Deformation Openness and Closedness of Various Classes of Compact Complex Manifolds; Examples

Abstract

We review the relations between compact complex manifolds carrying various types of Hermitian metrics (Kähler, balanced or {\it strongly Gauduchon}) and those satisfying the $\partial \bar{\partial }$-lemma or the degeneration at $E_1$ of the Frölicher spectral sequence, as well as the behaviour of these properties under holomorphic deformations. The emphasis will be placed on the notion of {\it strongly Gauduchon} (sG) manifolds that we introduced recently in the study of deformation limits of projective and Moishezon manifolds. Various examples of sG and non-sG manifolds are exhibited while a range of constructions already known in the literature are reviewed and reinterpreted from this new standpoint.

48 pages