Andrey Soldatenkov
This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperkähler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds
研究标记超kähler流形模空间的双曲性性质，证明Kobayashi伪距离消失，并推广Greb和Schwald的定理到任意维数和非紧曲线情况。
Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps
本文研究了带有 lagrangian 纤维的不可约 holomorphically symplectic 流形的抛物自同构，证明了相关 Betti 映射具有最大秩，从而得出诱导平移为有限阶的纤维集合是稠密的。
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperkähler manifold uniquely determines Hodge structures on all higher cohomology groups. We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.
Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class
本文针对具有平凡判别式的一般韦伊型阿贝尔四维流形的霍奇猜想给出新证明。借助其到\(K3^{[3]}\)型超凯勒六维流形的映射，通过超凯勒流形双有理几何及形变理论成果，得出六维流形切丛第二陈类拉回相关代数类结论。
On L-equivalence for K3 surfaces and hyperkähler manifolds
本文研究了K3曲面和超kähler流形中L等价与D等价的关系，通过Hodge理论证明了非常一般的L等价K3曲面是D等价的，并扩展到超kähler四维流形和sheaves模空间。
Hodge similarities, algebraic classes, and Kuga-Satake varieties
本文引入超凯勒流形超越格的霍奇相似性概念，证明特定 K3 曲面族平方的霍奇猜想，并利用 Kuga-Satake 构造的函子性推出广义 Kummer 形变型超凯勒流形的霍奇相似性均为代数的。
A note on the transcendental basepoint-free conjecture for Calabi-Yau manifolds
本文证明Calabi-Yau流形的超越无基点猜想在其Beauville-Bogomolov分解的超Kähler因子上成立时即成立，并利用Bakker-Lehn收缩定理在超Kähler流形上给出该猜想成立的一个充分条件。
Given a complex variety $X$, a linear algebraic group $G$ and a representation $ρ$ of the fundamental group $π\_1(X,x)$ into $G$, we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of the representation variety of $π\_1(X,x)$ into $G$ at $ρ$ using mixed Hodgediagrams and methods of $L\_\infty$ algebras. We apply it in two geometric situations: either when $X$ is compact K{ä}hler and $ρ$ is the monodromy of a variation of Hodge structure, or when $X$ is smooth quasi-projective and $ρ$ has finite image.
We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified group. M(G,A) has the structure of a hyperkahler orbifold. We show that when G is Sp(n) or SU(n), M(G,A) has a natural hyperkahler desingularization which we exhibit as a moduli space of G_C-bundles with an altered stability condition. In this way, we obtain the two known families of…
Summary of main work 1999-2012
Perverse-Hodge octahedron
本文提出反常霍奇八面体作为紧致超凯勒流形霍奇菱形的三维增强，并证明其存在性等价于对所有已知变形类型均成立的Nagai猜想。
Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.
The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non symplectic involution $α$. We first compute the Hodge numbers of a Calabi-Yau constructed in this way in a general setting and then we apply the results to several specific examples of non symplectic involutions, producing Calabi-Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$ and the involution $α$ is induced by a non symplectic involution on the K3 surface. In t…
In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor assumption on the polarization, we show that a very general polarized hyperkähler fourfold $F$ of $K3^{[2]}$-type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold …
The period-index problem for hyperkähler varieties: Lower and upper bounds
本文通过证明超kähler变种的周期-指数猜想的最优性，并在K3^n型变种上验证了非特殊共素Brauer类的猜想。
Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this to manifolds of $\mbox{K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford-Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.
Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the form Ωis closed with respect to the de Rham differential. The M is called HKT (hyperkaehler with torsion) if this form is closed with respect to the Dolbeault differential (this condition is weaker. Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploi…
Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.
On the transcendental lattices of Hyperkähler manifolds
本研究引入由K3型霍奇结构诱导的超凯勒流形概念，针对已知形变类型给出判断其与K3/阿贝尔曲面层模空间双有理等价的格论准则，并揭示O'Grady型超凯勒流形的特殊行为。
Derived equivalences of hyperkähler varieties
本文证明超 Kähler 簇上同调中的 Looijenga–Lunts–Verbitsky Lie 代数为导出不变量，并据此推出 Q-Hodge 结构同构、函子化有理 Mukai 格及自等价群像的计算结果。
Twists of intermediate Jacobian fibrations
本文通过引入解析相对雅可比层，研究OG10超凯勒拉格朗日量的截面、Tate–Shafarevich扭曲与周期，并用解析Brauer群计算扭曲群，证明三次四维流形的本原Hodge格与OG10流形第二上同调的特殊子格等距。
The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus $T$ admitting a symplectic embedding to a hyperkahler manifold $M$. If $M$ is generic in a $d$-dimensional family of deformations, then $\dim T\geq 2^{[(d+1)/2]}$.
A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds
该研究针对六维光滑紧不可约超凯勒流形，证明其第二贝蒂数的精确上界为23，还得出相关上同调消失条件及第二贝蒂数的可能取值，所用方法涉及分解定理、曲率不等式与双线性关系。
We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi--Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi--Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about the period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi--Yau type manifolds to their period domains is proved and applied to the cases of K$3$ surfaces, cubic fou…
Geometric Eisenstein series in non-abelian Hodge theory and hyperholomorphic branes from supersymmetry
本文通过几何Eisenstein级数研究非交换Hodge理论中的超几何sheaf理论，将BBB膜分解为cuspidal和Eisenstein部分，连接了物理对偶性与数学Langlands程序。
Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one
本文分类了Picard数为一的例外齐次簇上完全可约等变向量丛的零轨迹四重簇，通过霍奇数计算排除超凯勒情形，并确认仅存在Beauville--Donagi和Debarre--Voisin描述的两种超凯勒四重簇。
Let X be a hyperkahler manifold deformation equivalent to a Hilbert scheme of n points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X, and derive a generating series for deducing the number of canonical Hodge classes on X. The formula indicates the number of Hodge classes on X that remain Hodge under any deformation.
Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperkähler fourfold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp. of the cone over Gr(2,5) and a quadric). The associated hyperkähler fourfold is in both cases a smooth double cove…
We prove that the bimeromorphic class of a hyperkähler manifold deformation equivalent to O'Grady's six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the Kähler and the birational Kähler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.
A Strong Global Torelli theorem for OG6 type varieties
本文证明 OG6 型超凯勒流形由其次积分上同调及其 Hodge 结构和 Beauville--Bogomolov--Fujiki 形式唯一决定，并由此推出双亚纯等价即双全纯等价。
The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety. We give some examples of generalizations of classical results for variations of mixed Hodge structure, to the twistor setting. This supports a ``meta-theorem'' (which we state but don't prove) that one can everywhere replace the word ``Hodge'' by the word ``twistor''. We show that…
Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds
本文基于复Symplectic manifold上的Lagrangian纤维化，建立Hodge理论新结果，证明相对Hard Lefschetz定理及Maulik等猜想，展示Lagrangian纤维化诱导sl_3(C)或sl_4(C)作用。
A Gentle Introduction to the Non-Abelian Hodge Correspondence
本文以教学方式介绍非阿贝尔 Hodge 对应，阐明基本群表示与全纯丛之间的联系，并梳理其核心定理、几何结构与相关应用。
Lagrangian correspondences of nonabelian Hodge type and shifted twistor structures
本文建立平坦与Higgs完美复形的导出栈之间的Lagrange对应，并证明光滑射影簇上的解析Deligne-Hitchin-Simpson模栈具有规范平移预扭曲结构。
Beauville--Bogomolov--Fujiki positivity and Moishezonness of complex symplectic manifolds in Fujiki class $\mathscr C$
本文研究Fujiki C类中超Fujiki流形的BBF正性与Moishezon性的关系，证明Moishezon超Fujiki流形的相关性质，给出超Fujiki 4-fold等的对应判据并应用于光滑族的Moishezon轨迹描述。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。