face Q of P , let C Q denote the cone over Q × {1} in M R × R, and set C = C P . Note that if Q is empty, then C Q is the origin. Let v 0 , . . . , v n denote the primitive integer vectors on the rays of C corresponding to the vertices of P .
We present explicit descriptions of some of the coefficients below.

Degenerations of hypersurfaces
In this section, we use Theorem 3.2 together with the combinatorics of Section 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 to the intersection cohomology of a canonical compactification of such a hypersurface. The corresponding results for invariants with coefficients in Z appear in [32]. We will continue with the notation of Section 2 and Section 4.
Let M be a lattice and let N = Hom(M, Z). Let X • = {f = 0} ⊆ Spec K[M] ∼ = (K * ) n be a schön hypersurface, where f = u∈M γ u (t)x u for some γ u (t) ∈ C(t). The Newton polytope P of X • is the convex hull of S = {u ∈ M | γ u (t) = 0} in M R . Note that P may be viewed as a full-dimensional lattice polytope in its affine span with induced lattice structure, and X • ∼ = X ′ × (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 ω f : S → Z defined by ω f (u) = ord t γ u (t). Recall from Section 4.2 that we may consider the convex graph ν f associated to ω f , with corresponding regular lattice polyhedral subdivision S(ν f ).
Remark 5.1. Conversely, assume that ν : P → R is the convex graph of an integral height function on P . Since schönness is a generic condition, it follows that there exists a schön hypersurface Tropical geometry of hypersurfaces reduces to the study of Newton polytopes and polyhedral subdivisions [33,47]. In particular, Trop(X • ) admits a polyhedral structure Σ with cells γ in inclusion-reserving correspondence with positive-dimensional cells F of S(ν f ), such that the bounded cells of Σ correspond to the cells of S(ν f ) not contained in the boundary of P . For example, if F is a maximal dimensional cell in S(ν f ), then ν f | F is a Q-affine function and the corresponding Q-linear function is an rational point in N Q , also denoted ν f | F . Then γ = −ν f | F is a vertex in Σ. Write f = u∈M λ u t ω f (u) g u (t)x u for some λ u ∈ C and g u (t) ∈ C(t) such that g(1) = 1. Let F be a face of S(ν f ) corresponding to a cell γ in S(ν f ). Then (7) in Let M F denote the intersection of M with the translation of the affine span of F to the origin. Then equation in the right hand side of (7) determines a schön complex hypersurface In this case, Theorem 3.2 translates into the following corollary.
Corollary 5.3. Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Then the motivic nearby fiber of X • is given by Here σ(F ) denotes the smallest face of P containing F , and V • F is a schön complex hypersurface of a torus with good µ-action induced by multiplication by exp(−2π √ −1ν| F ).
Remark 5.4. In fact, the sum in Corollary 5.3 runs only over the positive dimensional faces in S(ν), since when F is a vertex of S(ν), V • F = ∅. Here we follow the convention that [V ] = 0 ∈ K µ 0 (Var C ) if V is empty. Example 5.5. Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Assume that ν is Q-affine. Then the associated polyhedral subdivision S(ν) of P is trivial and Corollary 5.
Example 5.6. Let V • = { u∈P ∩M λ u x u = 0} ⊆ (C * ) n be a schön complex hypersurface with Newton polytope P and dim P = n. Let ν be the convex graph of an integral height function on P , and assume that ν is Q-affine and ν(u) ∈ Z whenever λ u = 0. Then V • is invariant under multiplication by the associated element exp(−2π √ −1ν) ∈ (C * ) n , and we have an induced good µ-action on V • . For example, the hypersurfaces V • F with corresponding good µ-action in the statement of Corollary 5.3 all appear in this way.
In this case, the variety X • = {f = u∈P ∩M λ u t ν(u) x u = 0} ⊆ (K * ) n is schön, and may be viewed as a family f : X • → C * with isomorphic fibers. Explicitly, we may identify f −1 (1) with V • , and multiplication by exp(−2π √ −1λν) ∈ (C * ) n , for any λ ∈ C, gives an isomorphism between f −1 (1) and f −1 (exp(2π √ −1λ)). Considering 0 ≤ λ ≤ 1, we see that we may identify the good µ-action on V • above with the good µ-action on f −1 (1) induced by monodromy. Note that since the monodromy action is finite, the corresponding monodromy operator T on cohomology is semi-simple. As in Example 5.5, Corollary 5.3 states that Recall the definition of the weighted refined limit mixed h * -polynomial h * (P, ν; u, v, w) ∈ Z[Q/Z][u, v, w] from Definition 4.20. Our goal is to prove the following formula for the equivariant refined limit Hodge-Deligne polynomial.
