On rigid compact complex surfaces and manifolds

TL;DR

This paper explores various notions of rigidity in compact complex manifolds, classifies rigid surfaces, provides examples in higher dimensions, and proves the infinitesimal rigidity of certain branched coverings.

Contribution

It clarifies the relationships among different rigidity notions, classifies rigid surfaces, and establishes infinitesimal rigidity for Hirzebruch Kummer coverings of specific exponents.

Findings

Rigid curves are only the projective line.

Rigid surfaces not of general type are Del Pezzo or Inoue surfaces.

Hirzebruch Kummer coverings of exponent ≥ 4 are infinitesimally rigid.

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.