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 H¹(X,O). This implies for instance that any two compact complex parallelizable manifolds X and Y with isomorphic fundamental group have dim H¹(X,O) =dim H¹(Y,O). Moreover we show that for compact complex parallelizable manifolds H¹(X,O)=0 iff b₁(X)=0 iff X admits no small deformations, thus completely determining which compact complex parallelizable manifolds are rigid.