Theorem 5.7. Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Then the equivariant refined limit Hodge- We will also prove an equivalent statement involving intersection cohomology. We will use middle-perversity throughout. As explained in detail in [32,Section 6], given a projective variety X over 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 int (X ∞ , µ; u, v, w) associated to 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 • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Let ∆ P denote the normal fan to P with cones σ in inclusion-reserving bijection with the faces Q σ of P . Let X denote the closure of X • in the projective toric variety P(∆ P ) K over K corresponding to ∆ P . Then X admits a stratification X = σ∈∆ P X • σ , where X • σ is a schön hypersurface with Newton polytope Q σ and convex graph of an integral height function ν| Qσ . The theorem below is proved in the non-equivariant setting in [32, Theorem 6.1], but the proof goes through unchanged in our situation.
Theorem 5.8. [32, Theorem 6.1] Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Let X denote the closure of X • in the projective toric variety over K corresponding to the normal fan of P . Then the equivariant refined limit Hodge-Deligne polynomial obeys Our goal is to prove the following theorem.
Theorem 5.9. Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Let X denote the closure of X • in the projective toric variety over K corresponding to the normal fan of P . Then the equivariant refined limit Hodge-Deligne polynomial E int (X ∞ , µ; u, v, w) ∈ Z[Q/Z][u, v, w] associated to the intersection cohomology of X is given by is defined in terms of Stanley's g-polynomial (see Definition 4.3).
The following lemma is a formal consequence of Theorem 5.8. Details are provided in [32, Lemma 6.2].
It is well known that all interesting cohomology above occurs in middle dimension. We review the relevant facts below and refer the reader to [32,Sections 5.3,6] for details. We have weak Lefschetz theorems stating that the Gysin maps H m c (X • gen ) → H m+2 c ((C * ) n ) and IH m (X gen ) → IH m+2 (P(∆ P ) C ) are surjective for m ≥ n−1 and an isomorphism if m > n−1. One may deduce from Theorem 5.8 that the monodromy maps associated to the cohomology with compacts supports and intersection cohomology of (K * ) n and P(∆ P ) K respectively are trivial. Since X • gen is affine, H m c (X • gen ) = 0 for m < n − 1. Also, Poincaré duality for intersection cohomology means that the Gysin isomorphism above determines IH m (X gen ) for m < n − 1. We let H n−1 prim,c (X • ∞ ) and IH n−1 prim (X ∞ ) denote the kernels of the above Gysin maps with induced filtrations. These groups constitute the 'interesting' cohomology. Consider the contributions of these groups (up to a sign) to the equivariant refined limit Hodge-Deligne polynomial: E prim (X • ∞ , µ; u, v, w) := p,q,r α∈Q/Z h p,q,r (H n−1 prim,c (X • ∞ )) α αu p v q w r , E int,prim (X ∞ , µ; u, v, w) := p,q,r α∈Q/Z h p,q,r (IH n−1 prim (X ∞ )) α αu p v q w r .
By Lemma 5.10, it follows that Theorem 5.7 is equivalent to its specialization at w = 1: Before proceeding to the proof of Theorem 5.7 and Theorem 5.9, we will need the following theorem. In fact, it is a consequence of the proof that the theorem below is a special case of Theorem 5.7 and Theorem 5.9. An algorithm to compute the equivariant Hodge-Deligne polynomial below was given in [40,Section 2], extending an algorithm of Danilov and Khovanskiȋ in the non-equivariant setting [15]. The theorem below generalizes a formula of Borisov and Mavlyutov in the non-equivariant setting [9].
Theorem 5.11. Let V • = { u∈P ∩M λ u x u = 0} ⊆ (C * ) n be a schön complex hypersurface with Newton polytope P and dim P = n. Let ν be the convex graph of an integral height function on P , and assume that ν is Q-affine and ν(u) ∈ Z whenever λ u = 0. Then V • is invariant under multiplication by the associated element exp(−2π √ −1ν) ∈ (C * ) n , and we have an induced good µ-action on V • . The corresponding equivariant Hodge-Deligne Proof. Recall from Example 5.6 that the variety . Moreover, the action of monodromy on cohomology is semi-simple and hence E(X • ∞ , µ; u, v, w) = E(X • ∞ , µ; uw, vw) = E µ (V • ; uw, vw). If X = ∪ Q⊆P X • Q denotes the stratification in the statement of Theorem 5.8, then X • Q is defined by an equation of the form { u∈Q∩M λ u t ν(u) x u = 0}. It follows that if we restrict Theorem 5.7 to all X • that appear from complex schön hypersurfaces V • as above, then Theorem 5.7, Theorem 5.9 together with (9) and (10) are all still equivalent when restricted to this class of varieties over K.
We now complete the proofs of Theorem 5.7 and Theorem 5.9, which we have established are both equivalent to the equivalent statements (9) and (10).

Proof. By Corollary 5.3,
Substituting in Theorem 5.11 yields: Here the last equality follows from (7) in Theorem 4.21. This establishes (10) as desired.
Using Remark 5.1 and the above proof, we immediately deduce the following geometric description of the combinatorial invariants introduced in Section 4. As in Remark 2.4, the corresponding result for invariants with coefficients in Z can be found in [32, Corollary 5.11,Corollary 6.3].
Corollary 5.12. Let P be a full-dimensional lattice polytope in a lattice M and let ν : P → R be the convex graph of an integral height function on P . Let X • = {f = 0} ⊆ Spec K[M] be a schön hypersurface with Newton polytope P such that ν f = ν. Let X denote the closure of X • in the projective toric variety over K corresponding to the normal fan of P . Then h * (P, ν; u, v, w) = 1 + uvw 2 p,q,r α∈Q/Z h p,q,r (H dim P −1 prim,c (X • ∞ )) α αu p v q w r .
In particular, this specializes to: Moreover, l * (P, ν; u, v) = uv This specializes to: Example 5.13. Let P be a lattice polytope in a lattice N such that P contains the origin in its relative interior, and let {b i } denote the vertices of P . Let ν : P → R be the piecewise Q-affine function with value 0 at the origin and value 1 on the boundary of P . For an explicit example, see Example 4.19. Let X • = {f = 0} ⊆ Spec K[N] be a schön hypersurface with Newton polytope P such that ν f = ν. For example, if g is a schön complex polynomial with Newton polytope P , then take f = tg − λ for a generic choice of λ ∈ C * . Define the primitive spectrum of f to be: Consider the weighted h * -polynomial h * (P, ν; u) = 1+u , for some non-negative integers h * p,β . We consider an alternative way of encoding this polynomial. Specifically, for some sufficiently divisible positive integer N, define h(P ; t) := 1 + This was the definition of the weighted h * -polynomial given in [54]. Note that the h *polynomial h * (P ; t) is obtained from h(P ; t) by rounding up the exponents of t to the nearest integer. By [54, Corollary 2.12], we have the symmetry: h(P ; t) = t dim P h(P ; t −1 ).
Consider the complete fan Σ ′ in N R with cones given by the cones over the proper faces of P , and let Σ be a simplicial fan refinement of Σ ′ with the same rays. Then the the vertices {b i } of P constitute a choice of a non-zero lattice point on each ray of Σ. As in Remark 4.11, the triple Σ = (N, Σ, {b i }) is a stacky fan and we may consider the corresponding Deligne-Mumford stack X = X (Σ) with coarse moduli space the toric variety associated to the fan Σ [4]. The theory of orbifold cohomology, developed by Chen and Ruan [12,13], associates to X a finite-dimensional Q-algebra H * orb (X , Q), graded by Q. By [54,Theorem 4.3], together with Corollary 5.12, we see that the primitive spectrum of f coincides with the Betti polynomial of X : Equivalently, h(P ; t) = 1 + 2t 1/2 + t 2/3 + 4t + t 4/3 + 2t 3/2 + t 2 .

