Hermite几何中的Fino-Vezzoni猜想

The Fino-Vezzoni conjecture in Hermitian geometry

Fino-Vezzoni猜想是近年来非Kähler几何研究中的热点问题之一.该猜想的叙述是,任给紧复流形,若其上存在平衡度量又存在多重闭度量,则必存在Kähler度量.目前针对若干特殊的Hermite流形类,该猜想已经被验证.本文简要回顾关于该猜想的若干部分性结果,并对一类特殊的Lie复流形验证该猜想:若紧复流形的万有覆叠为带左不变复结构的Lie群,其Lie代数含有J-不变的余维为2的Abel理想,则该猜想在其上成立.这种情形是“几乎Abel”情形的一个自然推广.

The Fino-Vezzoni conjecture is an active research topic in non-Kähler geometry in recent years.It states that,if a compact complex manifold admits a balanced metric and a pluriclosed metric,then it must admit a Kahler metric.At present the conjecture is known for a number of special types of Hermitian manifolds.In this article,we prove the conjecture in the special case when the compact complex manifold has its universal cover being a Lie group equipped with a left-invariant complex structure,whose Lie algebra contains a J-invariant Abelian ideal of codimension 2.This is a natural generalization to the"almost Abelian"case.