No abstract

Abstract. In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Abstract. In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the γ k functions on the space of its hermitian metrics.

Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott-Chern cohomology class, a balanced metric which is hermitian Ricci-flat. T his can be viewed as a differential form level generalization of the classical Calabi-Yau equation. We establish the existence and uniqueness of the equation on complex tori, and prove certain uniqueness and openness on a general Kähler manifold.

In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.