Title: Kähler manifolds with trivial canonical class
Title: Framed bundles and instanton counting
Abstract. I will discuss the construction of moduli spaces of framed torsion-free sheaves on projective surfaces, in particular, sheaves that have a ``good framing" on a big and nef divisor. One shows that these moduli spaces are fine. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. I will compare with ADHM constructions in some particular cases. In a second part of my talk I will specialize to Hirzebruch surfaces and I will show out to compute the Poincaré series of the moduli spaces using instanton counting.
Title: Surfaces with genus 0, fundamental groups and moduli
Title: Divisorial Zariski decomposition on varieties whose first Chern class is zero
Abstract: We study the divisorial Zariski decomposition on varieties whose first Chern class is zero. We first prove that any exceptional divisor is contractible (up to a birational map that is an isomorphism in codimension one). We then characterize prime exceptional divisors on holomorphic symplectic manifolds.
Title: Schur quadrics and moduli of marked cubic surfaces
Abstract. This is a common work with C. Gruson and M. Meulien. Let S be a cubic surface in PV , dim V = 4 . A marking of S is an identification of the orthogonal of O(1) in Pic S , with the root lattice E6 . There is a theoretical way, by logarithmic derivatives, to embed this lattice in the space of "apolar" (with respect to S ) dual quadrics. We compute the images of the roots as suitably normalized "Schur quadrics". We can then associate to S a characteristic hyperplane of the same space, namely the quadrics containing the five planes (dual points), such that S is a linear combination of the cubes of these planes. I wish to present some steps of the (still unfinished) computation.
Title: Moduli of parabolic Higgs bundles and the Atiyah algebroid
Abstract: We will discuss the holomorphic Poisson geometry of the moduli space of parabolic Higgs bundles over a Riemann surface with marked points in terms of Lie algebroids. Time permitting we will also discuss the Hitchin system in this setting and show how this provides a global analogue of the Grothendieck-Springer resolution. This is a joint work with Johan Martens.
Title: The period map for cubic fourfolds versus the period map for certain symplectic fourfolds
Abstract: Beauville and Donagi showed that the period map for cubic fourfolds factors through the one for certain symplectic fourfolds. Voisin proved the injectivity of the former and Huybrechts proved the surjectivity of the latter. The domains of these period maps are not the same, and their relation was independently determined by Radu Laza and the speaker. In this talk we report on the finer geometric structure of this situation that is being obtained by Bart van den Dries.
Title: Fano fourfolds of degree ten and EPW sextics
Abstract: O'Grady constructed certain symplectic fourfolds as double covers of special sextic hypersurfaces, first considered by Eisenbud, Popescu and Walter. I will discuss a different construction of these symplectic fourfolds in terms of the geometry of Fano fourfolds of degree ten. (Joint work with Atanas Iliev.)
Title: Modular Galois covers associated to symplectic resolutions of singularities
Abstract: Let Y be a normal projective variety and p a morphism from X to Y, which is a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def(X) and Def(Y) are both smooth, of the same dimension, and p induces a finite branched cover f from Def(X) to Def(Y). We prove that f is Galois. When X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor, then the Galois group is a product of Weyl groups of finite type. We consider generalizations of the above set-up, where Y is affine symplectic, or a Calabi-Yau threefold with a curve of ADE-singularities, or a generalized Hitchin system.
Title: On self-correspondences of K3 surfaces via moduli of sheaves
Abstract: In series of our papers with Carlo Madonna
(2002-2007) we described self-correspondences of K3 surfaces via
moduli of sheaves with primitive isotropic Mukai vector, for Picard
number 1 and 2.
In my talk I want
to give some natural and functorial answer to this question for
arbitrary Picard number.
Moreover,
I shall characterise in terms of self-correspondences via moduli of
sheaves K3 surfaces with reflective Picard lattices, when the
automorphism group of the lattice is generated by reflections, up to
finite index. It is known since 1981 that the number of reflective
hyperbolic lattices is, in essential, finite.
Title: Moduli of irreducible symplectic manifolds
Abtract: In general the global Torelli theorem fails for irreducible symplectic manifolds, but moduli spaces still exist and we can still use the period map to study them. I shall describe joint work with Gritsenko and Hulek in which we obtain general type results for some of the moduli spaces and a better understanding of the failure of global Torelli.
Title: On the moduli space of mathematical instantons on P3
Abtract: We prove the irreducibility of the moduli space In of mathematical instanton vector bundles on P3 for odd n.
Title: Hyperkahler SYZ conjecture and multiplier ideal sheaves (minicourse)
Abtract: Let M be a compact, holomorphic symplectic Kaehler manifold, and L a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of L is effective. We explain how this argument can be modified for singular metrics and multiplier ideal sheaves. This is used to show that a power of a nef bundle is always effective.
Title: An introduction to Borcherds products (minicourse)
Abstract. I will explain generalities on Borcherds products and some nice examples of Borcherds products for odd unimodular lattices.
Here is the plan of the minicourse:
(1) lattices and definitions of vector-valued elliptic modular forms
(2) automorphic forms over domains of type IV
(3) the Borcherds theorem
(4) a theorem of Bruinier on the characterization of Borcherds products
(5) examples of vector-valued elliptic modular forms
(6) applications to moduli of K3 surfaces
(7) a family of elliptic modular forms for Γ0(4) and the corresponding Borcherds product for 2-elementary lattices
(8) a Borcherds product for an odd unimodular lattice
(9) the symmetry of the Borcherds product for an odd unimodular lattice
(10) interpretation as an automorphic form on the complexified Kähler cone of a Del Pezzo surface
(11) Some questions