Stable sheaves on hyper-Kähler manifolds = Faisceaux stables sur variétés hyper-Kähleriennes

In this thesis we study sheaves on hyper-Kähler manifolds, with final goal to solve a long standing conjecture due to Markman and O’Grady. Namely, we show that we can realize OG10 as a moduli space of stable bundles on a hyper-Kähler fourfold. We begin with stable sheaves and, more in general, stable complexes on K3 surfaces. It is a celebrated result, due to the work of many people, that moduli spaces parametrizing such stable complexes are projective hyper-Kähler manifolds. In the first chapter we provide a new conceptual proof of this fact, exploiting the powerful wall-crossing techniques made possible by the theory of Bridgeland stability conditions. The rest of the thesis is devoted to sheaves on higher dimensional hyper-Kähler manifolds. This is a relatively recent theory, which started with the breakthrough work by Taelman on the LLV algebra, and O’Grady’s works on modular sheaves. The second chapter of this thesis is devoted to a review of the main properties of the cohomology of a hyper-Kähler manifold, especially on Taelman’s work. The theory was later developed independently by Beckmann and Markman, who introduced a class of sheaves which we now call atomic. We review their works, together with O’Grady’s, in chapter three. In the rest of the thesis we consider two new examples of stable atomic sheaves on a hyper-Kähler fourfold. They are obtained by applying derived equivalences to Lagrangian surfaces. In both cases, their moduli spaces are ten dimensional (possibly singular) symplectic varieties, and are birational to OG10. In one of these cases, we are able to prove smoothness and therefore obtain a hyper-Kähler manifold of type OG10 as a moduli space of stable atomic sheaves.