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 $H^1(X,{\mathcal O})$. This implies for instance that any two compact complex parallelizable manifolds $X$ and $Y$ with isomorphic fundamental group have ${\rm dim}\ H^1(X,{\mathcal O})={\rm dim}\ H^1(Y,{\mathcal O})$. Moreover we show that for compact complex parallelizable manifolds $H^1(X,{\mathcal O})=0$ iff $b_1(X)=0$ iff $X$ admits no small deformations, thus completely determining which compact complex parallelizable manifolds are rigid.