Semi-infinite Hodge structures and mirror symmetry for projective spaces

Chen-Jiang decompositions for projective varieties, without Hodge modules

We give a new proof of a theorem by Pareschi, Popa and Schnell that the direct image of the canonical bundle of a smooth projective variety along a morphism to an abelian variety admits a Chen-Jiang decomposition, without using the theory of Hodge modules.

Explicit Hodge-type decomposition on projective complete intersections

We construct an explicit homotopy formula for the d-bar complex on a complete intersection subvariety V in CP^n. This formula can be interpreted as a Hodge-type decomposition for residual currents on V.

A brief note on the second fundamental form of the variation of Hodge structure of a smooth projective hypersurface

We determine the second fundamental form of a variation of Hodge Structure of a smooth projective hypersurface using the classical identification of the Hodge structure and the action of the infinitesimal variation of Hodge structure with pieces of the Jacobian Ideal of the hypersurface. This simple algebraic description also implies that, in this case, the variation of Hodge structure satisfies a new set of second order partial differential equations.Comments are welcome!

New mirror pairs of Calabi-Yau orbifolds

We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When $X$ is the Fermat quintic in $¶^4$, and $\tilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $Γ$ of the alternating group $A_5$, the $Γ$-Hilbert schemes $Γ$-$\Hilb(X)$ and $Γ$-$\Hilb(\tilde{X}^*)$ are smooth Calabi…

Rigid-analytic varieties with projective reduction violating Hodge symmetry

We construct examples of smooth proper rigid-analytic varieties admitting formal model with projective special fiber and violating Hodge symmetry for cohomology in degrees $\geq 3$. This answers negatively a question raised by Hansen and Li.

The E^3/Z3 orbifold, mirror symmetry, and Hodge structures of Calabi-Yau type

Starting from the Kähler moduli space of the rigid orbifold Z=E^3/\mathbb{Z}_3 one would expect for the cohomology of the generalized mirror to be a Hodge structure of Calabi-Yau type (1,9,9,1). We show that such a structure arises in a natural way from rational Hodge structures on Λ^3 \mathbb{K}^6, \mathbb{K}=\mathbb{Q}[ω], where ωis a primitive third root of unity. We do not try to identify an underlying geometry, but we show how special geometry arises in our abstract construction. We also show how such Hodge structure can be recovered as a polarized substructure of a bigger Hodge structure…

Pfaffian双重镜像的弦论Hodge数与同调射影对偶

Stringy Hodge numbers of Pfaffian double mirrors and Homological Projective Duality

本文通过引入修正版弦论E-函数，研究Pfaffian双重镜像的弦论Hodge数，推广Borisov-Libgober结果，并对Pfaffian簇的范畴crepant消解的Lefschetz分解给出数值预测。

Lectures on the Hodge theory of projective manifolds

This is a written-up version of eight introductory lectures to the Hodge theory of projective manifolds. The table of contents should be self-explanatory. The only exception is section 8 where I discuss, in a simple example, a technique for studying the class map for homology classes on the fibers of a map and one for approximating a certain kind of primitive vectors.

Limiting mixed Hodge structures on the relative log de Rham cohomology groups of a projective semistable log smooth degeneration

We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.

Hodge-Theoretic Mirror Symmetry for Toric Stacks

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant …

极限混合Hodge结构的投影超曲面周期

Periods of Limiting Mixed Hodge Structures of Projective Hypersurfaces

研究投影超曲面的一参数退化情形，证明极限混合Hodge结构的周期由对数、伽马和狄利克雷L函数的特殊值生成，方法基于GKZ系统的解析延拓。

射影超曲面族的退化与霍奇猜想

Degeneration of families of projective hypersurfaces and Hodge conjecture

该研究通过维数归纳法，结合光滑超曲面情形的既有结论、混合霍奇模的邻近循环函子等工具，证明了光滑复射影簇的霍奇猜想。

Hodge Structure of a Complete Intersection of Quadrics in a Projective Space

Given a smooth projective variety V of dimension n, one may say that V has motivic dimension less than d+1 if the cohomology of V comes from varieties of dimensions less than d+1 in some geometric way. In this paper, we show that a smooth complete interesection of k quadrics has a motivic dimension less than k.

