Streets-Tian Conjecture holds for 2-step solvmanifolds

TL;DR

This paper proves the Streets-Tian Conjecture for 2-step solvmanifolds, showing that such manifolds with Hermitian-symplectic metrics are necessarily K"ahler, using novel methods involving non-unitary frames.

Contribution

It confirms the Streets-Tian Conjecture for all 2-step solvmanifolds, extending the known results beyond dimension 2 with a new proof technique.

Findings

Conjecture holds for 2-step solvmanifolds

Use of non-unitary frames reveals hidden symmetries

Method may be applicable to other complex geometric problems

Abstract

A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be K\"ahlerian (i.e., admitting a K\"ahler metric). The conjecture is known to be true in dimension $2$ but is still open in dimensions $3$ or higher. In this article, we confirm the conjecture for all 2-step solvmanifolds, namely, compact quotients of 2-step solvable Lie groups by discrete subgroups. In the proofs, we adopted a method of using special {\em non-unitary} frames, which enabled us to squeeze out some hidden symmetries to make the proof go through. Hopefully the technique could be further applied.