Moduli Spaces on K3 Surfaces

Abstract

By definition, K3 surfaces are surfaces with vanishing first Betti number and trivial canonical bundle. Examples of K3 surfaces are provided by smooth complete intersections of type ( a 1 , …, a n −2 ) in ℙ n with ∑ a i = n + 1, Kummer surfaces and certain elliptic surfaces. In the Enriques classification K3 surfaces occupy, together with abelian, Enriques and hyperelliptic surfaces, the distinguished position between ruled surfaces and surfaces of positive Kodaira dimension. The geometry of K3 surfaces and of their moduli space is one of the most fascinating topics in surface theory, bringing together complex algebraic geometry, differential geometry and arithmetic. Following the general philosophy that moduli spaces of sheaves reflect the geometric structure of the surface it does not come as a surprise that studying moduli spaces of sheaves on K3 surfaces one encounters intriguing geometric structures. We will try to illuminate some aspects of the rich geometry of the situation. We present the material at this early stage in the hope that having explicit examples with a rich geometry in mind will make the more abstract and general results, where the geometry has not yet fully unfolded, easier to access. At some points we make use of results presented later (Chapter 9, 10). In particular, a fundamental result in the theory, namely the existence of a symplectic structure on the moduli space of stable sheaves, will be discussed only in Chapter 10.