Compact Complex Manifolds with Small Gauduchon Cone

TL;DR

This paper introduces sGG manifolds, a new class of compact complex manifolds characterized by the property that all Gauduchon metrics are strongly Gauduchon, and explores their properties, examples, and stability under deformations.

Contribution

It defines the sGG class of manifolds, provides numerical characterizations, and demonstrates their stability and differences from other classes through examples.

Findings

sGG manifolds have small Gauduchon cones.

They are stable under deformations and modifications.

Concrete nilmanifold examples illustrate their distinct properties.

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 K\"ahler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to explore the link between the {\it class} ${\cal C}$ 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-K\"ahler, 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.