K3 surfaces and
hyperkähler manifolds

K3 surfaces are known to be surfaces with a rich geometry and the study of their automorphisms has gained interest in the last decades. Given a K3 surface X, an automorphism is called symplectic (resp. non-symplectic) if its action on the nowhere vanishing 2-form generating the space $H^{2,0}(X)$ is trivial (resp. not trivial).

In this talk I will present some results on classification on automorphisms on K3 surfaces, both symplectic and non-symplectic. I will recall well-known results on symplectic automorphisms, due to Nikulin, and some more recent results on K3 surfaces admitting the action of a maximal symplectic group. I will then focus on classification of K3 surfaces admitting a non-symplectic automorphism and in particular the case where the moduli space is one-dimensional (joint work with M. Artebani, M.E. Valdés).

An irreducible holomorphic symplectic manifold X is a simply connected compact complex Kähler manifold such that $H^0(X, \Omega_X^2) is generated by a nowhere degenerate holomorphic symplectic form. Due to this feature an action on an irreducible holomorphic symplectic manifold is called symplectic if the induced action on the symplectic form is trivial. In a joint work with C. Onorati and D.C. Veniani we study symplectic birational transformations of finite order on irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O’Grady. More precisely we consider the induced isometries on the Beauville–Bogomolov–Fujiki lattice, classifying all possible invariant and coinvariant sublattices. As a consequence we give a rigidity result for symplectic automorphisms of manifolds of OG6 type.

Let $\mathcal{M}_{2k}$ denote the moduli space of $U\oplus \langle -2k\rangle$-polarized K3 surfaces. Geometrically, the K3 surfaces in $\mathcal{M}_{2k}$ are elliptic and contain an extra curve class, depending on $k\ge 1$. I will report on a joint work with M. Fortuna and M. Hoff, in which we compute the Kodaira dimension of $\mathcal{M}_{2k}$ for almost all $k$: more precisely, we show that it is of general type if $k\ge 220$ and unirational if $k\le 50$, $k\notin \{11,35,42,48\}$. After introducing the general problem, I will compare the strategies used to obtain both results. I will then show some examples arising from explicit geometric constructions.

I will present some results on the geometry of irreducible holomorphic symplectic manifolds of type OG10, like the shape of the monodromy group and of the Kähler cone. I will do this by working with a particular example of such manifolds, namely the symplectic compactification of the intermediate jacobian fibration associated to a cubic fourfold. As an application we will see that for a generic cubic fourfold the compactification is essentially unique.

The problem of determining when a K3 surface is realized as a Kummer surface was a problem introduced by Shioda and Inose, and solved by Morrison using foundational works of Nikulin about symplectic automorphisms on K3 surfaces. In the first part of the talk, we will generalize results about symplectic involutions to order three symplectic automorphisms: we will explicitly describe the induced action of these automorphisms on the K3-lattice (which is isometric to the second cohomology group of a K3 surface) and we will deduce the relation between the families of K3 surfaces admitting these automorphisms and the ones given by their quotients. In the second part, we will study the relation between Abelian surfaces admitting order 3 endomorphisms and K3 surfaces admitting order 3 symplectic automorphisms thus generalizing classical results about Shioda--Inose structures. This is joint work with Alice Garbagnati.

Several enumerative problems on a K3 surfaces, like counting Fourier--Mukai partners, Kummer structures or Enriques quotients, admit an answer in terms of lattice theory. The resulting counting formulas share a common pattern. I will give an overview of the known results, focusing especially on the enumeration of jacobian elliptic fibrations (joint work with Dino Festi).