Streets-Tian conjecture holds for 2-step solvmanifolds

Abstract

A Hermitian-symplectic metric is a Hermitian metric whose Kähler 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 Kählerian (i.e., admitting a Kähler 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 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. The proof of the main theorem also gives an explicit description of all Hermitian-symplectic metrics on any 2-step solvable Lie algebra equipped with a complex structure.