Indranil Biswas, Tomas L. Gomez
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.
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In this article we extend the proof given by Biswas and Gomez of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.
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.
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.
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.
We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to prove a Torelli type theorem in some cases.
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 $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.
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。
The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.
Let X and X' be compact Riemann surfaces of genus at least three. Let G and G' be nontrivial connected semisimple linear algebraic groups over C. If some components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated Deligne--Hitchin moduli spaces are biholomorphic, then X' is isomorphic to X or to the conjugate Riemann surface $\bar{X}$.
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'.
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 $X$.
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…
A Steiner bundle is a vector bundle on projective space arising as the cokernel of the map defined by a matrix of linear forms. These come up in various geometric settings, and by now they are the subject of a considerable literature. Starting with work of Dolgachev and Kapranov from 1993, several authors have considered the question of whether one can recover from the bundle the geometric data used to construct it. Here we prove such Torelli-type statements for the tautological bundles associated to sufficiently positive divisors on any very ample linear series. In an appendix, we give a new …
Fix integers $g\geq 3$ and $r\geq 2$, with $r\geq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $\MDH(X)$ denote the corresponding $\text{SL}(r, {\mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $\MDH(X)$ determines (up to an isomorphism) the unordered pair $\{X, \overline{X}\}$, where $\overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.
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 $\geq 4$.
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…
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…
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。
Let $(X,D)$ and $(X',D')$ be two compact Riemann surfaces of genus $g \geq 4$ with the set of marked points $D \subset X$ and $D' \subset X'$. Fix a parabolic line bundle $L$ with trivial parabolic structure. Let $\mathcal{N}_{\textnormal{Sp}}(2m,α,L)$ and $\mathcal{N}'_{\textnormal{Sp}}(2m,α,L)$ be the moduli spaces of stable symplectic parabolic Higgs bundles over $X$ and $X'$ respectively, with rank $2m$ and fixed parabolic structure $α$, with the symplectic form taking values in $L$. We prove that if $\mathcal{N}_{\textnormal{Sp}}(2m,α,L)$ is isomorphic to $\mathcal{N}'_{\textnormal{Sp}}(2…
We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of the finite fibers. We also bound the dimension of the infinite fibers. In particular, we obtain that the compactified Torelli map is injective away from curve with separating nodes if and only if the genus is at most 4.
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.
Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem
本文研究了由Persson引入的一类canonical polarized表面，这些表面作为Galois覆盖P²的八条一般线的分支。通过构造紧化的模空间，并利用KSBA稳定对描述稳定退化，证明了紧化的模堆光滑。此外，建立了通用的全球Torelli型结果：通过反不变部分的第二上同调群的Hodge结构和相关的G作用，可以恢复大部分Persson表面。
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.
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。
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。
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…
We study the Torelli locus T_g in the moduli space A_g of abelian varieties. We consider special subvarieties (Shimura subvarieties) contained in the Torelli locus. We review the construction of some non-trivial examples, and we discuss some conjectures, techniques and recent progress.
We introduce the notions of strong local Torelli and T-class for polarized manifolds, and prove that strong local Torelli implies global Torelli theorem on the Torelli spaces for polarized manifolds in the T-class. We discuss many new examples of projective manifolds for which such global Torelli theorem holds. As applications we prove that, in these cases, a canonical completion of the Torelli space is a bounded pseudoconvex domain in complex Euclidean space, and show that generic Torelli implies global Torelli on moduli space with certain level structure.
We prove a global Torelli theorem for pairs (Y,D), where Y is a smooth projective rational surface and D is an effective anti-canonical divisor which is a cycle of rational curves. This Torelli theorem was conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.
Using the notion of generalized divisors introduced by Hartshorne, we adapt the theory of adjoint forms to the case of Gorenstein curves. We show an infinitesimal Torelli-type theorem for vector bundles on Gorenstein curves. We also construct explicit counterexamples to the infinitesimal Torelli claim in the case of a reduced reducible Gorenstein curve.
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 …
We prove the infinitesimal Torelli theorem for general minimal complex surfaces X's with the first Chern number 3, the geometric genus 1, and the irregularity 0 which have non-trivial 3-torsion divisors. We also show that the coarse moduli space for surfaces with the invariants as above is a 14-dimensional unirational variety.
We prove that the period maps from the Torelli space and the moduli space with level $m$ structure of Calabi-Yau type manifolds to the corresponding period domain of polarized Hodge structures are injective. The proof is based on the construction of holomorphic affine structure on the Teichmüller space and the construction of the completion space of the Torelli space with respect to the Hodge metric.
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.
This paper is a combinatorial and computational study of the moduli space of tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were introduced recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which the aforementioned moduli spaces are objects and the Torelli map is a morphism. We compute the poset of cells of tropical M_g and of the tropical Schottky locus for genus at most 5. We show that tropical A_g is Hausdorff, and we also construct a finit…
A Teichmüller space $Teich$ is a quotient of the space of all complex structures on a given manifold $M$ by the connected components of the group of diffeomorphisms. The mapping class group $Γ$ of $M$ is the group of connected components of the diffeomorphism group. The moduli problems can be understood as statements about the $Γ$-action on $Teich$. I will describe the mapping class group and the Teichmuller space for a hyperkahler manifold. It turns out that this action is ergodic. We use the ergodicity to show that a hyperkahler manifold is never Kobayashi hyperbolic. This is my ICM submissi…
$d$-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。
In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, i.e. the Torelli space, is a complex affine manifold. As applications we prove that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the completion space is a bounded domain of holomorphy with a complete Kähler-Einstein metric. As a corollary we show that the period map from the moduli space of polarized Calabi-Yau manifolds with level $m$ structure is also injective.