Applications to the monodromy of complex polynomials
In this section, we apply Theorem 5.7 to deduce explicit combinatorial equations for some important invariants associated to the monodromy of complex polynomials. Recall that we identify the group Q/Z with the group S 1 Q of rational points on the circle {z ∈ C | |z| = 1}, sending [k] ∈ Q/Z to α = e 2π √ −1k ∈ S 1 Q . We will continue with the notation of Section 2, Section 4 and Section 5, and set M = Z n . 6.1. Degenerations of hypersurfaces in affine space. We first consider applications of the results of Section 5 for families of hypersurfaces of affine space. Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Assume that P ⊆ R n ≥0 , and for each (possibly empty) subset ∈ S}, Z S = R S ∩ Z n and P S = P ∩ 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, . . . , n}. Equivalently, P contains the origin and has non-zero 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 ∞ for the union of faces at infinity of P . Then the faces of P are precisely the faces at infinity together with {P S | S ⊆ {1, . . . , n}}. Let X denote the closure of X • in K n . Then X gen ⊆ C n is a smooth hypersurface and, as in Section 5, one has a weak Lefschetz result (see, for example, [15,Corollary 3.8]) stating that H 2(n−1) c (X gen ) = C with trivial monodromy action and the only other non-zero cohomology is in middle-dimension H n−1 c (X gen ). By Corollary 5.3, the motivic nearby fiber of X is given by (11) ψ where σ(F ) is the smallest face of P containing F . After rearranging terms, Theorem 5.7 implies that (12) uvw 2 E(X ∞ , µ; u, v, w) = (uvw 2 ) n + (−1) n−1 S⊆{1,...,n} (−1) n−|S| h * (P S , ν| P S ; u, v, w).
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 gen : where m F is the minimal positive integer such that m F ν| F is an affine function with respect to Z S . Here Vol(F ) is the normalized volume of F and 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: Specializing at u = v = w = 1 gives: where m F is the minimal positive integer such that m F ν| F is an affine function with respect to Z S .
Assume further that the restriction of ν to P ∞ is constant. Substitute in Definition 4.20 and observe that g([P S , P S ′ ]; t) = 1 for any S ⊆ S ′ by Example 4.5. By assumption, for every face Q at infinity, l * (Q, ν| Q ; u, v) = v dim Q+1 l * (Q; uv −1 ). In this case, we may rewrite (12) as: Using Theorem 4.6 and Example 4.8, we have: 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 Int(Q) ⊆ R n >0 , 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 = ∅, we compute that the linear coefficient of G(Q; t) is the number of vertices of P contained in R n >0 . One then computes that F (u, v) has the form where every term in β(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 non-trivial action of monodromy is (−w) n+1 l * (P, ν; u, v). We conclude that we have the following corollary. Corollary 6.3. Let X • ⊆ (K * ) n be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function (P, ν) and dim P = n. Let X ⊆ K n denote the closure of X • in K n . Assume that P ⊆ R n ≥0 is convenient, and the restriction of ν 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 gen ) of the Deligne weight filtration for r = n − 1, and we have the following explicit combinatorial formula for the equivariant limit mixed Hodge numbers of 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 gen ).
Example 6.4. With the notation of Corollary 6.3, let n = 2 and consider X ⊆ K 2 . Using Example 4.24 and Example 6.2, we compute , ∂P denotes the boundary of P , and h * 0,0,1 (P, ν) = In particular, the action of monodromy on Gr W 0 H 1 c (X gen ) is trivial and dim Gr W 0 H 1 c (X gen ) = b. The equivariant limit mixed Hodge numbers of Gr W 1 H 1 c (X gen ) are given by The Jordan normal form of the action of monodromy on ) α Jordan blocks of size 1 with eigenvalue α, and h 0,0 (Gr W 1 H 1 c (X ∞ )) α Jordan blocks of size 2 with eigenvalue α. Remark 6.5. Specializing by setting w = 1 above gives a formula for the equivariant limit mixed Hodge numbers: Substituting in the definitions and simplifying, the right hand side may be rewritten as follows: We present the following two examples of the above situation. Below we let f ∈ C[x 1 , . . . , x n ] be a complex polynomial with Newton polytope NP(f ). For every face Q of NP(f ), we let ∆ Q denote the convex hull of Q and the origin. We set P = ∆ NP(f ) . We assume that f is convenient in the sense that NP(f ) has non-zero intersection with each ray through a