Pluriclosed flow and Hermitian-symplectic structures

Abstract

We show pluriclosed flow preserves the Hermitian-symplectic structures. And we observe that it can actually become a flow of Hermitian-symplectic forms when an extra evolution equation determined by the Bismut-Ricci form is considered. Moreover, we get a topological obstruction to the long-time existence in arbitrary dimension.

Kähler structures for holomorphic submersions

“…1 The question is still open in general, though there are negative results which say that there are no such examples among complex surfaces ( [14,18]), nilmanifolds with invariant complex structures ( [8]), twistor spaces ( [20]), Moishezon manifolds and complex manifolds of Fujiki class ( [17,3]). See also [7,22] for some analytic approach to the general problem. We consider this question for complex manifolds that admit structures of holomorphic submersions.…”

“…Acknowledgement: The author is grateful to Weiyi Zhang for a careful reading of the draft of this paper and providing valuable feedback and corrections. He also thanks Jeffrey Streets and Gang Tian for very helpful comments and notifying him the papers [7,22].…”