A Study of K3 Surfaces with a View Towards Compact Hyperkähler Manifolds

Abstract

K3 surfaces are a special case of two very important class of objects: Hyperkähler manifolds and Calabi-Yau manifolds. This makes them integral to build an under- standing of a large part of geometry. This thesis builds up to a result that make them remarkable: the Torelli theorem. Torelli theorems allow us to capture algebraically, the essence of these surfaces, granting us a clean bijective correspondence between the surface (with certain choices of fixed data) and their Hodge structure. Since K3 sur- faces are K¨ahler, their cohomology groups admit a Hodge decomposition. We leverage our understanding of their Hodge structure to look at compact hyperkähler or irre- ducible holomorphic symplectic manifolds instead of working with the hyperk¨ahler structure of higher dimensional IHS manifolds. This is possible because the Hodge decomposition for the second cohomology of IHS manifolds resembles that of K3 sur- faces, with the intersection form replaced by the Beauville-Bogomolov-Fujiki form, and a weaker version of the Torelli theorem holds for these manifolds. The thesis lays down several non-trivial ideas used in this study in the first chapter, slowly progress- ing to delve deeper into divisors, bundles and K3 surfaces, which form the bulk of the study.