Deformations of balanced metrics

TL;DR

This paper investigates how small changes in complex structures affect the existence of balanced metrics on Hermitian non-Kähler manifolds, providing necessary conditions for their deformation.

Contribution

It introduces necessary conditions for the existence of smooth deformations of balanced metrics under small complex structure changes.

Findings

Necessary conditions for balanced metric deformations

Analysis of stability of balanced metrics under complex structure variations

Insights into Hermitian non-Kähler geometry

Abstract

Small deformations of the complex structure do not always preserve special metric properties in the Hermitian non-K\"ahler setting. In this paper, we find necessary conditions for the existence of smooth curves of balanced metrics $\{\omega_t\}_t$ which start with a fixed balanced metric $\omega$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.