Remarks on the extension of twisted Hodge metrics

不可分解循环与扭曲霍奇D猜想的情形

Indecomposable cycles and the case of the twisted Hodge D conjecture

本文引入扭曲循环及其上同调调节子，证明一般光滑射影曲面上调节子满射的猜想，并在椭圆四重流形上构造不可分解扭曲循环以证明该映射非零。

Twisted Hodge filtration: Curvature of the determinant

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the determinant of the twisted Hodge filtration $F^p_L=\oplus_{i\geq p}R^{n-i}Ω^i_{\mathcal{X}/S}(L)$ is (semi-) positive on the base $S$ if $L$ itself is (semi-) positive on $\mathcal{X}$.

通过Hodge模研究曲面上点的Hilbert概形的扭曲Hodge群与形变理论

Twisted Hodge groups and deformation theory of Hilbert schemes of points on surfaces via Hodge modules

本研究利用Hodge模理论计算紧复曲面上点的Hilbert概形的扭曲Hodge数，证明了相关猜想修正版，推广了经典公式，并确定了Douady空间形变理论的切空间与障碍空间。

Hodge CohFT的亚纯微分与扭曲DR层级

Meromorphic differentials and twisted DR hierarchies for the Hodge CohFT

本文将亚纯微分与扭曲DR层级的研究从平凡CohFT推广到Hodge CohFT，证明了其与非扭曲DR层级的对应关系，并由此导出Hodge积分之间的非平凡恒等式。

Extension of twisted Hodge metrics for Kähler morphisms

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over…

Autoequivalences of twisted K3 surfaces

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index one or two in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

扭曲参数素除子的霍奇滤过

Hodge filtrations for twisted parametrically prime divisors

该研究将霍奇滤过相关工作推广到$\mathbb{Q}$-除子情形，针对特定参数素全纯函数$f$和$α\geq 0$，给出了$\mathscr{M}(f^{-α})$上霍奇滤过的显式表达式。

Non vanishing loci of Hodge numbers of local systems

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…

来自扭曲不规则联络的新非阿贝尔 Hodge 图

New nonabelian Hodge graphs from twisted irregular connections

本文利用非阿贝尔 Hodge 图边重数满足的超度量不等式，在既有箭簇模性定理之外构造并刻画了新的非阿贝尔 Hodge 图。

Hodge metrics and the curvature of higher direct images

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct images bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.

Some remarks on Hodge symmetry

I make some remarks on Hodge symmetry, and prove for instance that if $k$ is a perfect field of characteristic $p>0$ and $X/k$ smooth, proper and Hodge-Witt scheme, and Hodge de Rham sequence of $X$ degenerates at $E_1$ and $X$ has torsion-free crystalline cohomology, then Hodge symmetry holds for $X$.

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

Some remarks on limit mixed Hodge structure and spectrum

We give some remarks on limit mixed Hodge structure and spectrum. These are more or less well-known to the specialists, and do not seem to be stated explicitly in the literature. However, they do not seem to be completely trivial to the beginners, and may be worth writing down explicitly.

A remark on the nilpotent orbit theorem for unipotent complex variations of Hodge structure

We present a new proof of a part of the nilpotent orbit theorem for unipotent complex variations of Hodge structure. In our proof, the $L^2$ extension theorem of Ohsawa-Takegoshi type plays an essential role.

扭曲霍奇菱形导致非傅里叶-向井函子的产生

Twisted Hodge diamonds give rise to non-Fourier-Mukai functors

本文利用扭曲霍奇菱形计算霍赫希尔德维数，构造并验证了奇数维光滑二次超曲面上大量非傅里叶-向井函子，并证明每个奇数维光滑二次超曲面至少存在一个这样的函子。

关于Hodge结构的B模型比较的一个注记

A remark on the B-model Comparison of Hodge structures

本文记录了一个关于半无限Hodge结构的B模型变形的比较定理，该定理被视为民间定理，其动机是应用于B模型范畴枚举不变量的研究。

Orbifold Hodge Numbers of Calabi-Yau Hypersurfaces

We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an $n$-dimensional simplicial Fano toric variety and then explicitly compute $h^{1,1}_{orb}$ and $h^{n-2,1}_{orb}$ for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.

The Künneth Formula for the Twisted de Rham and Higgs Cohomologies

We prove the Künneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying chain complexes under a certain elimination of indeterminacy.

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。

Proof of Caldararu's conjecture. An appendix to a paper by Yoshioka

We show that any orientation preserving Hodge isometry between the Hodge structures of two K3 surfaces X and X' twisted by Brauer classes $α$ resp. $α'$ can be lifted to a Fourier-Mukai equivalence between the derived categories of $α$ resp. $α'$-twisted sheaves.

