Laura Pertusi, Paolo Stellari
We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.
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Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $\mathcal{A}_X$. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines $X$ uniquely if $d > 3$ or if $d = 3$ and $n > 3$. This generalizes a result by D. Huybrechts and J. Rennemo, who proved the same statement under the additional assumption that $d$ divides $n+2$.
A categorical Torelli theorem for quartic del Pezzo surfaces
本文通过证明四次del Pezzo表面可由其Kuznetsov组件重建，解决了范畴Torelli问题，并验证了Kuznetsov组件的半正交不可分解性。
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。
Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
本文利用完备仿射锥的奇点范畴，在数值条件下统一证明了Fano及一般型（加权）超曲面的范畴Torelli定理，并通过形式等价恢复方程结构。
Categorical Torelli theorems for higher Picard rank Fano double covers
本文证明了四类高Picard秩Fano双覆盖的范畴Torelli定理，包括Verra四维流形，通过将Fano三维流形的Kuznetsov组件等价性转化为其K3分支除子的导出等价性，从而得出Fano三维流形的同构性。
Cyclic covers: Hodge theory and categorical Torelli theorems
本文研究带矩形Lefschetz分解基簇上的分歧循环覆盖，定义包含$\mathrm{D}^{\mathrm{b}}(Z)$的等变子范畴$\mathcal{A}_Z$，分解Kuznetsov分量，并推出低次数素Fano三维簇的范畴Torelli定理。
Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components
本文针对非埃卡特型三次三重体，通过非交换霍奇理论、布里奇兰模空间理论和周环理论三种途径，证明了库兹涅佐夫分量的等变范畴可决定原三次三重体的同构类，并进一步建立了等变无穷小范畴托雷利定理。
Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives
本文研究特殊Gushel-Mukai及相关Fano三维簇的无穷小Torelli问题，证明特殊Gushel-Mukai三维簇的无穷小周期映射不变部分单射，并给出周期映射微分核的几何描述。
Infinitesimal categorical Torelli theorems for Fano threefolds
本研究利用霍赫希尔德(上)同调工具，通过交换图关联光滑法诺簇的三类Torelli型定理，证明了一类素法诺三维簇的无穷小范畴Torelli定理，并无穷小地证明了Debarre-Iliev-Manivel猜想的重述形式。
Categorical Torelli theorems for Gushel-Mukai threefolds
本文利用 Kuznetsov 分量与 Grassmannian 重言子丛的额外数据重构一般普通 Gushel-Mukai 三重流形，证明其决定双有理同构类，并据此证明 Kuznetsov-Perry 猜想及重述周期映射纤维猜想。
A remark on the Torelli theorem for cubic fourfolds
本文利用Verbitsky证明的不可约全纯辛簇的整体Torelli定理，给出了三次四重Torelli定理的一个简短证明。
Torelli's theorem from the topological point of view
Torelli定理通过研究θ因子的交上同调层的卷积积得到证明。
We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of $\mathbb P^3$ branched along stable hyperplane arrangements.
Torelli theorem via Fourier-Mukai transform
我们表明，傅里叶变换的雅可比曲线互换“$δ$功能”的曲线和θ因子。Torelli定理是一个直接的结果。
Infinitesimal Torelli theorem for cyclic coverings of generalized flag varieties I
给出了G/P的循环覆盖的一个有效的无穷小Torelli定理，其中G是单代数群，P是极大抛物子群.
Torelli theorem for the moduli space of parabolic Higgs bundles
本文将Biswas和Gomez关于行列式固定的Higgs丛的模空间的Torelli定理的证明推广到抛物情形。
A p-adic analogue of the pseudonorm version of the birational Torelli type theorem is obtained via a comparison theorem of image closures. Among other results obtained, we have a criterion for existence of rational points of canonically polarized surfaces over finite fields.
A categorical invariant for cubic threefolds
我们证明了三次三倍的Torelli定理的一个分类版本。更精确地说，我们表明，非平凡的一个半正交分解的导出范畴的三次三重特征的同构类。
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.
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.
We show how to recover a general hypersurface in $\mathbb{P}^n$ of sufficiently large degree $d$ dividing $n+1$, from its finite order variation of Hodge structure. We also analyze the two other series of cases not covered by Donagi's generic Torelli theorem. Combined with Donagi's theorem, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.
