Special Hermitian metrics on compact solvmanifolds

TL;DR

This paper studies special Hermitian metrics on compact solvmanifolds, proving long-term solutions for pluriclosed flow in certain cases and proposing a new conjecture about balanced and SKT metrics.

Contribution

It introduces a new conjecture on the existence of balanced and SKT metrics and proves it for specific classes of nilmanifolds and solvmanifolds.

Findings

Long-time solutions for pluriclosed flow on certain solvmanifolds.

Validation of the conjecture for 6- and 8-dimensional nilmanifolds.

Validation of the conjecture for 6-dimensional solvmanifolds with trivial canonical bundle.

Abstract

We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant initial datum. Moreover, we state a new conjecture about the existence of balanced and SKT metrics on compact complex manifolds. We show that the conjecture is true for nilmanifolds of dimension 6 and 8 and for 6-dimensional solvmanifolds with holomorphically trivial canonical bundle.