coordinate vector [36]. Recall that for a complex hypersurface of a torus, schön is also called non-degenerate. Let Γ + (f ) = NP(f ) + R n ≥0 denote the Newton polyhedron of f . Observe that every bounded face of Γ + (f ) is a face of NP (f ). We let Γ f denote the union of the bounded faces of Γ + (f ). Remark 6.6. For any proper face Q of NP(f ) not containing the origin, consider a Q-affine function ν on ∆ Q with value m ± 1 at the origin, and value m on Q, for some integer m. Consider l * (Q, ν; u) ∈ Z[Q/Z][u] as a sum of integer valued polynomials indexed by α ∈ Q/Z. Then the coefficient of α = 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 ∈ C[x 1 , . . . , x n ] has no constant term, f is convenient and {f = 0} ⊆ (C * ) n defines a schön hypersurface. Then for a general choice of λ ∈ C * , X • = {f = λt} ⊆ (K * ) n defines a schön, convenient hypersurface with Newton polytope P , and ν 0 := ν is the piecewise Q-affine function on P with value 1 at the origin and value identically 0 on NP (f ). The cells of S(ν 0 ) are given by the union of {Q | Q ⊆ NP(f )} and {∆ Q | Q ⊆ Γ f }. By (11), we have the following equation for the motivic nearby fiber: (14) ψ By Corollary 6.3, the action of monodromy is trivial on the graded pieces Gr W r H n−1 c (X gen ) of the Deligne weight filtration for r = n − 1, and the equivariant limit mixed Hodge numbers of Gr W n−1 H n−1 c (X gen ) are given by: 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 gen ): By (13), we have a formula for the eigenvalues (with multiplicity) of the action of monodromy on the cohomology of X gen . Explicitly, α∈Q/Z dim H n−1 c (X ∞ ) α α is equal to (15) where 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)ν 0 | ∆ Q is an affine function with respect to the lattice given by intersecting M with the affine span of ∆ Q . When Q is empty, Vol(Q) = m(Q) = 1 and dim Q = −1.
Example 6.8 (Monodromy at infinity). Assume that f ∈ C[x 1 , . . . , x n ] is convenient. Assume that f is schön at infinity, meaning that {f | Q = 0} ⊆ (C * ) n defines a smooth hypersurface whenever Q is a face of P at infinity. Then for a general choice of λ ∈ C * , X • = {f t = λ} ⊆ (K * ) n defines a schön, convenient hypersurface with Newton polytope P , and ν ∞ := ν is the piecewise 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 S(ν ∞ ) are given by the union of {Q | Q ⊆ P ∞ } and {∆ Q | Q ⊆ P ∞ }. By (11), we have the following equation for the motivic nearby fiber: The motivic nearby fiber at infinity of a convenient polynomial f was introduced independently and from different perspectives in [40] and [46], and an explicit formula was given in [40,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 ∆ ∞ be the simplex P ∩ {x 1 + · · · + x n = ǫ} for fixed ǫ > 0 sufficiently small. Let S ∞ denote the regular, rational polyhedral subdivision obtained by intersecting S(ν ∞ ) with ∆ ∞ . That is, the cells of S ∞ are {Q ∞ := ∆ Q ∩ ∆ ∞ | Q ⊆ P ∞ }. 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 gen ): An algorithm to compute the equivariant limit mixed Hodge numbers above was given in [40,Section 5].
After specializing at v = w = 1, one obtains a formula for the spectrum at infinity of f [48] that is equivalent to [40,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: where 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)ν ∞ | ∆ Q is an affine function with respect to the lattice given by intersecting M with the affine span of ∆ Q . When Q is empty, Vol(Q) = m(Q) = 1 and dim Q = −1. The above is equivalent to a formula of Libgober and Sperber [37] for the zeta function at infinity of f .
Jordan block structure of monodromy at infinity. We continue with the notation of Example 6.8 above, and let f ∈ C[x 1 , . . . , x n ] be convenient and schön at infinity, and let X = {f t = λ} ⊆ K n for a general choice of λ ∈ C * . Recall that for every face Q of the Newton polytope NP(f ), we let ∆ Q denote the convex hull of Q and the origin. Recall that P = ∆ NP(f ) , and P ∞ denotes the union of the faces at infinity i.e. the faces of P not containing the origin. Recall that ν ∞ is the piecewise Q-affine function on P with value 0 at the origin and value identically 1 on the faces at infinity of P , and the cells of S(ν ∞ ) are given by the union of {Q | Q ⊆ P ∞ } and {∆ Q | Q ⊆ P ∞ }. Recall that ∆ ∞ is the simplex P ∩ {x 1 + · · · + x n = ǫ} for fixed ǫ > 0 sufficiently small, and S ∞ denotes the regular, rational polyhedral subdivision of ∆ ∞ obtained by intersecting S(ν ∞ ) with ∆ ∞ .
