A Variational Approach to SKT and Balanced Metrics

TL;DR

This paper introduces a variational method to identify Kähler metrics among SKT and balanced metrics on compact complex manifolds, providing new tools to detect non-Kählerian structures within these classes.

Contribution

It defines a novel functional whose critical points correspond to Kähler metrics, offering a new approach to obstruct Kählerianity in SKT and balanced manifolds.

Findings

Critical points of the functional are exactly Kähler metrics.

Provides new obstructions to Kählerianity in non-Kähler SKT and balanced manifolds.

Framework applicable to compact complex manifolds with specific metric types.

Abstract

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case we define a new functional whose critical points are proved to be precisely the K\"ahler metrics, if any, on the manifold. As general manifolds of either type need not admit K\"ahler metrics, this provides an approach to new obstructions to K\"ahlerianity within these two families of metrics.