scholarly journals Pseudoconvex domains on a Kähler manifold with positive holomorphic bisectional curvature

1976 ◽  
Vol 12 (1) ◽  
pp. 191-214 ◽  
Author(s):  
Osamu Suzuki
Keyword(s):  
Kähler Manifold ◽  
Pseudoconvex Domains ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature ◽  
Kahler Manifold

scholarly journals A Liouville Property of Holomorphic Maps

2013 ◽  
Vol 2013 ◽  
pp. 1-3
Author(s):  
Chengjie Yu
Keyword(s):  
Sectional Curvature ◽  
Kähler Manifold ◽  
Holomorphic Maps ◽  
Liouville Property ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature ◽  
Kahler Manifold ◽  
Simply Connected ◽  
Complete Kähler Manifold

We prove a Liouville property of holomorphic maps from a complete Kähler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kähler manifold with a certain assumption on the sectional curvature.

scholarly journals A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

2013 ◽  
Vol 2013 (679) ◽  
pp. 223-247 ◽  
Author(s):  
Burkhard Wilking
Keyword(s):  
Lie Algebra ◽  
Ricci Flow ◽  
Algebraic Approach ◽  
Kähler Manifold ◽  
Curvature Condition ◽  
Bisectional Curvature ◽  
Harnack Inequalities ◽  
Kahler Manifold ◽  
Positive Bisectional Curvature ◽  
Compact Kähler Manifold

Abstract We consider a subset S of the complex Lie algebra 𝔰𝔬(n, ℂ) and the cone C(S) of curvature operators which are nonnegative on S. We show that C(S) defines a Ricci flow invariant curvature condition if S is invariant under AdSO(n, ℂ). The analogue for Kähler curvature operators holds as well. Although the proof is very simple and short, it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact Kähler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to ℂℙn. Moreover, the methods can also be applied to prove Harnack inequalities.

scholarly journals Complex Hessian Equations on Some Compact Kähler Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-48 ◽  
Author(s):  
Asma Jbilou
Keyword(s):  
Smooth Function ◽  
A Priori ◽  
Kähler Manifold ◽  
Kahler Manifolds ◽  
Continuity Method ◽  
Bisectional Curvature ◽  
Hessian Equation ◽  
Holomorphic Bisectional Curvature ◽  
Hessian Equations ◽  
Kähler Form

On a compact connected2m-dimensional Kähler manifold with Kähler formω, given a smooth functionf:M→ℝand an integer1<k<m, we want to solve uniquely in[ω]the equationω̃k∧ωm-k=efωm, relying on the notion ofk-positivity forω̃∈[ω](the extreme cases are solved:k=mby (Yau in 1978), andk=1trivially). We solve by the continuity method the corresponding complex elliptickth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.

scholarly journals Bounded p.s.h. functions and pseudoconvexity in Kähler manifold

1998 ◽  
Vol 149 ◽  
pp. 1-8 ◽  
Author(s):  
Takeo Ohsawa ◽  
Nessim Sibony
Keyword(s):  
Kähler Manifold ◽  
Pseudoconvex Domains ◽  
Real Hypersurfaces ◽  
Plurisubharmonic Functions ◽  
Kahler Manifold

Abstract.It is proved that the C2-smoothly bounded pseudoconvex domains in Pn admit bounded plurisubharmonic exhaustion functions. Further arguments are given concerning the question of existence of strictly plurisubharmonic functions on neighbourhoods of real hypersurfaces in Pn.

scholarly journals Relative non-pluripolar product of currents

2021 ◽  
Author(s):  
Duc-Viet Vu
Keyword(s):  
Kähler Manifold ◽  
Monotonicity Property ◽  
Necessary Condition ◽  
Positive Current ◽  
Lelong Numbers ◽  
Kahler Manifold ◽  
Closed Positive Current ◽  
Positive Currents ◽  
Compact Kähler Manifold

AbstractLet X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ T 1 , … , T m be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ T 1 , … , T m . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ T 1 , … , T m when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.

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