Following [40], one may use results of Broughton and Sabbah to read off the Jordan normal form of the action of monodromy on H n−1 c (X gen ) from the equivariant limit Hodge-Deligne polynomial. We outline the argument below. By Poincaré duality, we may work with usual cohomology H n−1 (X gen ) rather than cohomology with compact supports. We write H n−1 (X gen ) = H n−1 (X gen ) =1 ⊕ H n−1 (X gen ) =1 , where H n−1 (X gen ) =1 denotes the 1eigenspace of H n−1 (X gen ), and H n−1 (X gen ) =1 denotes the sum of the α-eigenspaces, for α = 1. The induced monodromy weight filtration M • on H n−1 (X gen ) may be written as and H n−1 (X gen ) =1 respectively. Let N = log T u denote the nilpotent operator where T u is the unipotent part of monodromy acting on H n−1 (X gen ), and consider the induced action of N on H n−1 (X gen ) =1 and H n−1 (X gen ) =1 . As explained in detail in [40], using results of Broughton [10], we have the following special case of a deep result of Sabbah. Recall the definition of the weight filtration of a nilpotent operator from Definition 2.6. Using Poincaré duality together with the symmetries of Remark 6.9, the corollary below follows immediately from Example 6.8. Corollary 6.13. Let f ∈ C[x 1 , . . . , x n ] be convenient and schön at infinity, and let X = {f t = λ} ⊆ K n . Then we have following formulas for the equivariant limit mixed Hodge numbers of H n−1 (X gen ): where the sum runs over all (possibly empty) faces at infinity.
Specializing the formulas in Corollary 6.13 by setting u = v yields: The following result now follows directly from Theorem 6.12. For every face at infinity Q of P , it follows from Remark 4.9 that we may write l Q,i t i (1 + t + · · · + t n−1−dim Q−2i ), for some non-negative integers { l Q,i | 0 ≤ i ≤ ⌊ n−1−dim Q 2 ⌋}. We define Corollary 6.14. Let f ∈ C[x 1 , . . . , x n ] be convenient and schön at infinity, and let J ∞ k,α be the number of Jordan blocks of size k with eigenvalue α for the action of monodromy at infinity on H n−1 (f −1 (t)) for t fixed and sufficiently large. Then where the sum runs over all (possibly empty) faces at infinity. This reproves the main results of [40]. Namely, the first two equations are equivalent to [ Example 6.17. Following on with Example 6.10, if f = n i=1 x m i i ∈ C[x 1 , . . . , x n ], for some m i ∈ Z >0 , then the action of monodromy is semi-simple, and all Jordan blocks have size 1. With the notation of Example 6.10, Example 6.18. As in Example 6.11, let f = a 0 x 5 + a 1 x 6 + a 2 x 5 y 2 + a 3 xy 5 + a 4 y 5 + a 5 xy 2 + a 6 x 2 y ∈ C[x, y] for a general choice of a 0 , . . . , a 6 ∈ C * . The action of monodromy on H 1 (X gen ) has 22 Jordan blocks of size 1, 4 with eigenvalue 1, 3 with eigenvalues exp(2π √ −1/6), exp(2π √ −1/3), exp(4π √ −1/3) and exp(10π √ −1/6) respectively, 2 with eigenvalue −1 and 1 with eigenvalues exp(π √ −1/5), exp(3π √ −1/5), exp(7π √ −1/5) and exp(9π √ −1/5) respectively, as well as a single Jordan block of size 2 with eigenvalue −1.

Monodromy of Milnor fibers.
Finally, we present some local versions of the formulas above (see Example 6.7).
Consider a polynomial f ∈ C[x 1 , . . . , x n ] and consider the polynomial map f : C n → C. Assume that f −1 (0) ⊆ C n has an isolated singularity at the origin. Restricting f : C n → C to a small ball about the origin in C n , and replacing C with a sufficiently small punctured disc about the origin, gives a locally trivial fibration called the Milnor fibration. Fix a fiber F 0 , called the Milnor fiber. A fundamental result of Milnor asserts that F 0 has the homotopy type of a wedge of (n − 1)-spheres [39]. In particular, H m (F 0 ) = 0 unless m = 0, n − 1. Here the monodromy action on H 0 (F 0 ) = C is trivial. In [57], Steenbrink introduced a mixed Hodge structure (F • , M • ) on H n−1 (F 0 ). The weight filtration has the following description in terms of the induced monodromy map T = T s T u on H n−1 (F 0 ) (cf. Theorem 6.12). We write H n−1 (F 0 ) = H n−1 (F 0 ) =1 ⊕ H n−1 (F 0 ) =1 , where H n−1 (F 0 ) =1 denotes the 1-eigenspace of H n−1 (F 0 ), and H n−1 (F 0 ) =1 denotes the sum of the α-eigenspaces, Recall that Γ + (f ) = NP(f ) + R n ≥0 denotes the Newton polyhedron of f , and Γ f denotes the union of the bounded faces of Γ + (f ). Recall that for every bounded face Q of Γ + (f ), ∆ Q denotes the convex hull of Q and the origin. We assume that f is convenient i.e. Γ + (f ) has non-zero intersection with each ray through a coordinate vector. We assume that f is schön at 0, meaning that {f | Q = 0} ⊆ (C * ) n defines a smooth hypersurface whenever Q is a bounded face of Γ + (f ). Note that this is a weaker condition than the condition that f is schön in Example 6.7. Let P + be the union of {∆ Q | Q ⊆ Γ f }, and let ν 0 be the piecewise Q-affine function on P + with value 1 at the origin and value 0 on Γ f . Then the lattice polyhedral decomposition S(ν 0 ) of P + has cells {∆ Q | Q ⊆ Γ f } and {Q ⊆ Γ f }. Although we will not need this, we note that even though P + is not convex, one can apply all the combinatorial constructions and results of Section 4. With the notation of Example 6.7, the following formula for the motivic Milnor fiber was given in [41,Theorem 4.3] (cf. (14)): where σ(∆ Q ) is the smallest face of P + containing ∆ Q . Specializing the above expression via the equivariant Hodge-Deligne map and applying Theorem 5.11 yields the following formula for the equivariant Hodge-Deligne polynomial: where h * (Q; u, v) is the weighted limit mixed h * -polynomial with respect to the convex graph that is identically zero. This gives the following expression for p,q α∈Q/Z h p,q (H n−1 (F 0 )) α αu p v q : Remark 6.19. The formula below should be compared to (15). Note that E(F 0 , µ; 1, 1) = α∈Q/Z m (−1) m dim H m (F 0 ) α α determines the eigenvalues (with multiplicity) of the action of monodromy. Specializing the above formula at u = v = 1, and applying Example 4.12, gives the following formula: where 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)ν 0 | ∆ Q is an affine function with respect to the lattice given by intersecting M with the affine span of ∆ Q . When Q is empty, Vol(Q) = m(Q) = 1 and dim Q = −1. This is equivalent to a famous formula of Varchenko [62] for the zeta function of monodromy of F 0 . As in Remark 2.4, we obtain a formula for dim H n−1 (F 0 ) originally due to Kouchnirenko [36].
As in previous sections, expanding the definitions above, simplifying and using Remark 6.6, yields the following 'non-negative' formulas for the equivariant mixed Hodge numbers of F 0 . Let ∆ 0 be the simplex P + ∩ {x 1 + · · · + x n = ǫ} for fixed ǫ > 0 sufficiently small. Let S 0 denote the regular, rational polyhedral subdivision obtained by intersecting S(ν 0 ) with ∆ 0 . That is, the cells of S 0 are {Q 0 := ∆ Q ∩ ∆ 0 | Q ⊆ Γ f }. Theorem 6.20. Let f ∈ C[x 1 , . . . , x n ] be a complex polynomial such that f −1 (0) admits an isolated singularity at the origin. Assume further that f is convenient and schön at 0. With the notation above, we have the following formulas for the equivariant mixed Hodge numbers of the cohomology of the associated Milnor fiber F 0 : where the sum runs over all (possibly empty) bounded faces of the Newton polyhedron of f .