Let (M,ω) be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu–Yau and Tosatti–Yang that M is necessarily projective has ample canonical bundle. In this paper, we show any irreducible subvariety of general type, thus confirming in particular case celebrated conjecture Lang. Moreover, can extend the theorem to quasinegative curvature building on earlier...

We improve Chen-Ricci inequalities for a Lagrangian submanifold Mn of dimension n (n 2) in a 2n -dimensional complex space form M̃2n(4c) of constant holomorphic sectional curvature 4c with a semi-symmetric metric connection and a Legendrian submanifold Mn in a Sasakian space form M̃2n+1(c) of constant φ -sectional curvature c with a semi-symmetric metric connection, respectively.

We obtain a conceptually new differential geometric proof of P.F. Klembeck’s result (cf. [9]) that the holomorphic sectional curvature kg(z) of the Bergman metric of a strictly pseudoconvex domain Ω ⊂ C approaches −4/(n + 1) (the constant sectional curvature of the Bergman metric of the unit ball) as z → ∂Ω.

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmüller space Teich(Sg) of a closed surface Sg of genus g is uniformly bounded away...

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow...