Compact complex manifolds with small Gauduchon cone

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kahler manifolds, known as Fujiki classC manifolds. Our main idea is to explore the link between the classC property and the closed positive currents of bidegree (1, 1)that the manifold supports, a fact leading to the study of semicontinuity properties under deformations of the complex structure of the dual cones of cohomology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly non-Kahler, manifolds defined by the condition that every Gauduchon metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be small in a certain sense. We term them sGG manifolds and find numerical characterizations of them in terms of certain relations between various cohomology theories (De Rham, Dolbeault, Bott-Chern, Aeppli). We also produce several concrete examples of nilmanifolds demonstrating the differences between the sGG class and well-established classes of complex manifolds. We conclude that sGG manifolds enjoy good stability properties under deformations and modifications.