Torelli's theorem for high degree symmetric products of curves

Torelli theorem via Fourier-Mukai transform

We show that the Fourier transform on the Jacobian of a curve interchanges "$δ$ functions" at the curve and the theta divisor. The Torelli theorem is an immediate consequence.

A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve

Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.

Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem

Birational Calabi-Yau threefolds in the same deformation family provide a `weak' counterexample to the global Torelli problem, as long as they are not isomorphic. In this paper, it is shown that deformations of certain desingularized weighted complete intersections provide examples of families containing birational varieties. The constructed examples are shown to be non-isomorphic using a specialization argument.

On birational Torelli theorems

Let $G$ be a simple simply-connected connected linear algebraic group over $\mathbb{C}$. We proved a $2$-birational Torelli theorem for the moduli space of semistable principal $G$-bundles over a smooth curve of genus $\geq 3$, which says that if two such moduli spaces are $2$-birational then the curves are isomorphic. We also proved a $3$-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus $\geq 4$.

A Torelli theorem for the moduli space of Higgs bundles on a curve

Let X be a smooth projective complex curve, and let M be the moduli space of stable Higgs bundles on X (with genus g>1), with rank n and fixed determinant ξ, with n and deg(ξ) coprime. Let X' and ξ' be another such curve and line bundle, and M' the corresponding moduli space. We prove that if M and M' are isomorphic as algebraic varieties, then X and X' are isomorphic.

Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical…

A Torelli theorem for moduli spaces of principal bundles on curves defined over $\mathbb R$

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$--bundles on $X$ determine uniquely the isomorphism class of $X$.

余数与微分Torelli对于等奇异曲线

Residues and Infinitesimal Torelli for Equisingular Curves

本文基于余数理论，为等奇异平面曲线提供微分Torelli定理，并扩展Green--Voisin哲学到奇异和等奇异情况，构建统一的Torelli型问题框架。

A Torelli theorem for moduli spaces of principal bundles over a curve

Let X and X' be compact Riemann surfaces of genus at least 3, and let G and G' be nonabelian reductive complex groups. If one component M_G^d(X) of the moduli space for semistable principal G-bundles over X is isomorphic to another component M_{G'}^{d'}(X'), then X is isomorphic to X'.

The Infinitesimal Torelli Theorem for hypersurfaces in abelian varieties

Given a compact Kähler manifold, the Infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example for hyperelliptic curves of genus greater than $2$. Nevertheless the Infinitesimal Torelli Theorem holds for many other classes of manifolds. We will prove it for smooth hypersurfaces in simple abelian varieties with sufficiently high self-intersection giving an effective bound on a result by Green in this particular case.

Symmetric spaces uniformizing Shimura varieties in the Torelli locus

An algebraic subvariety Z of A_g is totally geodesic if it is the image via the natural projection map of some totally geodesic submanifold X of the Siegel space. We say that X is the symmetric space uniformizing Z. In this paper we determine which symmetric space uniformizes each of the low genus counterexamples to the Coleman-Oort conjecture obtained studying Galois covers of curves. It is known that the counterexamples obtained via Galois covers of elliptic curves admit two fibrations in totally geodesic subvarieties. The second result of the paper studies the relationship between these fib…

曲线上向量丛模空间乘积的Torelli定理

Torelli theorem for the product of moduli spaces of vector bundles over a curve

本文证明了亏格至少为4的光滑射影曲线上半稳定向量丛模空间（及模叠）乘积的Torelli型定理，并利用分解定理计算了其自同构群。

Torelli theorem for the parabolic Deligne-Hitchin moduli space

We prove that, given the isomorphism class of the parabolic Deligne-Hitchin moduli space over a smooth projective curve, we can recover the isomorphism class of the curve and the parabolic points.

Theta-duality on Prym varieties and a Torelli Theorem

Let p:C' -> C be an unramified double covering of irreducible smooth curves and let P be the attached Prym variety. We prove the schematic theta-dual equalities in the Prym variety T(C')=V^2 and T(V^2)=C', where V^2 is the Brill-Noether locus of P associated to p considered by Welters. As an application we prove a Torelli Theorem analogous to the fact that the g-th symmetric product of a curve D of genus g determines the curve.

将Torelli映射扩展到模曲线空间的替代紧化上

Extending the Torelli map to alternative compactifications of the moduli space of curves

本文研究了具有轴状奇点的曲线，构造了替代紧化并证明Torelli映射可扩展，同时识别了Smyth意义下的替代紧化中轴状区域。

A Torelli theorem for curves over finite fields

We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.

