On Rigidity and the Albanese Variety for Parallelizable Manifolds

Jörg Winkelmann

We investigate the Albanese variety, cohomology groups and deformations of complex parallelizable manifolds. In particular, we give a precise method to determine the dimension of H^1(X,O). This implies for instance that any two compact complex parallelizable manifolds X and Y with isomorphic fundamental group have dim H^1(X,O)=dim H^1(Y,O). Moreover we show that for compact complex parallelizable manifolds H^1(X,O)=0 iff b_1(X)=0 iff X admits no small deformations, thus completely determining which compact complex parallelizable manifolds are rigid.