On rigid compact complex surfaces and manifolds

Abstract

This article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the relations among them. Only for curves these notions coincide and the only rigid curve is the projective line. For surfaces we prove that a rigid surface which is not minimal of general type is either a Del Pezzo surface of degree ≥5 or an Inoue surface. We give examples of rigid manifolds of dimension n≥3 and Kodaira dimensions 0, and 2≤k≤n. Our main theorem is that the Hirzebruch Kummer coverings of exponent n≥4 branched on a complete quadrangle are infinitesimally rigid. Moreover, we pose a number of questions.