Generalized complex structures on 4-manifolds

Abstract

We study the existence of generalized complex structures on closed smooth fourmanifolds in this thesis. We first prove an integrability theorem of generalized complex structures in terms of almost bihermitian structures. After reviewing the results of type zero and type two generalized complex structures, we prove the necessary and sufficient topological conditions for the existence of type one generalized complex structures on compact four-manifolds. We also obtain a nonexistence result of T2-invariant untwisted generalized complex structures with mixed types on the four torus T4. Finally, we prove a result about finite group actions on tamed almost complex four-manifolds.