Japanese-European Symposium on Symplectic Varieties and Moduli Spaces – Abstracts
October 26th to 30th, RIMS, Kyoto University, Japan Addington (Short Course): Cubic 4-folds
I will give three expository lectures on cubic 4-folds. I will try to discuss some classical geometry (Pfaffian cubics, cubics containing a plane, nodal cubics), some hyperkaehler geometry (the variety of lines and Lehn's symplectic 8-fold), some lattice theory (Torelli theorem, Hassett's special cubics), and the ways in which Kuznetsov's K3 category clarifies these topics.
Lehn (Short Course): Coisotropic subvarieties: deformation theory and applications
I will explain the notion of coisotropic subvarieties in symplectic manifolds and some basic material regarding their deformation theory. The motivation for this comes from Voisin's recent results on the structure of Chow ring of irreducible holomorphic symplectic varieties on the one side and from birational geometry of symplectic varieties on the other side. This is work in progress, jointly with B. Bakker and G. Pacienza.
Abe: Strange Duality for Height Zero Moduli Spaces of Sheaves on P^2
Drezet defined the height of the moduli space of sheaves on P^2. In the talk, we review the definition of height, and discuss the strange duality for the moduli spaces of sheaves on P^2, one of which has height zero.
Baker: Hyperbolicity and torsion in families of abelian varieties
The hyperbolicity of the moduli space A_g of abelian varieties implies that nontrivial families of abelian varieties exist over a quasiprojective curve B only when the Euler number $\chi(B)$ is negative. Conjecturally the existence of torsion sections of higher order requires the genus (or even the gonality) to be larger. We investigate a general picture for understanding such phenomena---and prove the conjecture in a special case---by bounding Seshadri constants of the Hodge bundle along special subvarieties of the moduli space. Time permitting we will discuss a related result: that families of elliptic curves over a base of fixed gonality are determined up to isogeny by their p- torsion local systems. This is joint work with J. Tsimerman.
Camere: Algebraic moduli spaces of projective hyperkähler manifolds.
In this talk I will give an application of Amerik--Verbitsky's theory of MBM classes to the construction of algebraic moduli spaces of projective hyperkähler manifolds. In particular, I will focus my attention on lattice polarized hyperkähler manifolds and on pairs of fourfolds of K3^[2]- type with a non-symplectic automorphism of prime order. This second topic is joint work with S.
Boissière and A. Sarti.
Fu: Chow rings of hyperkähler varieties: Beauville-Voisin conjecture and the motivic hyperkähler resolution conjecture
I want to talk about a particular feature of the intersection theory of hyperkähler varieties. The first part will be about Beauville's weak splitting principle, particularly for generalized Kummer varieites. The second part concerns a joint work with Zhiyu Tian. Ruan's conjecture on hyperkähler resolution says that the orbifold cohomology ring of an smooth projective orbifold is isomorphic to the cohomology ring of its hyperkähler resolution of singularities. We would like to formulate the motivic analogue of this conjecture on the Chow ring (or more generally the Chow motive as an algebra in the category of Chow motives) of hyperkähler varieties. The particular interesting cases that I want to discuss are the Hilbert-Chow morphisms for the symmetric product of K3 surfaces and the similar generalized Kummer varieties. This work could be seen as a continuation of the work of C. Vial.
Hayashi: Universal Covering Calabi-Yau Manifolds of the Hilbert Schemes of n Points of Enriques Surfaces
For an Enriques surface E, let E[n] be the Hilbert scheme of n points of Enriques surface. E[n] has a Calabi-Yau manifold X as the universal covering space π : X → E[n] of degree 2. I will show that for n ≥ 3, the number of distinct Enriques quotients of X is one. Furthermore X admits no Enriques quotient of X other than the original.
Matsushita: On certain finite map from an irreducible symplectic manifold.
Let X be an irreducible symplectic manifold and Def(X) the Kuranishi space of X. It is conjectured that there is a point t of Def(X) such that the fibre X_t at t admits a Lagrangian fibration X_t -> P^n.
We will show that if the conjecture holds, there exists a dense closed subset V of Def(X) which has the following properties:
For every point s of V, the fibre X_s at s admits a finite morphism X_s -> P^n x P^n.
Menet: Integral cohomology of quotients and Beauville-Bogomolov forms
Singular irreducible holomorphically symplectic varieties (SIHS) generalize naturally the hyperkähler manifolds, since they appear in many situations, for example as moduli spaces of sheaves on K3 or abelian surfaces. One of the main tool for the study of hyperkähler manifolds is the Beauville-Bogomolov form. After the generalization of this form to the SIHS varieties by Namikawa and Matsushita, an interesting question was to provide concrete examples of such BB- forms. Since the BB-form is related to the integral cohomology of the variety by the Fujiki formula, central questions concerning integral cohomology appeared. To be more precise, let X/G be the quotient of a complex manifold by a group of prime order. Although, it is easy to calculate the cohomology of X/G with coefficients in a field, there is not general technique in calculations for the integral cohomology of X/G. In this talk, we will show some new techniques to calculate the integral cohomology of X/G and apply it to provide concrete examples of BB-forms on SIHS varieties.
Nagai: Symmetric product of a semistable degeneration of surfaces
We study the geometry of relative symmetric products of a semistable degeneration of surfaces, in particular their singularity. We will discuss some relationship to the corresponding degeneration of Hilbert schemes.
Ouchi: Lagrangian embeddings of cubic fourfolds containing a plane
For a cubic 4-fold X not containing a plane, Lehn et al constructed an irreducible holomorphic symplectic 8-fold which contains X as a Lagrangian submanifold via twisted cubic curves on X. In this talk, I will talk about Lagrangian embeddings of cubic 4-folds containing a plane. The desired irreducible holomorphic symplectic 8-fold can be constructed as a moduli space of Bridgeland stable objects on the derived categories of twisted K3 surfaces.
Perego: Moduli spaces of sheaves over non-projective K3 surfaces
I will report on a revised version of a joint work with M. Toma about the geometry of moduli spaces of slope-stable sheaves over non-projective K3 surfaces. More precisely, I will show that once the Mukai vector v and the polarization \omega are fixed in a given way (in particular the polarization is v-generic and the rank and the first Chern class of the sheaves are prime to each other), the moduli space of the slope-stable sheaves (with respect to \omega) of Mukai vector v are compact, connected complex manifolds carrying a holomorphic symplectic form, and they are all deformation equivalent to a Hilbert scheme of points on a projective K3 surface. The approach we uses is based on moduli spaces of slope-stable twisted sheaves and deformations along twistor lines.
Piyaratne: Fourier-Mukai theory and stability conditions on abelian varieties
The notion of Fourier-Mukai transform for abelian varieties was introduced by Mukai in early 1980s. Since then Fourier-Mukai theory turned out to be extremely successful in studying stable sheaves and complexes of them, and also their moduli spaces. I will explain how the Fourier-Mukai techniques are useful to show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions on abelian threefolds. First we reduce the requirement of the Bogomolov-Gieseker type inequalities to a smaller class of tilt stable objects which are essentially minimal objects of the conjectural stability condition hearts for a given smooth projective threefold. Then we establish the existence of Bogomolov-Gieseker type inequalities for these minimal objects of abelian threefolds by showing certain Fourier-Mukai transforms give equivalences of abelian categories which are double tilts of coherent sheaves. Part of this is a joint work with Antony Maciocia.
Riess: On Beauville's conjectural weak splitting property
We present a result on the Chow ring of irreducible symplectic varieties. The main object of interest is Beauville's conjectural weak splitting property, which predicts the injectivity of the cycle class map restricted to a certain subalgebra of the rational Chow ring (the subalgebra generated by divisor classes). For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations. After deducing that this implies the weak splitting property in many new cases, we present parts of the proof.
Ueda: Moduli of relations of quivers
The derived category of coherent sheaves on an algebraic variety admitting a tilting object is described by a quiver with relations, and one can study not necessarily commutative deformations
of the variety by deforming the relations. In the talk, we will discuss our joint work with Tarig Abdelgadir and Shinnosuke Okawa on moduli spaces of relations of quivers associated with the projective plane, the quadric surface, and cubic surfaces.
Uehara: Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimensions
We study the group of autoequivalences of the derived categories of coherent sheaves on smooth projective elliptic surfaces with non-zero Kodaira dimensions. We find a description of it when each reducible fiber is a cycle of (−2) -curves.
Organisers:
Giovanni Mongardi Hisanori Ohashi Malte Wandel