Schedule

The following is a preliminary schedule of the talks.

Wednesday:

• 11:00 - 12:00: Peternell

• 14:00 - 15:00: Debarre

• 15:30 - 16:30: Tommasi

• 16:45 - 17:45: Hitchin

• 18:00: Poster session

Thursday

• 9:30 - 10:30: Huybrechts

• 11:00 - 12:00: Saccà

• 14:00 - 15:00: Voisin

• 15:30 - 16:30: Ottem

• 16:45 - 17:45: Farkas

• 18:30: Reception at the "Königlicher Pferdestall" (Appelstr. 7, 30167 Hannover).

Friday

• 9:30 - 10:30: Kondo

• 11:00 - 12:00: Casalaina Martin

• 13:30 - 14:30: Gritsenko

• 15:00 - 16:00: Grushevsky

Abstracts

S. Casalaina-Martin: Moduli spaces of cubic hypersurfaces.

In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent.

O. Debarre: Complete curves in the moduli space of polarized K3 surfaces and hyper-Kähler manifolds.

Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron, we prove that the moduli space of polarized K3 surfaces of degree 2e contains complete curves for all e >= 62 and for some sporadic lower values of e (starting at 14).

We also construct complete curves in the moduli spaces of polarized hyper-Kähler manifolds of K3^[n]-type or Kum_n-type for all n>=1 and polarizations of various degrees and divisibilities. This is joint work with Emanuele Macrì.

G. Farkas: The Kodaira dimension of M_22 and M_23: geometric and tropical aspects.

It is one of the landmark results in algebraic geometry of the 20th century that the moduli space M_g of curves of genus g is a variety of general type when g>23. I will discuss joint work with Jensen and Payne proving that both moduli spaces M_22 and M_23 are of general type, highlighting both the geometrical and the novel tropical aspects related to this circle of ideas.

V. Gritsenko: Moduli of polarised irreducible holomorphic symplectic varieties and automorphic forms.

In our joint papers with Klaus Hulek and Gregory Sankaran we obtained many results on the geometric type of moduli spaces of polarised irreducible holomorphic symplectic manifolds of type Hilb^n(K3). In this talk, I collect open questions from the theory of automorphic forms related to the moduli spaces and present some new results on the second infinite series of IHSV, polarised generalised Kummer varieties.

S. Grushevsky: Differentiating Siegel modular forms, and the moving slope of A_g.

The slope of the effective cone of the moduli space A_g of complex principally polarized abelian varieties has attracted a lot of attention since the 1980s, in relation to the Kodaira dimension of A_g and to understanding the structure of the rings of Siegel modular forms. Beyond identifying the effective divisor of minimal slope, if such exists, it is interesting to determine the moving slope --- the minimal slope where there are two divisors of that slope with different support. By studying holomorphic differentials operators that map Siegel modular forms to Siegel modular forms (of independent interest), we recover the known moving slopes for g<=4 and obtain a bound for the moving slope for g=5. Joint work with T. Ibukiyama, G. Mondello, R. Salvati Manni

N. Hitchin: Vector bundles, multivector fields and theta functions.

An invariant polynomial of degree d on the Lie algebra of a group G defines holomorphic symmetric tensors on the moduli space of stable G-bundles over a curve C, parametrized by the space H^¹(C,K^(-d+1)). Furthermore, in a talk given in Hannover in 2009, I showed that an invariant exterior form of degree d on the Lie algebra defines in a similar fashion skew-symmetric tensors parametrized by the same space. These have the important property that they commute using the symmetric or skew-symmetric form of the Schouten bracket.

The talk will investigate these tensors for low rank vector bundles E, and in particular small genus of C, and relate their properties to the associated theta divisor for E, linking the question to some classical algebraic geometry.

D. Huybrechts: Chow groups of surfaces of lines in cubic fourfolds.

I will discuss Chow groups of two types of non-Lagrangian surfaces in the Fano variety of lines in a cubic fourfold. There are certain similarities with Chow groups of K3 surfaces that will be highlighted.

S. Kondo: Kummer surfaces and quadric line complexes in characteristic two.

We give an analogue in characteristic 2 of the classical theory of quadric line complexes and Kummer surfaces.

This is a joint work with Toshiyuki Katsura.

J. Ottem: Coniveau and strong coniveau classes on rationally connected varieties

A cohomology class of a smooth complex variety of dimension n is said to be of "coniveau" at least c if it vanishes on the complement of a closed subvariety of codimension at least c, and of "strong coniveau" at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most n-c. These notions give rise to two filtrations on the cohomology groups of a variety, which are known to coincide in many cases (for instance, they agree on the rational cohomology of any smooth projective variety). However, we show that they differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties. The difference between the two filtrations also give rise to new birational invariants. In this talk, I will present some new examples where the two filtrations differ.

T. Peternell: Stein and affine complements in projective and Kähler manifolds.

Given a projective manifold X and a smooth hypersurface in X, we study conditions under which the complement X \ Y is Stein or affine. Particular emphasis is laid on the case when X is the projectivization of the so-called canonical extension of the tangent bundle of a projective manifold M and Y is the projectivization of the tangent bundle itself. There is an interesting conjectural connection to the nefness of the tangent bundle of M (joint work with Andreas Höring).

G. Saccà: Moduli spaces as Irreducible Symplectic Varieties

Recent developments have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. This builds on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

O. Tommasi: Geometry of fine compactified Jacobians in genus 1.

The degree d universal Jacobian parametrizes degree d line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian. We focus on the case of genus 1 and obtain a combinatorial classification for fine universal compactified Jacobians, which enables us to construct new examples of them. The description we obtain for universal fine compactified Jacobians of genus 1 also yields a formula for their rational cohomology.

This is joint work with Nicola Pagani (Liverpool).

C. Voisin: Strong representability of 1-cycles on rationally connected threefolds.

We show that for any smooth projective rationally connected threefold X, there exists a smooth projective variety M, which can be taken to be a surface, and a family of 1-cycles on X parameterized by M, such that the induced morphism from Alb(M) to J(X) is an isomorphism.