Tropes, Torelli and theta characteristics

Our main result is an effective version of the Torelli theorem in genus $3$ and any characteristic not $2$: the configuration of the odd theta characteristics of a curve $C$ of genus $3$ determines a del Pezzo surface $S$ of degree two and that $C$ is recovered as the normalization of a certain curve on $S$. This surface is not the one usually associated to $C$; we describe the distinction. We also give an explicit geometrical description of the quadratic twist (observed by Serre) that arises in the statement of the Torelli theorem.

Counter-examples of high Clifford index to Prym-Torelli

We give examples of curves of arbitrarily high Clifford index such that the Prym map is not injective at any of their étale double covers.

The infinitesimal Torelli problem

Let $g \geq 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We verify the well known fact that the map induced on tangent spaces by the Torelli map is dual to the multiplication map $\mathrm{Sym}^2 H^0(C, ω_C) \rightarrow H^0(C, ω_C^{\otimes 2})$.

Degeneration of intermediate Jacobians and the Torelli theorem

Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.

Torelli theorem for the moduli spaces of rank 2 quadratic pairs

Let $X$ be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of $α$-polystable quadratic pairs on $X$ of rank 2.

Torelli theorem for graphs and tropical curves

Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Fina…

Extended Torelli map to the Igusa blowup in genus 6, 7, and 8

It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.

关于在两点处分歧的五次平面曲线覆盖的通用托雷利定理

Generic Torelli for coverings of plane quintics ramified in two points

本文证明了限制在两点分歧的五次平面曲线覆盖族上的分歧Prym映射具有通用单射性，从而建立了该情形的通用托雷利定理。

向量丛与主 G-丛模叠的 Torelli 定理

Torelli theorem for moduli stacks of vector bundles and principal G-bundles

本文证明了曲线上固定秩和行列式的向量丛模叠的同构类唯一决定曲线同构类与向量丛秩，并将 Torelli 定理推广到亏格至少为3的曲线上非阿贝尔约化群的主 G-丛模叠。

包含在Torelli轨迹中的Mumford型Shimura曲线

Mumford-type Shimura curves contained in the Torelli locus

该研究构造了两类亏格4的Shimura曲线族的显式方程，其雅可比簇为定义在有理数域上、自同态代数为整数环且带额外Hodge张量的Mumford型阿贝尔四维簇，是该类阿贝尔簇的首批显式实例。

Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

We prove a generic Torelli theorem for Jacobian elliptic surfaces, provided that the geometric genus is large compared to the irregularity. The result is effective to the extent that defining equations for the base curve are recovered from the period data.

A Generalized Torelli Theorem

Given a smooth projective curve $C$ of genus $g$ over the complex numbers, Torelli's thoerem asserts that the pair $(J(C),W^{g-1})$ determines $C$, where $W^{g-1}$ is an image of the $g-1$st symmetric power of $C$ inside the Jacobian under an Abel-Jacobi map. We show that the theorem holds with $g-1$ replaced by an integer $d$ in the range $1\le d\le g-1$.

A Torelli-like Theorem for Smooth Plane Curves

The Information-Theoretic Schottky Problem treats the period matrix of a compact Riemann Surface as a compressible signal. In this case, the period matrix of a smooth plane curve is characterized by only 4 of its columns, a significant compression.

Torelli theorem for the moduli space of framed bundles

Let X be an irreducible smooth complex projective curve of genus g>2, and let x be a fixed point. A framed bundle is a pair (E,ϕ), where E is a vector bundle over X, of rank r and degree d, and ϕ:E_x\to C^r is a non-zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ>0, which gives rise to the moduli space of τ-semistable framed bundles M^τ. We prove a Torelli theorem for M^τ, for τ>0 small enough, meaning, the isomorphism class of the one-pointed curve (X,x), and also the integer r, are uniquely determined by the isomorphism class of the varie…

Mumford型Shimura曲线在非双曲情形下的非存在性

Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus

研究证明在非双曲曲线的Teichmüller族中，不存在具有严格最大Higgs场的Shimura曲线，特别是Mumford型Shimura曲线不在该族中。

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the c…

将Andreotti对Torelli定理的证明推广到节点曲线

Extending Andreotti's proof of Torelli's theorem to nodal curves

本文推广Andreotti的证明到节点曲线，证明稳定半阿贝尔对同构可函子地诱导典范三元组间的射影等价，恢复紧化Torelli定理，并修复经典证明的技术缺陷。

Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.

On the tropical Torelli map

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.

$d$-椭圆轨迹与 Torelli 映射

$d$-elliptic loci and the Torelli map

本文证明紧型稳定映射模空间中的 Gromov–Witten 虚基本类，与 $A_g$ 上椭圆同源因子特殊闭链的 Torelli 拉回相等。

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmueller space. The main tool for these results is the …

模栈与向量丛扭曲模空间的Torelli定理

Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves

本文证明了向量丛模空间的两个Torelli定理：从固定行列式模栈恢复曲线，及从扭曲模空间恢复曲线和Brauer类，方法涉及特殊线性线性化、Hitchin构造和Kummer覆盖等。

A birational Torelli theorem with a Brauer class

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $β_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( β_{C'}) = β_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are i…

Theta对偶性与超椭圆曲线循环覆盖的Prym-Torelli定理

Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves

本文通过theta对偶性重构方法，研究超椭圆曲线奇素数度循环覆盖的Prym映射的通用单射性，识别出额外剩余轨迹，并在不同度下给出单射性条件。

通过拟稳定除子的图的 Torelli 定理

A Torelli theorem for graphs via quasistable divisors

本文证明图的拟稳定除子偏序集可完全决定图的双连通分支（至多收缩分离边），从而给出 Torelli 定理的组合版本，并将其推广到热带曲线。

On Brill-Noether loci over Quot schemes and a Torelli theorem

We prove a non abelian Torelli type result for smooth projective curves by working in the derived category of some associated polarized Quot schemes and defining Brill-Noether loci and Abel-Jacobi maps on them.

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.

On the Torelli problem and Jacobian Nullwerte in genus three

We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three.

Torelli 轨迹与 Newton 多边形

The Torelli locus and Newton polygons

本文围绕阿贝尔簇模空间中的 Torelli 轨迹展开，综述其几何性质及其在正特征下光滑曲线 Jacobian 的算术不变量与 Newton 多边形研究中的应用。

Global Prym-Torelli for double coverings ramified in at least 6 points

We prove that the ramified Prym map $\mathcal P_{g, r}$ which sends a covering $π:D\longrightarrow C$ ramified in $r$ points to the Prym variety $P(π):=\text{Ker}(\text{Nm}_π)$ is an embedding for all $r\ge 6$ and for all $g(C)>0$. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that $\mathcal P_{g, 2}$ and $\mathcal P_{g, 4}$ have positive dimensional fibers.

Generic Torelli theorem for Prym varieties of ramified coverings

In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and g>5. We also show that a very generic Prym variety of dimension at least 4 is not isogenous to a Jacobian.

通过Fujita分解研究模态射的刚性

Rigidity of modular morphisms via Fujita decomposition

本文利用曲线族Hodge丛的Fujita分解性质，证明了Torelli、Prym、Spin-Torelli态射及射影曲线模叠间覆盖映射的刚性，即它们不可形变。

Torelli Groups and Geometry of Moduli Spaces of Curves

In this paper we give an exposition of Dennis Johnson's work on the first homology of the Torelli groups and show how it can be applied, alone and in concert with Saito's theory of Hodge modules, to study the geometry of moduli spaces of curves. For example, we show that the picard groups of moduli spaces of curves with a fixed level structure are finitely generated, classify all "natural" normal functions defined over moduli spaces of curves with a fixed level, and also "compute" the height paring between cycles over moduli spaces of curves which are homologically trivial and disjoint over th…

曲线上向量丛的 Quot 概形的 Torelli 定理

Torelli theorems for Quot schemes of vector bundles on curves

本文证明曲线上向量丛的 Quot 概形（$d\geq 2$）之间所有同构均为自然同构（除一个例外），从而给出自同构群的完整描述，并建立从 Quot 概形重建曲线的 Torelli 型定理。

Torelli theorem for the moduli spaces of pairs

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,ϕ) over X consists of an algebraic vector bundle E over X and a holomorphic section ϕof E. There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover…

Complete intersections: Moduli, Torelli, and good reduction

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.

Torelli theorem for moduli spaces of SL(r,C)-connections on a compact Riemann surface

Let $X$ be any compact connected Riemann surface of genus $g \geq 3$. For any $r\geq 2$, let $M_X$ denote the moduli space of holomorphic $SL(r,C)$-connections over $X$. It is known that the biholomorphism class of the complex variety $M_X$ is independent of the complex structure of $X$. If $g=3$, then we assume that $r\geq 3$. We prove that the isomorphism class of the variety $M_X$ determines the Riemann surface $X$ uniquely up to isomorphism. A similar result is proved for the moduli space of holomorphic $GL(r,C)$-connections on $X$. We also show that the Torelli theorem remains valid for t…

Logarithmic vector fields along smooth plane cubic curves

We study the sheaves of logarithmic vector fields along smooth cubic curves in the projective plane, and prove a Torelli-type theorem in the sense of Dolgachev-Kapranov for those with non-vanishing j-invariants.