E. Getzler, E. Looijenga
We give formulas for the Dolbeault numbers of \Mbar_{3,1}, using the first author's calculations of the weights of the cohomology of M_{2,2} and M_{2,3} and the second author's calculation of the weights of the cohomology of M_{3,1}.
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We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space \Mbar_4 of stable complex curves of genus 4, with its Hodge structure.
We give three new proofs of a theorem of C. Sabbah asserting that the weight filtration of the limit mixed Hodge structure at infinity of cohomologically tame polynomials coincides with the monodromy filtration up to a certain shift depending on the unipotent or non-unipotent monodromy part.
Given a family of Laurent polynomials, we will construct a morphism between its (proper) Gauss-Manin system and a direct sum of associated GKZ systems. The kernel and cokernel of this morphism are very simple and consist of free O-modules. The result above enables us to put a mixed Hodge module structure on certain classes of GKZ systems and shows that they have quasi-unipotent monodromy.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic pair on $X$ is a couple $(E,ϕ)$, where $E$ is a holomorphic bundle over $X$ of rank $n$ and degree $d$, and $ϕ\in H^0(E)$ is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which $E$ has fixed determinant.
We recall the construction of the Hodge character and we show, using a result due to F. Bittner, that these can be constructed using classical pure Hodge theory only, sideskipping Deligne's construction of functorial mixed Hodge structures on the cohomology of algberaic varieties. We then give a proof, based on the weak factorization theorem, that the nearby fibre is well-defined in the Grothendieck group of varieties. Finally, various formulas for numerical invariants of Hodge-theoretic origin are presented which are associated to one-parameter degenerations of projective manifolds.
We prove a generalized Mariño-Vafa formula for Hodge integrals over $\Mbar_{g, γ-μ}(\cB G)$ with $G$ an arbitrary finite abelian group. Then we use this formula to study the local Gromov-Witten theory of an orbi-curve with cyclic stack points in a Calabi-Yau three-orbifold.
Let $X$ be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $Sym^{n}X$ when the cohomology of $X$ is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials $μ_{X^{n}}^{S_{n}}\left(t,u,v\right)$, codifying the permutation action of $S_{n}$ as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products $μ_{Sym^{n}X}\left(t,u,v\right)$. These formulas are then applied to the case of linear alg…
In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems introduced by Hernández Ruiperez and Tejero Prieto.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,ϕ)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a holomorphic map $ϕ\colon E_{2} \to E_{1}$. There is a concept of stability for triples which depends on a real parameter $σ$. In this paper, we determine the Hodge polynomials of the moduli spaces of $σ$-stable triples with $\mathrm{rk}(E_1)=\mathrm{rk}(E_2)=2$, using the theory of mixed Hodge structures (in the cases that they are smooth and compact). This gives in particular the Poincar{é} pol…
We compute the Hodge-Deligne polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2,C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2,C), that is Id, -Id, diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the Hodge-Deligne polynomial under fibrations.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,ϕ)$ on $X$ consists of two holomorphic vector bundles $E_1$ and $E_2$ over $X$ and a holomorphic map $ϕ:E_2 \to E_1$. There is a concept of stability for triples which depends on a real parameter $σ$. In this paper, we determine the Hodge polynomials of the moduli spaces of $σ$-stable triples with $\rk(E_1)=3$, $\rk(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polyn…
Newton Polyhedrons and Hodge Numbers of Non-degenerate Laurent Polynomials
研究无穷远牛顿多面体为单纯形的非退化便利洛朗多项式，利用堆叠扇上的逐锥多项式函数环刻画消失圈的极化混合霍奇结构，计算其霍奇菱形，并以初等方法解决相应Birkhoff问题。
We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.
We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights. For line arrangements, we prove in a new way the fact due to Budur and Saito that the spectrum is determined by the weak combinatorial data, and show that such a result fails for the Hodge-Deligne polynomials.
