No abstract

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with non-positive bisectional curvature and quasi-negative Chern-Ricci curvature is ample.

We prove that compact complex manifolds with admitting metrics with negative Chern curvature operator either admit a dd c -exact positive (1, 1) current, or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow.

We prove that compact complex manifolds admitting metrics with negative Chern curvature operator either admit a $d d^c$-exact positive $(1,1)$ current or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow.