On rigid compact complex surfaces and manifolds

A compact complex manifold $X$ is rigid if it has no nontrivial deformations. The only rigid complex curve is the projective line; for dimension 2 we prove:
Theorem. Let $S$ be a compact complex surface, which is rigid, then:

• $S$ is minimal of general type, or
• $S$ is a Del Pezzo surface of degree $ \ge 5$, or
• $S$ is an Inoue surface.

We explain different concepts of rigidity, their relations and give new examples and pose open questions.
This is joint work with F. Catanese