Compact Complex Manifolds with Small Gauduchon Cone

Abstract

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 Kähler manifolds, known as Fujiki {\it class} $\mathcal{C}$ manifolds. Our main idea is to explore the link between the {\it class} $\mathcal{C}$ 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 cohomology 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.

The title has been changed, the abstract and the introduction have been rewritten. To appear in the Proceedings of the London Mathematical Society

References in corpus (2)

• Geometry of nilmanifolds with left-invariant complex structure and deformations in the large
• On Bott-Chern cohomology of compact complex surfaces

Cited by in corpus (3)

• Non-Kähler Mirror Symmetry of the Iwasawa Manifold
• Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds
• The Lee--Gauduchon cone on complex manifolds