Deformation Openness and Closedness of Various Classes of Compact   Complex Manifolds; Examples
  We review the relations between compact complex manifolds carrying various types of Hermitian metrics (K\"ahler, balanced or {\it strongly Gauduchon}) and those satisfying the $\partial\bar\partial$-lemma or the degeneration at $E_1$ of the Fr\"olicher 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. 
Dan Popovici
math.AG math.CV math.DG