Introduction to the Moduli Spaces of Sheaves on K3 Surfaces

The minicourse will consist of 5-6 lectures and its aim will be to introduce the moduli spaces of semistable coherent sheaves on a smooth projective variety X, mainly focusing on the case when X is a K3 surface. The reason for choosing K3 surfaces is that the moduli spaces of sheaves in this case carry a rich geometric structure – they are holomorphic symplectic varieties. Therefore they serve as one of the main sources of examples of compact hyperkähler manifolds, and the minicourse may be considered a complement to Verbitsky’s lecture course on hyperkähler geometry. The theory of moduli spaces of sheaves is technically quite complicated, and I will mainly avoid giving all the details of the proofs, focusing mainly on the ideas and methods of working with the moduli spaces. The tentative list of topics to be discussed:

1. General introduction to the notion of (semi-)stability for coherent sheaves, existence of coarse moduli spaces.

2. Sheaves on projective surfaces, their discrete invariants and moduli spaces. The case of surfaces with trivial canonical bundle. Examples of moduli spaces.

3. Symplectic structure on the moduli spaces of sheaves on K3 surfaces.

4. Local structure of the singularities of the moduli spaces of sheaves on K3 surfaces, brief discussion of O’Grady’s exceptional variety OG10.

5. Calabi-Yau varieties arising as moduli spaces of sheaves on Enriques surfaces. e livros e artigos nele citados.