Automorphisms of O'Grady's sixfolds

We study automorphisms of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to the O’Grady’s sixfold. We classify non-symplectic and symplectic automorphisms using lattice theoretic criterions related to the lattice structure of the second integral cohomology. Moreover we introduce the concept of induced automorphisms. There are two birational models for O'Grady's sixfolds, the first one introduced by O'Grady, which is the resolution of singularities of the Albanese fiber of a moduli space of sheaves on an abelian surface, the second one which concerns in the quotient of an Hilbert cube by a symplectic involution. We find criterions to know when an automorphism is induced with respect to these two different models, i.e. it comes from an automorphism of the abelian surface or of the Hilbert cube.

Numerical invariants for measurable cocycles

The theory of numerical invariants for representations can be generalized to measurable cocycles. This provides a natural notion of maximality for cocycles associated to complex hyperbolic lattices with values in groups of Hermitian type. Among maximal cocycles, the class of Zariski dense ones turns out to have a rigid behavior. An alternative implementation of numerical invariants can be given by using equivariant maps at the level of boundaries and by exploiting the Burger-Monod approach to bounded cohomology. Due to their crucial role in this theory, we prove existence results in two different contexts. Precisely, we construct boundary maps for non-elementary cocycles into the isometry group of CAT(0)-spaces of finite telescopic dimension and for Zariski dense cocycles into simple Lie groups. Then we approach numerical invariants. Our first goal is to study cocycles from complex hyperbolic lattices into the Hermitian group SU(p,q). Following the theory recently developed by Moraschini and Savini, we define the Toledo invariant by using the pullback along cocycles, also by involving boundary maps. For cocycles Γ × X → SU(p,q) with 1cohomology, their image is contained in a finite dimensional algebraic subgroup of PU(p,∞). Finally, we classify Zariski dense measurable cocycles Γ × X → G from finitely generated groups into Hermitian groups not of tube-type. Precisely, we show that the pullback of the Kahler class completely determines the cohomology class of such cocycles.

A cohomological approach to Ruelle-Pollicott resonances and speed of mixing of Anosov diffeomorphisms

Given a transitive Anosov diffeomorphism on a closed manifold it is known that, for smooth enough observables, the system is mixing w.r.t. the measure of maximal entropy. Therefore, it makes sense to investigate the speed of decay of correlations and to look for the so-called Ruelle-Pollicott resonances, in order to determine a complete asymptotics for the decay of correlations. In this thesis we are able to find the first terms of that asymptotics and to prove an estimate for the speed of decaying of correlations. The proof is based on a surprising connection between the action of a transfer operator on suitable anisotropic Banach spaces of currents and the action induced by the Anosov map on the de Rham cohomology.

Automorphisms of irreducible holomorphic symplectic manifolds and related problems

We study automorphisms and the mapping class group of irreducible holomorphic symplectic (IHS) manifolds. We produce two examples of manifolds of K3[2] type with a symplectic action of the alternating group A7. Our examples are realized as double EPW-sextics, the large cardinality of the group allows us to prove the irrationality of the associated families of Gushel-Mukai threefolds. We describe the group of automorphisms of double EPW-cubes. We give an answer to the Nielsen realization problem for IHS manifolds in analogy to the case of K3 surfaces, determining when a finite group of mapping classes fixes an Einstein (or Kähler-Einstein) metric. We describe, for some deformation classes, the mapping class group and its representation in second cohomology. We classify non-symplectic involutions of manifolds of OG10 type determining the possible invariant and coinvariant lattices. We study non-symplectic involutions on LSV manifolds that are geometrically induced from non-symplectic involutions on cubic fourfolds.