Streets-Tian Conjecture holds for 2-step solvmanifolds
Shuwen Chen, Fangyang Zheng
TL;DR
The paper addresses the Streets-Tian conjecture, which asks whether a compact complex manifold admitting a Hermitian-symplectic metric must be Kähler. Focusing on Lie-complex manifolds with a left-invariant complex structure and, in particular, $G/\Gamma$ where $G$ is $2$-step solvable, the authors develop a framework using non-unitary frames to expose hidden symmetries and reduce to invariant metrics via averaging. They derive strong algebraic restrictions on the Lie algebra structure constants under a Hermitian-symplectic metric, and construct a frame change $\psi_i = \varphi_i + p_i \sigma_i$ that yields a new Kähler form $\tilde{\omega}$ with $d\tilde{\omega}=0$, thereby producing a left-invariant Kähler metric. Consequently, any compact quotient of a $2$-step solvable group with a left-invariant Hermitian-symplectic structure is Kähler, confirming the Streets-Tian conjecture in this broad class. The methods also hint at extensions toward the Fino-Vezzoni conjecture in the $2$-step solvable case and highlight the utility of non-unitary frames for uncovering structural symmetries.
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 {\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.