通过叠曲线解开扭曲特征簇

Untwisting the twisted character variety via a stacky curve

本文通过构造叠曲线将扭曲特征簇等价于其特征簇的分量，并利用非阿贝尔霍奇对应及单纯概形方法计算其tautological类的权重。

The L-functions of twisted Witt extensions

The L-function of a non-degenerate twisted Witt extension is proved to be a polynomial. Its Newton polygon is proved to lie above the Hodge polygon of that extension. And the Newton polygons of the Gauss-Heilbronn sums are explicitly determined, generalizing the Stickelberger theorem.

A remark on potentially semi-stable representations of Hodge-Tate type (0,1)

In this note we complement a part of a theorem of Fontaine-Mazur. We show that if $(V,ρ)$ is an irreducible finite dimensional representation of the Galois group $Gal({\bar K}/K)$ of a finite extension of $K\Q_p$, of Hodge-Tate type $(0,1)$ then it is potentially semi-stable if and only if it is potentially crystalline. This was proved by Fontaine-Mazur for dimension two and $p\geq 5$ by their classfication theorem.

Some remarks on decomposition theorem for proper Kähler morphisms

We explain a correct proof of the decomposition theorem for direct images of constant Hodge modules by proper Kähler morphisms of complex manifolds. We also give some examples showing certain difficulty in the non-constant Hodge module case.

A remark on the Generalized Hodge Conjecture

Let X be a smooth, projective variety over the field of complex numbers. On the space H of its rational cohomology of degree i we have the arithmetic filtration F^p. On the other hand, on the space of cohomology of degree i of X with complex coefficients we have the Hodge filtration G^p. It is well known that F^p is contained in the intersection of G^p with H, and that, in general, this inclusion is strict. In this paper we propose a natural substitute S^p for the Hodge filtration space G^p such that the intersection of S^p with H is the space F^p of the arithmetic filtration. In particular, S…

A remark on the non-rationality Problem for generic cubic fourfolds

It is proved that the non-rationality of a generic cubic fourfold follows from a conjecture on the non-decomposability in the direct sum of non-trivial polarized Hodge structures of the polarized Hodge structure on transcendental cycles on a projective surface.

邻近周期层的 Hodge-Grothendieck 类与单值化不变量

Hodge-Grothendieck classes and monodromy invariants of nearby cycles sheaves

研究邻近周期层的 Hodge-Grothendieck 类与单值化不变量，评述邻近周期函子并证明广义局部不变周期结果。

阿贝尔曲面的导出同源与同源

Derived isogenies and isogenies for abelian surfaces

本文通过引入扭曲导出等价，利用Tate同源定理和推广的Shioda技巧，证明了在特征≠2,3的代数闭域上阿贝尔曲面的导出同源等价于主同源。

Twisted genera of symmetric products

We prove very general formulae for the generating series of (Hodge) genera of symmetric products with coefficients, which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series for Intersection cohomology Hodge numbers and Goresky-MacPherson Intersection cohomology signatures of symmetric products of complex pro…

$L^2$-Extension of Adjoint bundles and Kollár's Conjecture

We give a new proof of Kollár's conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of Albanese maps. Our technique is the $L^2$-method and we give a concrete construction and proofs of the conjecture. The $L^2$ point of view allows us to generalize Kollár's conjecture to the context of non-abelian Hodge theory.

Some remarks on the semi-positivity theorems

We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semi-positivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semi…

A remark on singularities of primitive cohomology classes

Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle dimensional primitive Hodge classes. In this note, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle dimensional cohomology classes and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class. In a final section, we write-up a classical inductive argument relating the Hodge conjectur…

关于由向量丛扭曲的 Higgs 丛的若干评注

Remarks on Higgs bundles twisted by a vector bundle

本文研究紧 Riemann 曲面上秩二向量丛扭曲的 Higgs 丛，通过 Hecke 变换刻画相伴线丛扭曲对，并建立其模空间与谱覆盖上秩一无挠 Higgs 层正像之间的谱对应。

有界核心划分与博雷-韦-博特定理

Bounded core partitions and Borel-Weil-Bott

本文基于博雷-韦-博特定理，通过分析分解，提出两种计算相关Hodge数的公式，并引入新的整数钩积统计量和半标准 Young表，改进了Hodge数正性的已有界限。

A remark on braid group actions on coherent sheaves

A construction of braid group actions on coherent sheaves using mixed Hodge modules and some well known constructions from geometric representation theory is given.

Hodge genera of algebraic varieties, II

We study the behavior of Hodge-theoretic genera under morphisms of complex algebraic varieties. We prove that the additive $χ_y$-genus which arises in the motivic context satisfies the so-called ``stratified multiplicative property", which shows how to compute the invariant of the source of a proper surjective morphism from its values on various varieties that arise from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic analogue of the Riemann-Hurwitz formula. We also study the contribution of monodromy to the $χ_y$-genus of a smooth projective fami…