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

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

We give a proof of the Hard Lefschetz Theorem for orbifolds that does not involve intersection homology. We use a foliated version of the Hard Lefschetz Theorem due to El Kacimi.

We establish an interior C 2 estimate for k+ 1 convex solutions to Dirichlet problems of k-Hessian equations. We also use such estimate to obtain a rigidity theorem for k+ 1 convex entire solutions of k-Hessian equations in Euclidean space.

We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of BersNirenberg [8]. We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.

The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k . We also discuss some applications of our estimates.