Complex Manifolds With Negative Curvature Operator

Source abstract

Abstract 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.