Schiffer variations and the generic Torelli theorem for hypersurfaces

通过Kuznetsov分量研究加权超曲面的IVHS与范畴Torelli定理

IVHS via Kuznetsov components and categorical Torelli theorems for weighted hypersurfaces

本文通过Kuznetsov分量的Hochschild–Serre代数研究加权超曲面的范畴Torelli定理，重构Jacobi环并重建无穷小Hodge结构变分，证明广义Veronese双锥和$\mathbb{P}^n$的$k$层覆盖等类别的范畴Torelli定理。

Hochschild cohomology versus the Jacobian ring, and the Torelli theorem for cubic fourfolds

The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism type. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. In Voisin's original proof of the global Torelli theorem for smooth cubic fourfolds the Jacobian ring does not intervene. In this paper we present a proof of the global Torelli theorem for cubic fourfolds that relies on the Jacobian ring and the (derived) global Torelli theorem for K3 surfaces. It emphasizes, once again, the relation between K3 surfaces and smooth cubic fourfolds. More generally, fo…

The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space

Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of length $ 1 $ depending on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then, we pro…

Lines on the secant cubic hypersurfaces of Severi varieties

The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomo…

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also g…

On Deformations of Nodal Hypersurfaces

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

On Massey products and Rational homogeneous varieties

We study the equivalence between infinitesimal Torelli theorem for smooth hypersurfaces in rational homogeneous varieties with Picard number one and the theory of generalized Massey products. This equivalence shows that the differential of the period map vanishes on an infinitesimal deformation if and only if certain twisted differential forms are elements of the Jacobian ideal of the hypersurface. We also prove an infinitesimal Torelli theorem result for smooth hypersurfaces in log parallelizable varieties.

Moduli of Linear Sections of a General Hypersurface

We investigate the global variation of moduli of linear sections of a general hypersurface. We prove a "generic Torelli" result for a large proportion of cases, and we obtain a complete picture of the global variation of moduli of line slices of a general hypersurface.

$μ$-constant monodromy groups and Torelli results for marked singularities, for the unimodal and some bimodal singularities

This paper is a sequel to [He7]. There a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. One can consider it as a global $μ$-constant stratum or as a Teichmüller space for singularities. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_Z$. Here $G_Z$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_…

Serre代数、矩阵分解与超曲面的范畴Torelli定理

Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

本文为Fano簇的Kuznetsov分量构造Serre代数，对Fano超曲面建立其子代数与Jacobi环的联系，并由此证明一类Fano超曲面的范畴Torelli定理。

Jacobian环与微分Torelli定理

Jacobian rings and the infinitesimal Torelli Theorem

本文研究Jacobian环，并识别非退化超曲面在环面中的混合Hodge组件，通过Laurent多项式计算微分的核，从而完成微分Torelli定理的证明，特别是在n≥4的情况下。

$μ$-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Als…

Sur l'application des périodes d'une Variation de Structure de Hodge attachée aux familles d'hypersurfaces á singularités simples

Let $n$ be a positive even integer and $d$ a positive integer . To every complete family $Z$ of n dimensional degree d hypersurfaces in the projective space with isolated A-D-E singularities we construct according to an idea of Carlson-Toledo a Deligne-Mumford stack $\bar Z$ whose moduli space is $Z$ such that the monodromy representation extends. We study the corresponding periods mapping and establish an infinitesimal Torelli theorem along the isosingular strata of lZ$ under transversality assumptions. We apply this result to prove Steiness of the universal covering space of $\bar Z$.

The moduli of smooth hypersurfaces with level structure

We construct the moduli space of smooth hypersurfaces with level $N$ structure over $\mathbb{Z}[1/N]$. As an application we show that, for $N$ large enough, the stack of smooth hypersurfaces over $\mathbb{Z}[1/N]$ is uniformisable by a smooth affine scheme. To prove our results, we use the Lefschetz trace formula to show that automorphisms of smooth hypersurfaces act faithfully on their cohomology. We also prove a global Torelli theorem for smooth cubic threefolds over fields of odd characteristic.

函数域上超曲面族的沙法列维奇猜想

Shafarevich's conjecture for families of hypersurfaces over function fields

本文证明函数域上光滑拟射影复代数簇$\mathcal{S}$上，$\mathbb{P}_{\mathbb C}^{n+1}$中次数为$d$的光滑射影超曲面的霍奇泛型非平凡族仅有限，且有限性一致，还将结果推广到高余维完全交等满足无穷小托雷利定理的代数簇。

Focal Varieties of Curves of Genus 6 and 8

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.

Hochschild-Serre 代数的群作用与范畴重构

Group action of Hochschild-Serre algebra and categorical reconstruction

本文研究 Serre 函子在 Hochschild-Serre 代数上的作用，并应用于证明加权射影空间中超曲面的 Kuznetsov 分量及孤立奇点拟齐次多项式的范畴 Torelli 定理，实现范畴重构。