Let X be a nonsingular complex projective variety that is acted on by a reductive group $G$ and such that $X^{ss} \neq X_{(0)}^{s}\neq \emptyset$. We give formulae for the Hodge--Poincaré series of the quotient $X_{(0)}^s/G$. We use these computations to obtain the corresponding formulae for the Hodge--Poincaré polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify some hyperplanes on which all polynomials in multivariable Bernstein ideal vanish.
The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers of congruence covers in any dimension, generalizing results of E. Hironaka…
We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.
Explicit formulas for mixed Hodge polynomials of character varieties of nilpotent groups
本文利用Florentino等人的公式，为幂零群到辛群和正交群的特征簇的混合Hodge多项式给出了显式分拆公式。
When $k<n$, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which $(n,d,k)=(3,d,1)$ and $d$ is even, obtaining from them the usual Poincaré polynomials.
Equivariant Hodge polynomials of heavy/light moduli spaces
本文研究Hassett重/轻权模空间及其光滑轨迹的等变Hodge-Deligne多项式，给出变量替换公式，用已知模空间的S_n-等变生成函数计算(S_m×S_n)-等变生成函数。
Hodge-Deligne polynomials of character varieties of abelian groups
本文计算了复拟射影 F-簇对角作用的商 X^r/F 的混合 Hodge-Deligne 多项式，并应用于推导自由阿贝尔群 G-特征簇的显式公式，在 GL(n,C) 和 SL(n,C) 情形下利用分拆组合学给出了更具体的表达式。
Weighted Ehrhart theory via mixed Hodge modules on toric varieties
本文利用环簇上混合 Hodge 模的示性类解释加权 Ehrhart 理论，构造广义 Ehrhart 多项式，并给出交上同调符号差的组合公式。
Given a polynomial map $f:\Bbb C^{n+1}\to\Bbb C$, one can attach to it a geometrical variation of mixed Hodge structures (MHS) which gives rise to a limit MHS. The equivariant Hodge numbers of this MHS are analytical invariants of the polynomial map and reflect its asymptotic behaviour. In this paper we compute them for a class of generic polynomials, in terms of Hodge numbers attached to isolated hypersurface singularities and Hodge numbers of cyclic coverings of projective space branched along a hypersurface.
Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,n}$
通过Hassett模空间墙穿越，将曲线模空间上同调计算归结为最小权重情形，并给出亏格1的显式多项式公式，实现高效计算。
It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using Gårding's theory for hyperbolic polynomials, we obtain a mixed Hodge-index type theorem for classes of type $(1,1)$. The new feature is that this Hodge-index type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate and even negative along some directions.
Geometric monodromies, mixed Hodge numbers of motivic Milnor fibers and Newton polyhedra
本文引入多项式上同调群中局部与全局单值理论，结合环面几何与动机Milnor纤维，用Newton多面体描述等变混合Hodge数。
Motivated by a connection between the topology of (generalized) configuration spaces and chromatic polynomials, we show that generating functions of Hodge-Deligne polynomials of quasiprojective varieties and colorings of acyclic directed graphs with the complete graph $K_n$ as the underlying undirected graph. In order to do this, we combine an interpretation of Crew-Spirkl for plethysms involving chromatic symmetric polynomials using colorings of directed acyclic graphs to plethystic exponentials often appearing Hodge-Deligne polynomials of varieties. Applying this to a recent result of Floren…
We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian $G=G(2,N)$ parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of $G$ in projective space. The analysis of the Hodge index inequalities involves estimates on certain Racah polynomials.
