The Streets--Tian conjecture on compact homogeneous quotients

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). We study Hermitian-symplectic metrics on homogeneous complex manifolds with compact isotropy, and on their compact discrete quotients. If the transitive group is connected and unimodular, we show that an invariant Hermitian-symplectic form gives an invariant Kähler metric, with no lattice assumption. By averaging, the same conclusion holds for compact quotients, even if the initial metric has no invariant lift. The proof uses the top exterior power of a Hermitian-symplectic form to bound the volume functional on an invariant Aeppli slice. Applying this bound to midpoint metrics gives a uniform trace estimate. From this, volume maximization produces a balanced metric in the slice, which is also pluriclosed and hence Kähler. We also describe the resulting Kähler representative and give examples showing the role of the closed lift.