Generalized complex geometry is a newly emerging field that unites two areas of geometry, symplectic and complex, revealing surprising new aspects in both. Largely motivated by physics, it provides a mathematical context for studying certain string theoretic topics. Since it is a relatively new field, mathematicians are still learning how known geometric objects fit into the realm of generalized complex geometry. One such object is Penrose's twistor space. In this dissertation, we study the generalization of twistor theory for K3 surfaces, hyperkahler, and quaternionic Kahler manifolds. We use generalized complex geometry to construct a manifold fibered over the product of two copies of one-dimensional complex projective space that arises from a family of complex and symplectic structures on a K3 surface. We call this manifold a generalized twistor space. After proving that it has an integrable generalized complex structure, we describe properties on this space analogous to those in classical twistor theory. We then extend this construction to all hyperkahler manifolds of higher dimension. Finally, we consider the quaternionic Kahler analogue of this generalized twistor space. We produce a candidate for a generalized almost complex structure on the space and conjecture that the structure is integrable.Doctor of Philosoph
