Compact Complex Manifolds with Small Gauduchon Cone

International audienceThis 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 Kähler manifolds, known as Fujiki class C manifolds. Our main idea is to explore the link between the class C property and the closed positive currents of bidegree (1, 1) that the manifold supports, a fact leading to the study of semi-continuity properties under deformations of the complex structure of the dual cones of co-homology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly non-Kähler, 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 characterisations 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