# **Titles and abstracts**

**Andrea Cattaneo**:
* Finite group actions and real structures on irreducible holomorphic symplectic manifolds. *

For a given complex manifold, we can look at the set of its real forms, namely those real manifolds whose complexification is the manifold we started with. It is then an interesting question whether this set (when non-empty) is finite or not. This lead naturally to the study of involutions, and more generally of finite group actions, either by holomorphic or anti-holomorphic maps.
In the case of irreducible holomorphic symplectic manifolds, we will show that the number of distinct faithful finite group actions by holomorphic and anti-holomorphic automorphisms is finite, which in particular implies the finiteness of the number of distinct real forms.
This is a joint work with Lie Fu.

**Soheyla Feyzbakhsh**: * Applications of Bridgeland stability conditions in classical algebraic
geometry. *

I will explain how wall-crossing with respect to Bridgeland stability
conditions provides a new upper bound for the number of global sections of sheaves on
K3 surfaces. This, in particular, extends and completes a program proposed by Mukai to
reconstruct a K3 surface from a curve on that surface. Furthermore, the upper bound
characterizes special vector bundles on curves on K3 surfaces, which have the maximum
number of global sections for the minimum degree. Therefore, it gives an explicit
expression for Clifford indices of curves on K3 surfaces.

**Paola Frediani**: *On Shimura subvarieties of A_g contained in the Prym locus.*

I will present some results obtained in collaboration with E. Colombo, A. Ghigi and M. Penegini on Shimura subvarieties of A_g generically contained in the Prym locus.
I will explain the construction of 1-dimensional families of double covers compatible with a fixed group action on the
base curve C such that the quotient of C by the group is the projective line. I will give a simple criterion for the image of these families under the Prym map to be a Shimura curve.
I will show that this criterion allows us to construct several examples of Shimura curves generically contained in the Prym locus in A_g for g<13. Finally, I will give a lower bound
for the maximal dimension of a totally geodesic (and hence Shimura) subvariety of A_g contained in the Prym locus.

**Alice Garbagnati**: * Classification of elliptic fibrations on K3 surfaces admitting a non-symplectic involution.*

The surfaces which can admit more than one non trivial elliptic
fibrations are necessarly K3 surfaces, thus the problem of classifing
elliptic fibrations on surfaces is striclty related with the special
geometry of the K3 surfaces.
We discuss this problem and, using generalizations of a geometric
method
introduced by Oguiso, we complete the classification of the elliptic
fibrations on K3 surfaces which admit a non-symplectic involution
acting
trivially on the Neron-Severi group, extending previous works by
Oguiso, Kloosterman,Comparin-Garbagnati. We provide a geometric
construction of the fibrations classified, relating some of them with
conic bundles of rational elliptic surfaces. This description allows us
to write the Weierstrass equations of the elliptic fibrations on the K3
surfaces explicitly and to study their specializations. This talk is
inspired by joint works with C. Salgado.

**Robert Laterveer**:
* K3 surfaces and Gushel-Mukai sixfolds.*

Even-dimensional Gushel-Mukai varieties display remarkable similarities to cubic fourfolds.
For one thing, they have a Hodge diamond "of K3 type". What's more, there is a divisor in the moduli space of Gushel-Mukai sixfolds (and fourfolds) where the Gushel-Mukai variety can be related to an actual K3 surface, both on the level of Hodge theory and on the level of derived categories (the K3 surface is "generalized dual" to the Gushel-Mukai variety, in the language of Kuznetsov-Perry).
In my talk, I will discuss the following result:

Theorem: let X be a Gushel-Mukai sixfold generalized dual to a K3 surface Y. Then the transcendental Chow motives of X and Y are isomorphic.

This has interesting intersection-theoretic consequences for Gushel-Mukai sixfolds X as in the theorem: their Chow ring behaves like the one of a K3 surface.

**Radu Laza**: *Birational Geometry of the moduli space of K3 surfaces.*

I will discuss a program, joint with K. O'Grady, to investigate the birational geometry of locally symmetric varieties of K3 type (similar considerations apply to the case of ball
quotients). The motivation for our study is the search for geometric compactifications for the moduli of polarized K3 surfaces. Namely, as a consequence of Torelli theorem, the moduli
of polarized K3 surfaces (with canonical singularities) can be identified to a locally symmetric variety D/\Gamma. As such, there are natural `arithmetic compactifications, e.g. the
Baily-Borel (BB) compactification. Unfortunately, the BB compactification has obscure geometric meaning. Consequently, it is natural to compare it with more geometric
compactifications, such as those given by GIT. I will explain that there is a natural continuous interpolation between BB and GIT compactifications, and that there is a rich geometric
and arithmetic structure behind this picture. In particular, I will show that the Borcherds-Gritsenko relations provide an explanation to some surprising geometric behavior. The focus
of the talk will be on the quartic K3 case. Some new results on degree 6 K3s (with F. Greer) will be briefly discussed.

**Christian Liedtke **: * A Nron-Ogg-Shafarevich criterion for K3 Surfaces.*

Let R be complete local ring with field of fractions K and residue field k, let X be a K3 surface over K, and assume that X has potential semi-stable reduction (which is
automatic if char(k)=0). If char(K)=0, then we show that the following are equivalent: 1) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for one l different from p
2) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for all l different from p 3) the p-adic Galois representation on H^2(\bar{X},Q_p) is crystalline 4) the surface
has good reduction after an unramified extension of K For example, if R is a power series over the complex numbers, then this says that a family of K3 surfaces over a pointed disk can
be filled in smoothly over the origin (that is, has good reduction) if and only the monodromy representation on H^2 is trivial. However, in the arithmetic situation, where the residue
field k might not be algebraically closed, then an unramified base change might be needed. We show by example that sometimes, a non-trivial base-change is necessary. In any case, we
show that if 1) to 3) hold, then there always exists a model over R whose special fiber X_0 has at worst canonical singularities. Then, good reduction of X is equivalent to having an
isomorphism between H^2 of X and the minimal resolution of singularities of X_0, such that this isomorphism is compatible with the natural Galois-actions (or F-isocrystal structures).
In my talk, I will introduce all the above notions, which will not give me much time to explain proofs. Part of this is joint with Matsumoto, part of this is joint with Chiarellotto
and Lazda.

**Antonio Rapagnetta**: * A class of examples of singular irreducible symplectic varieties. *

By the Bogomolov decomposition theorem, irreducible holomorphic symplectic manifolds play a central role in the classification of compact Kähler manifolds with numerically trivial canonical bundle. Very recently, Höring and Peternell completed the proof of the existence of a singular analogue of the Bogomolov decomposition theorem. In view of this result, singular irreducible symplectic varieties (following Greb, Kebekus and Peternell) are singular analogue of irreducible holomorphic symplectic manifolds. In a joint work with Arvid Perego, still in progress, we show that all moduli spaces of sheaves on projective K3 surfaces are singular irreducible symplectic varieties. We compute their Beauville form and the Hodge decomposition of their second integral cohomology, generalizing previous results, in the smooth case, due to Mukai, O'Grady and Yoshioka.

**Alessandra Sarti**: * Nikulin configurations on Kummer surfaces.*

A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to results
of Nikulin this means that the K3 surface is a Kummer surface and the abelian surface in the Kummer structure is determined by the 16 curves. A classical question of Shioda is about
the existence of non isomorphic Kummer structures on the same Kummer K3 surface. The question was positively answered and studied by several authors, and it was shown that the number
of non-isomorphic Kummer structures is finite, but no explicit geometric construction of such structures was given. In the talk I will show how to construct explicitly non isomorphic
Kummer structures on some generic Kummer K3 surfaces. This is a joint work with X. Roulleau.

**Matthias Schütt **: * Zariski K3 surfaces.*

It is a classical fact in algebraic geometry that
unirational curves are rational, and the same holds true for surfaces in characteristic zero, but not in positive characteristic or in higher dimension. In this talk I will report on
joint work with T. Katsura to construct Zariski K3 surfaces, i.e. K3 surfaces admitting a purely inseparable map of degree p from the projective plane. In particular, we will prove
that any supersingular Kummer surface is Zariski in certain characteristics.

**Paolo Stellari **: *
Cubic fourfolds, noncommutative K3 surfaces and stability conditions.*

We illustrate a new
method to induce stability conditions on semiorthogonal decompositions and apply it to the Kuznetsov component of the derived category of cubic fourfolds. We use this to generalize
results of Addington-Thomas about cubic fourfolds and to study the rich hyperkaehler geometry associated to these hypersurfaces. This is the content of joint works with Arend Bayer,
Howard Nuer, Martí Lahoz, Emanuele Macrì and Alex Perry.

**Ronald van Luijk**:
* Automorphism groups of K3 surfaces over fields that are not algebraically closed.*

A consequence of the Torelli Theorem for K3 surfaces is that the automorphism group of a complex algebraic K3 surface is finite if and only if the automorphism group of its Picard lattice modulo the Weyl group of that lattice is finite. In other words, whether or not the K3 surface has finite automorphism group depends only on its Picard lattice. Over general fields this is no longer true. We will give examples of what can go wrong, and we also give some analogs of finiteness statements that are still correct. This is joint work with Martin Bright and Adam Logan.