Torelli theorem for the moduli spaces of rank 2 quadratic pairs
设$X$是光滑的射影复曲线。证明了在一定条件下，秩为2的$X$上的$α$ -多稳定二次对的模空间的Torelli型定理成立。
The global Torelli theorem: classical, derived, twisted
调查写的程序的AMS会议代数几何，西雅图，2005年。根据这次会议上发表的讲话，但对最近的事态发展补充了一些意见。
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 $\geq 4$.
Algebraic surfaces of general type with $q=0$, $p_g=2$ and $K^2=1$ were described by Enriques and then studied in more detail by Horikawa. In this paper we consider a $16$-dimensional family of special Horikawa surfaces which are certain bidouble covers of $\mathbb{P}^2$. The construction is motivated by that of special Kunev surfaces which are counterexamples for infinitesimal Torelli and generic global Torelli problem. The main result of the paper is a generic global Torelli theorem for special Horikawa surfaces. To prove the theorem, we relate the periods of special Horikawa surfaces to the…
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.
Infinitesimal Torelli theorem for surfaces with c_1^2=3, χ=2, and the torsion group Z/3
本文证明了第一陈数为3，几何亏格为1，不规则性为0且具有非平凡3-扭除子的一般极小复曲面X的无穷小Torelli定理.我们还证明了具有上述不变量的曲面的粗模空间是一个14维单有理簇。
We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact Kähler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof (arXiv:1106.5573). As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana (arXiv:math/0402243) and Huybrechts (arXiv:math/0106014) respectively.
A Generalized Torelli Theorem
给定复数上亏格为$g$的光滑射影曲线$C$，Torelli定理断言对$(J(C),W^{g-1})$确定$C$，其中$W^{g-1}$是$C$在Abel-Jacobi映射下的Jacobian内的$g-1$的第一次对称幂的像.我们证明了在$1\le d\le g-1$的范围内用整数$d$代替$g-1$的定理成立。
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…
Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles
在抛物权系统是一般的假设下，证明了光滑复射影代数曲线上半稳定抛物Higgs丛模空间的Torelli定理.当亏格至少为2时，利用这一结果，我们还证明了秩至少为2的半稳定抛物丛的模空间的一个Torelli定理.证明中的关键输入是J. C.胡图比斯，可积系统和代数曲面，杜克大学数学。83（1996），19- 49。
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.
Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkählerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared for a seminar talk) was to extract the correct statement and to give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs.
We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the Teichmüller space and Hodge metric completion of the Torelli space. A canonical global holomorphic section of the holomorphic $(n, 0)$ class on the Teichmüller space is constructed.
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.
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
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.
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 $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.
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。
We show that the general Enriques surface can be recovered from the Kuznetsov component of its bounded derived category of coherent sheaves.
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.
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。
Theta-duality on Prym varieties and a Torelli Theorem
设p：C' -> C是不可约平滑曲线的非分歧双重覆盖，P是其附着的Prym簇.本文证明了Prym簇T（C '）=V^2和T（V^2）= C'中的示意性θ-对偶等式，其中V^2是Welters所考虑的与p相关的P的Brill-Noether轨迹.作为应用，我们证明了一个类似于亏格为g的曲线D的第g次对称积决定曲线的Torelli定理。
Torelli theorem for stable curves
研究了从稳定曲线的模空间到主极化稳定半阿贝尔对的模空间的Torelli态射。我们给出了它的纤维的两个特征，描述了它的内射轨迹，并给出了有限纤维的基数的一个精确的上界。我们还约束了无限纤维的维度。特别地，我们得到了紧化Torelli映射是离点曲线单射的当且仅当亏格至多为4。
We prove an equivalence between the infinitesimal Torelli theorem for top forms on a hypersurface contained inside a Grassmannian $\mathbb G$ and the theory of adjoint volume forms presented in L. Rizzi, F. Zucconi, "Generalized adjoint forms on algebraic varieties", Ann. Mat. Pura e Applicata, in press. More precisely, via this theory and a suitable generalization of Macaulay's theorem we show that the differential of the period map vanishes on an infinitesimal deformation if and only if certain explicitly given twisted volume forms go in the generalized Jacobi ideal of $X$ via the cup produc…
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.
