Torelli Theorem Research Articles

New perspectives on categorical Torelli theorems for del Pezzo threefolds

Let Yd be a del Pezzo threefold of Picard rank one and degree d≥2. In this paper, we apply two different viewpoints to study Yd via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component:(i)Brill–Noether reconstruction. We show that Yd can be uniquely recovered as a Brill–Noether locus of Bridgeland stable objects in its Kuznetsov component.(ii)Exact equivalences. We prove that up to composing with an explicit auto-equivalence, any Fourier–Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree 2≤d≤4 can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier–Mukai type auto-equivalences of the Kuznetsov component of Yd.In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel.

A categorical Torelli theorem for hypersurfaces

AbstractLet be a smooth Fano hypersurface of dimension and degree . The derived category of coherent sheaves on contains an interesting subcategory called the Kuznetsov component . We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines uniquely if or if and . This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that divides .

Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

New counterexamples to the birational Torelli theorem for Calabi–Yau manifolds

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 ) (n^2-1) -folds which, for n ≥ 2 n \geq 2 even, admit an isometry between their middle cohomologies. These varieties also satisfy an L \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 pushforwards of a general ( 1 , 1 ) (1,1) -section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.

Derived Equivalence for Elliptic K3 Surfaces and Jacobians

Abstract We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners, which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.

Categorical Torelli theorems for Gushel–Mukai threefolds

AbstractWe show that a general ordinary Gushel–Mukai (GM) threefold can be reconstructed from its Kuznetsov component together with an extra piece of data coming from tautological subbundle of the Grassmannian . We also prove that determines the birational isomorphism class of , while determines the isomorphism class of a special GM threefold if it is general. As an application, we prove a conjecture of Kuznetsov–Perry in dimension 3 under a mild assumption. Finally, we use to restate a conjecture of Debarre–Iliev–Manivel regarding fibers of the period map for ordinary GM threefolds.

On Fourier–Mukai type autoequivalences of Kuznetsov components of cubic threefolds

AbstractWe determine the group of all Fourier–Mukai type autoequivalences of Kuznetsov components of smooth complex cubic threefolds, and provide yet another proof for the Fourier–Mukai version of the categorical Torelli theorem for smooth complex cubic threefolds.

Cyclic coverings of genus curves of Sophie Germain type

Abstract We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $\frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$ -type abelian varieties combined with theta-duality techniques.

Torelli theorems for some Steiner bundles

Torelli theorems for some Steiner bundles

Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{a\in Z}$, where $Z\subset C$ is the ramification locus of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that corresponds to $X\to C$. (2) For each $a\in Z$ the map $\overline {\nabla }_{v_a}:H^{2,0}(X)\to H^{1,1}_{\rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${\mathbb {C}}[\eta _a]$ and its kernel is $H^0(X,\Omega ^2_X(-E_a))$, where $E_a=f^{-1}(a)$. (3) The classes $[\eta _a]$ form an orthogonal basis of $H^{1,1}_{\rm prim}(X)$ and $[\eta _a]$ is represented by a meromorphic $2$-form $\eta _a$ in $H^0(X,\Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $h\ge q+3$. (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{\rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${\mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$. (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

A BALL QUOTIENT PARAMETRIZING TRIGONAL GENUS 4 CURVES

Abstract We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$ .

Projective normality and basepoint‐freeness thresholds of general polarized abelian varieties

AbstractFor a polarized abelian variety , Z. Jiang and G. Pareschi introduce an invariant , called the basepoint‐freeness threshold. Using this invariant, we show that a general polarized abelian variety of dimension is projectively normal if and the type of is not . This bound is sharp since it is known that any polarized abelian variety of type is not projectively normal. We also give an application of to the infinitesimal Torelli theorem for .

A note on the Nielsen realization problem for 𝐾3 surfaces

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a K 3 K3 surface. The first theorem is that the natural map π 0 ( D i f f ( K 3 ) ) → A u t ( H 2 ( K 3 ; Z ) ) \pi _{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb {Z})) has a section over its image. The second is that there exists a subgroup G G of π 0 ( D i f f ( K 3 ) ) \pi _{0}(Diff(K3)) of order two over which there is no splitting of the map D i f f ( K 3 ) → π 0 ( D i f f ( K 3 ) ) Diff(K3) \to \pi _{0}(Diff(K3)) , but there is a splitting of H o m e o ( K 3 ) → π 0 ( H o m e o ( K 3 ) ) Homeo(K3) \to \pi _{0}(Homeo(K3)) over the image of G G in π 0 ( H o m e o ( K 3 ) ) \pi _{0}(Homeo(K3)) , which is non-trivial. The third is that the map π 1 ( D i f f ( K 3 ) ) → π 1 ( H o m e o ( K 3 ) ) \pi _{1}(Diff(K3)) \to \pi _{1}(Homeo(K3)) is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for K 3 K3 surfaces.

The Moduli Space of Cubic Threefolds with a Non-Eckardt Type Involution via Intermediate Jacobians

Abstract There are two types of involutions on a cubic threefold: the Eckardt type (which has been studied by the first named and the third named authors) and the non-Eckardt type. Here we study cubic threefolds with a non-Eckardt type involution, whose fixed locus consists of a line and a cubic curve. Specifically, we consider the period map sending a cubic threefold with a non-Eckardt type involution to the invariant part of the intermediate Jacobian. The main result is that the global Torelli Theorem holds for the period map. To prove the theorem, we project the cubic threefold from the pointwise fixed line and exhibit the invariant part of the intermediate Jacobian as a Prym variety of a (pseudo-)double cover of stable curves. The proof relies on a result of Ikeda and Naranjo–Ortega on the injectivity of the related Prym map. We also describe the invariant part of the intermediate Jacobian via the projection from a general invariant line and show that the two descriptions are related by the bigonal construction.

A Torelli theorem for graph isomorphisms

It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result refines tools used in algebraic geometry to examine the fibers of the compactified Torelli map.

Infinitesimal categorical Torelli theorems for Fano threefolds

Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A categorical variant of the infinitesimal Torelli theorem for X states that the morphism η:H1(X,TX)→HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the aforementioned three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we prove infinitesimal categorical Torelli theorems for a class of prime Fano threefolds. We then prove, infinitesimally, a restatement of the Debarre–Iliev–Manivel conjecture regarding the general fibre of the period map for ordinary Gushel–Mukai threefolds.

On irreducible symplectic varieties of K3[n]-type in positive characteristic

We show that there is a good notion of irreducible symplectic varieties of 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.

Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds

We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove that the infinitesimal Torelli theorem does not hold for hyperelliptic Fano threefolds of Picard rank 1, index 1, degree 4, and study the action of the automorphism group on cohomology. The results of this paper are used to prove Lang-Vojta’s conjecture for the moduli of such Fano threefolds in a follow-up paper.

Stability conditions on Kuznetsov components

We introduce a general method to induce Bridgeland stability conditions on semiorthogonal components of triangulated categories. In particular, we prove the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of Fano threefolds and of cubic fourfolds. As an application, in the appendix, written jointly with Xiaolei Zhao, we give a variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.

Finite subgroups of automorphisms of K3 surfaces

AbstractWe give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.