We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, rigidity and degeneracy theorems without assuming any pointwise curvature signs both the domain target manifolds are proved, in which key roles played by total integration of function first eigenvalue second Ricci an almost nonpositivity notion sectional introduced our previous work. also apply...

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M, Q) of dimension 4n ≥ 8. Using a Penrose transform we show that, when (M, Q) is compact and admits a compatible quaternionic-Kähler metric of negative scalar curvature, Θ admits no global non-trivial holomorphic sections with respect to any of its holomorphic structures...

It was proved in Chen's paper [3] that every real hypersurface the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$\delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is mean and $\delta(2)$ a $\delta$-invariant introduced by him. In this paper, we study non-Hopf hypersurfaces satisfying equality case inequality under condition along each integral curve Reeb vector ...

LetM be a compact, holomorphic symplectic Kähler manifold, and L a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of L is effective. This result is related to the hyperkähler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.

Let M be a complete noncompact Kähler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by Od(M) the space of holomorphic functions of polynomial growth of degree at most d on M. In this paper we prove that dimCOd(M) ≤ dimCO[d](C), for all d > 0, with equality for some positive integer d if and only if M is holomorphically isometric to C. We also obtain ...

In [Y], Yau proposed to study the uniformization of complete Kähler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete Kähler manifold M with positive bisectional curvature is biholomorphic to C. See also [GW], [Si]. For this sake, it was further asked in [Y] whether or not the ring of the holomorphic functions with polynomial growth, which we...

It is a classical fact that the cotangent bundle T M of a differentiable manifold M enjoys a canonical symplectic form Ω. If (M, J, g, ω) is a pseudo-Kähler or para-Kähler 2n-dimensional manifold, we prove that the tangent bundle TM also enjoys a natural pseudo-Kähler or para-Kähler structure (J̃, g̃,Ω), where Ω is the pull-back by g of Ω and g̃ is a pseudoRiemannian metric with neutral signature ...

In a previous paper, [1], we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted L-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of L-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sect...

Following [KNTY] we study a so-called bc-ghost system of zero conformal dimension from the viewpoint of [TUY] and [U2]. We show that the ghost vacua construction results in holomorphic line bundles with connections over holomorphic families of curves. We prove that the curvature of these connections are up to a scale the same as the curvature of the connections constructed in [TUY] and [U2]. We...