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 E_{6}
. 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 **P**^{3}

**Abtract: **We prove the irreducibility of the
moduli space *I*_{n} of mathematical instanton
vector bundles on **P**^{3} 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 Γ_{0}(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