Matsushima-Shimura theorem

Statement

Let $X$ be an irreducible compact quotient of a polydisc. Then:

• $H^{1}(X,T_{X})=0$
• As a corollary, $X$ is locally rigid as a complex manifold

Relation with other results

Calabi-Vesentini theorem

This draws the same conclusions from slightly more relaxed hypotheses.

Further information: Calabi-Vesentini theorem