For a simply connected complex algebraic variey $X$, by the mixed Hodge structures $(W_{\bullet}, F^{\bullet})$ and $(\tilde W_{\bullet}, \tilde F^{\bullet})$ of the homology group $H_{*}(X;\mathbb Q)$ and the homotopy groups $π_{*}(X)\otimes \mathbb Q$ respectively, we have the following mixed Hodge polynomials $$MH_X(t,u,v):= \sum_{k,p,q} \operatorname{dim} \Bigl ( Gr_{F^{\bullet}}^{p} Gr^{W_{\bullet}}_{p+q} H_k (X;\mathbb C) \Bigr) t^{k} u^{-p} v^{-q},$$ $$\quad \, \, MH^π_X(t,u,v):= \sum_{k,p,q} \operatorname{dim} \Bigl (Gr_{\tilde F^{\bullet}}^{p} Gr^{\tilde W_{\bullet}}_{p+q} (π_k(X) \ot…
We study the nonnegativity of stringy Hodge numbers of a projective variety with Gorenstein canonical singularities, which was conjectured by Batyrev. We prove that the $(p,1)$-stringy Hodge numbers are nonnegative, and for threefolds we obtain new results about the stringy Hodge diamond, which hold even when the stringy $E$-function is not a polynomial. We also use the Decomposition Theorem and mixed Hodge theory to prove Batyrev's conjecture for a class of fourfolds.
Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they are nonnegative. We prove this for a class of `mild' isolated singularities (the allowed singularities depend on the dimensio…
We define notions of higher order spectra of a complex quasi-projective manifold with an action of a finite group $G$ and with a $G$-equivariant automorphism of finite order, some of their refinements and give Macdonald type equations for them.
Let $k$ be an algebraically closed field of positive characteristic. For any integer $m \geq 2$, we show that the Hodge numbers of a smooth projective $k$-variety can take on any combination of values modulo $m$, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.
A complete classification of negative stringy Hodge numbers
该研究确定了Batyrev非负性猜想的维数阈值：其在维数≤4时成立，维数≥5时存在反例，为相关领域提供了关键结论。
Hodge-Riemann polynomials
本文研究丰富向量丛上满足 Hodge-Riemann 关系的 Hodge-Riemann 多项式，证明其多种性质并证实 Ross-Toma 的预测，进而证明 Schur 多项式乘积的导数序列具有 Schur 对数凹性，验证了 Ross-Wu 猜想。
Relations for quadratic Hodge integrals via stable maps
该研究遵循Faber-Pandharipande方法，利用到$ \mathbb{P}^1 $的稳定映射模空间虚局部化公式计算霍奇积分关系，证明其某些生成函数为多项式，为二次霍奇积分研究提供新结果。
In this paper we study the topology of the moduli spaces of representations of degree $2$ for free monoids. We calculate the virtual Hodge polynomials of the character varieties for several types of $2$-dimensional representations. Furthermore, we count the number of isomorphism classes for each type of $2$-dimensional representations over any finite field ${\Bbb F}_q$, and show that the number coincides with the virtual Hodge polynomial evaluated at $q$.
Let $\mathcal{X}_ΓG:=\mathrm{Hom}(Γ,G)/\!/G$ be the $G$-character variety of $Γ$, where $G$ is a complex reductive group and $Γ$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_ΓG$, and present some applications, focusing on the cases when $Γ$ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere.
We prove a version of the Hodge-Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.
We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursion recovers the Witten-Kontsevich theorem when restricted to the top degree terms, and also the combinatorial factor of the lambda_g formula as the lowest degree terms.
We prove that given any rational Hodge isometry $ψ:H^2(S_1,\mathbb{Q})\rightarrow H^2(S_2,\mathbb{Q})$ between any two Kähler $K3$ surfaces $S_1$ and $S_2$ the cohomology class of $ψ$ in $H^{2,2}(S_1\times S_2)$ is a polynomial in Chern classes of coherent analytic sheaves over $S_1 \times S_2$. Consequently, the cohomology class of $ψ$ is algebraic whenever $S_1$ and $S_2$ are algebraic.
Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincaré duality, the Hard Lefschet…
We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of the theorem. This yields the "Hodge decomposition" of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomi…
We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-α}$ associated to the effective $\mathbb{Q}$-divisor $D=α\cdot Z$, where $0<α\le1$ and $Z=(f=0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic functio…
Weighted Ehrhart theory via equivariant toric geometry
本文通过等价 toric 几何方法，提出了一种广义加权 Ehrhart 理论的 K-理论和几何解释，并证明了相关的互易和纯度公式，统一了经典的 Ehrhart 理论和最近的组合公式。
The distribution of the spectral numbers of an isolated hypersurface singularity is studied in terms of the Bernoulli moments. These are certain rational linear combinations of the higher moments of the spectral numbers. They are related to the generalized Bernoulli polynomials. We conjecture that their signs are alternating and prove this in many cases. One motivation for the Bernoulli moments comes from the comparison with compact complex manifolds.
I demonstrate how certain identities for Macdonald's polynomials established by Garsia, Haiman and Tesler, together with the conjecture of Hausel, Letellier and Villegas imply explicit relations between mixed Hodge polynomials of different character varieties.
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial $f$ in $\C^{n+1}$, where $f$ satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\C^{n+1}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th …
Stringy Irregular Hodge Numbers for Non-Simplicial Toric Landau--Ginzburg Data
将轨道折叠不规则Hodge理论推广至非单纯环带数据，构造滤过Koszul--de Rham复形，证明其射影crepant单纯细分模型共享同一Hodge多项式，从而定义弦论不规则Hodge数。
For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…
The Getzler's formula relates the S_n-equivariant Hodge-Deligne polynomial of the space of ordered tuples of distinct points on a given variety X with the Hodge-Deligne polynomial of X. We obtain the analogue of this formula for the case when X has a nontrivial automorphism group. Collecting together all strata of $\mathcal{M}_2$ with different automorphism groups, we derive a formula for the S_n-equivariant Euler characteristic of $\mathcal{M}_{2,n}$.
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in $\mathbb{P}^3$ is improved and applications to deformation theory of nodal surfaces are given.
We describe a class of isolated nondegenerate hypersurface singularities that give a polynomial contribution to Batyrev's stringy E-function. These singularities are obtained by imposing a natural condition on the facets of the Newton polyhedron, and they are strictly canonical. We prove that Batyrev's conjecture concerning the nonnegativity of stringy Hodge numbers is true for complete varieties with such singularities, under some additional hypotheses on the defining polynomials (e.g. convenient or weighted homogeneous). The proof uses combinatorics on lattice polytopes. The results form a s…
The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and conjectured that they are nonnegative. We compute explicit formulae for the contribution of an A-D-E singularity to the stringy E-function in arbitrary dimension. With these results we can say when the stri…
Calculating Hodge and weight filtrations on localizations
本文介绍Macaulay2包MixedHodgeModules，用于计算多项式环局部化上的Hodge与权重滤过及相关不变量，并支持计算多种复形。
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theo…
For each smooth Fano threefold $X$ with Picard number 1 we consider a weak Landau--Ginzburg model, that is a fibration over $\mathbb C^1$ given by a certain Laurent polynomial. In the spirit of L. Katzarkov's program we prove that the number of irreducible components of the central fiber of its compactification is $h^{1,2}(X)+1$. In particular, it does not depend on the compactification. The question of dependence on the model is open; however we produce examples of different weak Landau--Ginzburg models for the same variety with the same number of components of the central fiber.
A Hodge Theoretic Generalization of $\mathbb{Q}$-Homology Manifolds II: Local Complete Intersections
研究复代数簇$Z$的奇点不变量$\HRH(Z)$，聚焦局部完全交子簇，将其与伯恩斯坦 - 佐藤多项式等著名不变量联系，超曲面情形可由这些不变量完全刻画$\HRH(Z)$，高余维情形更复杂。
We compute the Hodge numbers of the moduli space of semi-stable sheaves on the complex projective plane supported on quintic curves and having Euler characteristic 3. For this purpose we study the fixed-point set for a certain torus action on the moduli space.
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