On strongly Gauduchon metrics of compact complex manifolds

TL;DR

This paper investigates the properties and existence of strongly Gauduchon metrics on compact complex manifolds, focusing on their cohomology cones, behavior under modifications, and conditions for existence on fibred manifolds.

Contribution

It introduces the study of cohomology cones generated by strongly Gauduchon metrics and proves an existence theorem for such metrics on certain fibred manifolds.

Findings

Cohomology cones of strongly Gauduchon metrics are analyzed.

Existence of strongly Gauduchon metrics is established for manifolds fibred over curves with fibers satisfying ddbar-lemma.

Conditions for the existence of these metrics under proper modifications are identified.

Abstract

In this paper, we study strongly Gauduchon metrics on compact complex manifolds. We study the cohomology cones SG in the de Rham cohomology groups generated by all strongly Gauduchon metrics and its direct images under proper modifications. We also study the moduli of strongly Gauduchon manifolds. We prove an existence result of strongly Gauduchon metrics on compact complex manifold which is fibred over a compact complex curve. In particular, if a compact complex manifold X has a topologically essential fibration over a compact complex curve, and if the generic fibres satisfy ddbar-lemma, then X admits strongly Gauduchon metrics.