Stability of Strongly Gauduchon Manifolds under Modifications

TL;DR

This paper proves that compact complex manifolds with strongly Gauduchon metrics remain stable under modifications, extending previous work on deformation limits and contrasting with the instability of Kähler manifolds.

Contribution

It establishes the stability of strongly Gauduchon manifolds under modifications, a property not shared by Kähler manifolds, using currents and regularisation techniques.

Findings

Strongly Gauduchon manifolds are stable under modifications.

Stability property contrasts with Kähler manifolds.

Method involves direct and inverse images of positive currents.

Abstract

In our previous works on deformation limits of projective and Moishezon manifolds, we introduced and made crucial use of the notion of strongly Gauduchon metrics as a reinforcement of the earlier notion of Gauduchon metrics. Using direct and inverse images of closed positive currents of type $(1, \, 1)$ and regularisation, we now show that compact complex manifolds carrying strongly Gauduchon metrics are stable under modifications. This stability property, known to fail for compact K\"ahler manifolds, mirrors the modification stability of balanced manifolds proved by Alessandrini and Bassanelli.