Continuity of the tropical Prym-Torelli map
本文对Röhrle等人提出的连续热带Prym簇给出新内蕴构造，明确将其构造为整数环面，并证明热带Prym-Torelli映射是连续的。
Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic $\text{SL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree $d$. We prove that the isomorphism class of the variety ${\mathcal M}_X$ determines the Riemann surface $X$ uniquely up to an isomorphism, although the biholomorphism class of ${\mathcal M}_X$ is known to be independent of the complex structure of $X$. The isomorphism class of t…
The article proves the Infinitesimal Torelli theorem for surfaces subject to the following conditions: 1) the canonical bundle of a surface is ample and generated by its global sections, 2)the geometric genus $p_g \geq 4$, 3) the irregularity $q=0$ . The main novelty is a realization of the Kodaira-Spencer classes lying in the kernel of the cohomology cup-product controlling the derivative of the period map of weight 2 in the category of the coherent sheaves of a surface.
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…
A Torelli-type theorem for hyperkähler quotients
本文针对来自四元数向量空间$\boldsymbol{\rm H}^n$的超双曲商空间，在假设Kirwan映射满射下证明周期映射双射的托雷利型定理，将Kronheimer的ALE引力瞬子托雷利型定理推广至代数几何领域，适用于toric超双曲簇与Nakajima quiver簇。
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.
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.
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。
We prove a global Torelli theorem for the moduli space of marked triples (X,m,A), consisting of an irreducible holomorphic symplectic manifold X, a marking m of its second integral cohomology, and a stable and rigid sheaf A of Azumaya algebras on the cartesian product X^d, d>0, such that the second Chern class of A is invariant under a finite index subgroup of the monodromy group of X. The main example involves the rank 2n-2 sheaf over the cartesian square of holomorphic symplectic manifolds of K3[n]-type considered in the work of the first author arXiv:1105.3223. The result will be used in th…
In this paper we study totally geodesic subvarieties $Y \subset \mathsf{A}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained in the Torelli locus, then $\dim Y \leq (7g -2)/3$.
We construct a moduli space of adequately marked Enriques surfaces that have a supersingular K3 cover over fields of characteristic $p \geq 3$. We show that this moduli space exists as a scheme locally of finite type over $\mathbb{F}_p$. Moreover, there exists a period map from this moduli space to a period scheme and we obtain a Torelli theorem for supersingular Enriques surfaces.
We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing di…
We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they can be recovered from the attached logarithmic vector bundles.
Let $X$ be a smooth compact complex surface subject to the following conditions: (i) the canonical line bundle $\mathcal{O}_X(K_X) $ is very ample, (ii) the irregularity $q(X): = h^1(\mathcal{O}_X) =0$, (iii) $X$ contains no rational normal curves of degree $\leq (p_g-1)$, (iv) the multiplication map $m_2: Sym^2(H^0(\mathcal{O}_X(K_X))) \longrightarrow H^0 (\mathcal{O}_X (2K_X))$ is surjective. It is shown that the Infinitesimal Torelli holds for such $X$. Our proof is based on the study of the cup-product $$ H^1 (Θ_X) \longrightarrow (\mathcal{O}_X(K_X))^{\ast} \otimes H^1 (Ω_X) $$ where $Θ_X…
Let $ \mathcal{D} = \{D_{1}, ..., D_{\ell}\} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \mathbf{P}^{n} $, with degrees $ d_{1}, ..., d_{\ell} $; let $ Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $ \mathcal{D} $ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$ d_{i} $ hypersurfaces of $ Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $. Then, when $ n = 2 $, by describing the moduli spaces containing $ Ω_{\mathbf{P}^{2}…
We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.
We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperkähler metric or twistor deformations.
A generic Torelli Theorem for certain log Calabi--Yau threefolds
本文证明了满足特定纤维为环面对爆破且$H^3(Y,\mathbb{Z})=0$条件的光滑三维对数Calabi--Yau对的族的通用Torelli定理，并应用于Fano-LG对应中的Landau--Ginzburg模型。
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…
Topological Hyperbolicity of Moduli spaces of Elliptic Surfaces
该研究引入拓扑双曲性概念刻画复簇基本群的大性，受沙法列维奇猜想启发研究极化流形模空间的该性质，给出两类模空间的相关结果作为支撑证据。
Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let Ω^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that Ω^{1}_{P^n}(log D) admits a resolution of lenght 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when all the D_{i}'s have the same degree d and l >= {{n+d} \choose {d}}+3: indeed, we recover the components of D as unstable smooth hypersurfaces of Ω^{1}_{P^n}(log D). Finally we analyze the cases of one q…
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
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