Towards the Global Torelli Theorem
本文证明所有维数下K-平凡簇的有限性结果，利用极化K3曲面的整体托雷利定理建立纤维同构的霍奇理论充分条件，并应用于双有理等价纤维稠密的射影族，得出所有纤维同构的结论。
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 Deligne--Hitchin moduli space, II
设X和X'是亏格至少为3的紧致黎曼曲面.设G和G'是C上的非平凡连通半单线性代数群.若Deligne-Hitchin模空间的某些分支$M_{DH}^d(X,G)$和$M_{DH}^{d'}(X',G')$是双全纯的，则X'同构于X或共轭黎曼曲面$\bar{X}$ .
We show that two smooth projective curves C_1 and C_2 of genus g which have isomorphic symmetric products are isomorphic unless g=2. This extends a theorem of Martens.
A Torelli theorem for moduli spaces of principal bundles over a curve
设X和X'是亏格至少为3的紧致Riemann曲面，G和G'是非交换约化复群.若X上半稳定主G-丛的模空间的一个分支M_G^d（X）同构于另一个分支M_{G'} ^{d '}（X'），则X同构于X '.
Infinitesimal Derived Torelli Theorem for K3 surfaces
本文证明了两个光滑射影K3曲面的一阶变形在Fourier-Mukai变换下是等价的当且仅当曲面的全上同调群存在一个特殊的等距群，该等距群保持Mukai对、无穷小权-2分解和正四维空间的定向。这推广了Torelli定理的衍生版本。一路上，我们显示的兼容性行动的霍赫希尔德同调和奇异上同调的任何傅立叶-Mukai函子。
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 $X$.
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。
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 …
Torelli theorem for the moduli spaces of pairs
设X是复数上亏格至少为2的光滑射影曲线。X上的一个对（E，E）由X上的一个代数向量丛E和E的一个全纯截E组成。有一个依赖于实参数τ的对的稳定性的概念。本文证明了行列式固定且秩至少为2的τ-稳定对的模空间的第三上同调群是极化纯Hodge结构，并且它们同构于具有自然极化的H^1（X）（除了极少数例外情况）。这意味着这样的模空间的Torelli定理。我们得到了秩至少为2且行列式固定的稳定丛的模空间的第三上同调群是一个极化纯Hodge结构，它同构于H^1（X）。我们还证明了行列式不固定的对和丛的相应模空间的Torelli定理。
A global Torelli theorem for rigid hyperholomorphic sheaves
本文证明了标记三元组模空间（X，m，A）的整体Torelli定理，该模空间由不可约全纯辛流形X，其第二整上同调的标记m和Azumaya代数在笛卡儿积X^d，d>0上的稳定刚性层A组成，使得A的第二Chern类在X的单值群的有限指标子群下不变.主要的例子涉及秩2n-2层的卡方的全纯辛流形的K3[n]-型的工作中考虑的第一作者arXiv：1105.3223。所得结果将用于作者即将进行的K3曲面广义变形的研究中。
A Torelli theorem for the moduli space of Higgs bundles on a curve
设X是光滑射影复曲线，M是X上稳定Higgs丛的模空间（亏格g>1），秩n，行列式固定，n与deg（<$）互质.设X'和X'是另一个这样的曲线和线丛，M'是相应的模空间。证明了如果M与M'同构为代数簇，则X与X'同构.
Torelli theorem for the Deligne--Hitchin moduli space
固定整数$g\geq 3$和$r\geq 2$，如果为$g=3$，则为$r\geq 3$。给定一个亏格为$g$的紧致连通Riemann曲面$X$，用$\MDH(X)$表示相应的$\text{SL}(r, {\mathbb C})$ Deligne-Hitchin模空间.我们证明了复解析空间$\MDH(X)$决定（直到同构）无序对$\{X, \overline{X}\}$，其中$\overline{X}$是由$X$上的相反几乎复结构定义的黎曼曲面。
We produce counterexamples to the birational Torelli theorem for Calabi-Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non birational pairs of Calabi-Yau $(n^2-1)$-folds which, for $n \geq 2$ even, admit an isometry between their middle cohomologies. These varieties also satisfy an $\mathbb L$-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi-Yau pairs, namely all those which are realized as pushforwar…
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