Studies of closed/open mirror symmetry for quintic threefold through log mixed Hodge theory

We correct the definitions and descriptions of the integral structures in [U14]. The previous flat basis in [ibid] is characterized by the Frobenius solutions and integral in the first approximation by mean of the graded quotients of monodromy filtration, but not integral in the strict sense. In this article, we use the integral structure of Iritani in [I11] for A-model. Using this precise version, we study open mirror symmetry for quintic threefolds through log mixed Hodge theory, especially the recent result on Neron models for admissible normal functions with non-torsion extensions in the j…

Hodge classes of type (2, 2) on Hilbert squares of projective K3 surfaces

We give a basis for the vector space generated by rational Hodge classes of type (2,2) on the Hilbert square of a projective K3 surface general in its rank, which is a subspace of the singular cohomology ring with rational coefficients: we use Nakajima operators and an algebraic model developed by Lehn and Sorger as main tools. We then obtain a basis of the lattice generated by integral Hodge classes of type (2,2) on the Hilbert square of a projective K3 surface general in its rank: we exploit lattice theory, a theorem by Qin and Wang and a result by Ellingsrud, Göttsche and Lehn.

On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces

Mirrors $X^{\vee}$ of quasi-smooth Calabi-Yau hypersurfaces $X$ in weighted projective spaces ${\Bbb P}(w_0, \ldots, w_d)$ can be obtained as Calabi-Yau compactifications of non-degenerate affine toric hypersurfaces defined by Laurent polynomials whose Newton polytope is the lattice simplex spanned by $d+1$ lattice vectors $v_i$ satisfying the relation $\sum_i w_i v_i =0$. In this paper, we compute the stringy $E$-function of mirrors $X^\vee$ and compare it with the Vafa's orbifold $E$-function of quasi-smooth Calabi-Yau hypersurfaces $X$. As a result, we prove the equalities of Hodge numbers …

$L^2$-Hodge理论构造Calabi-Yau光滑射影超曲面的Frobenius流形

$L^2$-Hodge theoretic construction of Frobenius manifolds for Calabi-Yau smooth projective hypersurfaces

本文通过Li-Wen的$L^2$-Hodge理论，为Calabi-Yau光滑射影超曲面的上同调构造了Frobenius流形结构，并与Barannikov-Kontsevich的构造进行了精确比较。

System of Hodge Bundles and Generalized Opers on Smooth Projective Varieties

Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define notion of generalized opers on $X$, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle $E$ together with a connection $\nabla$ over $X$ implies semistability of the pair $(E, \nabla)$.

Hodge-theoretic splitting mechanisms for projective maps (in appendix, a letter from P. Deligne)

According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…

Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds

This proceedings note introduces aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of Calabi-Yau threefolds over the thrice-punctured sphere with b^3 = 4, or equivalently h^{2,1} = 1, and the related issues of geometric realization of these variations. The presentation parallels that of the first author's talk at the BIRS workshop.

Hodge sheaves underlying flat projective families

We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms …

The Hodge structure of semiample hypersurfaces and a generalization of the monomial-divisor mirror map

We solved the long-standing problem of describing the cohomology ring of semiample hypersurfaces in complete simplicial toric varieties. Also, the monomial-divisor mirror map is generalized to a map between the whole Picard group and the space of infinitesimal deformations for a mirror pair of Calabi-Yau hypersurfaces. This map is compatible with certain vanishing limiting products of the subrings of the chiral rings, on which the ring structure is related to a product of the roots of $A$-type Lie algebra.

Real non-abelian mixed Hodge structures for quasi-projective varieties: formality and splitting

We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, th…

堆叠 Clarke 镜面对的不规则 Hodge 数

Irregular Hodge numbers of stacky Clarke mirror pairs

本文证明堆叠 Clarke 镜面对扭曲上同调的不规则 Hodge 滤过分次片段对偶，并推广相关猜想与 Hodge 数对偶结果。

复超曲面奇点范畴上的霍奇结构

The Hodge structure on the singularity category of a complex hypersurface

本文证明复仿射超曲面奇点范畴的非交换霍奇结构等同于相应射影超曲面的经典霍奇结构，并由此建立霍奇猜想与dg范畴类比之间的等价性。

Non-genericity of infinitesimal variations of Hodge structures arising in some geometric contexts

We prove that the infinitesimal variations of Hodge structure arising in a number of geometric situations are non-generic. In particular, we consider the case of generic hypersurfaces in complete smooth projective toric varieties, generic hypersurfaces in weighted projective spaces and generic complete intersections in projective space and show that, for sufficiently high degrees, the corresponding infinitesimal variations are non-generic.

Generalized periods and mirror symmetry in dimensions n>3

The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such variety can be expressed in terms of certain invariants of a new generalization of variation of Hodge structures attached to the dual variety. To formulate the general principles of Mirror Symmetry in arbitrary dimension it is necessary to introduce the ``extended moduli space of complex structures'' M. An analog M\to H*(X,C)[n] of the classical period map is de…

Period- and mirror-maps for the quartic K3

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with th…

Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the fami…

Hodge numbers of Landau-Ginzburg models

We study the Hodge numbers of Landau-Ginzburg models as defined by Katzarkov, Kontsevich and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau-Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold $X$ there is a natural LG mirror $(Y,\mathsf{w})$ so that $h^{p,q}(X) = f^{3-q,p}(Y,\mathsf{w})$.

射影结构与Hodge理论

Projective structures and Hodge theory

该研究在紧Riemann曲面上构造了与周期映射第二基本形式相关的Hodge射影结构$p_h$，通过射影结构与射影曲线模空间$(1,1)$-形式的对应，证明其与单值化给出的自然射影结构$p_u$不同。

循环开-闭映射与霍奇结构的变分

The cyclic open--closed map and variations of Hodge structures

本文在半正情形下为体形变的相对Fukaya范畴构造循环开-闭映射，证明其是半无限霍奇结构极化变分的态射，并给出同构判据，从而在Calabi-Yau镜像对中由同调镜像对称推出枚举镜像对称。

拟射影曲线上扩张的非阿贝尔霍奇对应

The non-abelian Hodge correspondence for extensions on quasi-projective curves

本文为拟射影曲线上的对数平坦丛与对数希格斯丛建立精确等价，推广了温和辛普森-望月对应至幂零留数情形。

Rational Integrals of the second kind on a complex projective manifold and its primitive cohomology

Let X be a complex algebraic manifold of dimension n+1 embedded in a sufficiently higher dimensional complex projective space, and Y a generic hyperplane section of X. We describe the mixed Hodge structure on H^p(X-Y,C) and the Hodge filtration of the middle primitive cohomology group H^n(Y,C)_0 of Y in terms of rational integrals on X. (Key words: Primitive cohomology, Rational integral of the 2nd kind, Generalized Poincare residue map, Hodge filtration, Mixed Hodge structure)

The Generalized Hodge conjecture for 1-cycles and codimension two algebraic cycles

In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…

Quantum periods - I. Semi-infinite variations of Hodge structures

We introduce a generalization of variations of Hodge structures living over moduli spaces of non-commutative deformations of complex manifolds. Hodge structure associated with a point of such moduli space is an element of Sato type grassmanian of semi-infinite subspaces in H^*(X,C)[[h^{-1},h]]. Periods associated with such semi-infinite Hodge structures serve in order to extend mirror symmetry relations in dimensions greater then three.

积分Hodge/塔特猜想在与投影超曲面的乘积上的失效

On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces

本文通过细化无 ramifications cohomology方法，证明了Enriques面与光滑奇维投影超曲面的乘积上积分Hodge/塔特猜想的失效，并推广了Shen的结果到一般代数闭域。

Hodge-type conjecture for higher Chow groups

Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain smooth projective varieties over the algebraic closure of the rational number field. We also verify the conjecture on the surjectivity in the case of the complement of a union of general hypersurfaces in…

Hodge Theory and Deformations of Affine Cones of Subcanonical Projective Varieties

We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a distinguished graded component of the module of first order deformations of $A_X$, and later on we show how to identify the whole primitive cohomology of $X$ as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over $X$. In the particular case of a projective smooth hypersurface $X$ we recover Griffiths' isomorp…

On the Hard Lefschetz property of stringy Hodge numbers

For projective varieties with a certain class of 'mild' isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property. This result fits nicely with a 6-dimensional counterexample of Mustata and Payne for the Hard Lefschetz property for stringy Hodge numbers in general. We also give such an example, ours is a hypersurface singularity.

Towards Mirror Symmetry for Varieties of General Type

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers d…

On the Hodge conjecture for hypersurfaces in toric varieties

We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.

Q-因子性与霍奇-杜布瓦理论

Q-factoriality and Hodge-Du Bois theory

该研究证明了正规射影簇及正规簇解析芽的$\text{Q}$-因子性缺陷的霍奇理论公式，其结果涵盖Samuel猜想局部解析版本、有理同调三维簇刻画及霍奇-杜布瓦数在翻转下的不变性等。

Semi-regular varieties and variational Hodge conjecture

We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $π:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset \mathcal{X}_o$, the cohomology class corresponding to $Z$ remains a Hodge class (as $\mathcal{X}_o$ deforms along $B$) if and only if $Z$ remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety $Z$ of dimension $n$ is a semi-regular sub-variety of a smooth projective hypersur…

非阿贝尔霍奇理论与Higgs丛的模空间

Non-Abelian Hodge Theory and Moduli Spaces of Higgs Bundles

本文研究非阿贝尔霍奇理论与Higgs丛模空间，探讨了Hitchin纤维的几何性质及拓扑镜像对称性，并通过计算庞加莱多项式验证了相关对称性。

Quantum Mirror Symmetry for Borcea-Voisin Threefolds

Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoretic sense, but their mirror symmetry at the quantum level have not previously been shown. We prove a Givental-style quantum mirror theorem for certain Borcea-Voisin threefolds: by means of certain birational models, we show that their Gromov-Witten J-functions are related by a mirror map to solutions of the multi-parameter Picard-Fuchs equations coming from the variation of Hodge structures of their mirror partners.

θ丛、Quillen联络与霍奇理论射影结构

Theta bundle, Quillen connection and the Hodge theoretic projective structure

该研究针对亏格≥2紧黎曼曲面模空间上霍奇理论射影结构族，借助霍奇线丛行列式对偶的复联络刻画，构造对应联络并给出三种等价描述，完善射影结构与联络的对应关系。

Schematic homotopy types and non-abelian Hodge theory

In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X \mapsto (X\otimes \mathbb{C})^{sch}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a \textit{Hodge decomposition} on $(X\otimes\mathbb{C})^{sch}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb{C}^{\times δ}$ on the object $(X\otimes…

A-model Implications of Extended Mirror Symmetry

Associativity of the quantum product ensures flatness of the Dubrovin connection and is the basis for Hodge-theoretic mirror symmetry of Calabi-Yau threefolds. We use ring and module structure on cohomology pertaining to a Lagrangian submanifold to define an extension of the A-model Variation of mixed Hodge structure that matches the predictions from extended mirror symmetry. Our construction makes contact with axioms for open Gromov-Witten theory recently proposed by Solomon-Tukachinsky.

Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.

某些有理椭圆射影四维簇的拓扑刻画与 Hodge 结构

Topological characterization and Hodge structures of some rationally elliptic projective fourfolds

研究嵌入 $\mathbb{P}^8$ 的有理椭圆射影四维簇，借助法丛 Chern 数等刻画其拓扑并证明同构于 $\mathbb P^4$ 的条件，同时验证 Hodge 猜想及非正 Hodge 层级。

Hodge theory for tropical varieties

In this paper we prove that the cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. On the way to establish these results, we introduce and prove other results of independent interest. This includes a generalization of the results of Adiprasito-Huh-Katz, Hodge theory for combinatorial geometries, to any unimodular quasi-projective fan having the same support as the Bergman fan of a matroid, a tropi…

Hypergeometric Equations and Weighted Projective Spaces

We compute the Hodge numbers of the variation of Hodge structure of the middle cohomology (with compact support) of the Landau-Ginzburg model dual to a weighted projective space. We state a conjectural formula for the Hodge numbers of general hypergeometric variations. We show that a general Landau-Ginzburg fibre is birational to a Calabi-Yau variety if and only if a general anticanonical section of the weighted projective space is Calabi-Yau. We analyse the 104 weighted 3-spaces with canonical singularities, and show that a general anticanonical section is not a K3 surface exactly in those 9 …

关于非阿贝尔霍奇位形I

On the non-abelian Hodge locus I

本文研究了准射影变种上整数局部系统背后的极化Hodge结构，证明了非阿贝尔霍奇位形的代数性及有限性定理。