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# Compact Kähler manifolds with positive orthogonal bisectional curvature

**Article**

*in*Mathematical Research Letters 24(3):767-780 · October 2017

*with*75 Reads

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Abstract

In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact Kähler manifold with positive orthogonal bisectional curvature must be biholomorphic to Pn.

- ... By Proposition 1.1 and Theorem 4.1, we know M is compact. Due to Gu-Zhang [10] (see also Chen [3] and Feng-Liu-Wan [7]), M is biholomorphic to CP n . It exists a Fubini-Study metric ω F S (a special Kähler-Einstein metric). ...Preprint
- Jun 2019

In this paper, we prove that any complete shrinking gradient K\"ahler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient K\"ahler-Ricci solitons with nonnegative orthogonal bisectional curvature. - Article
- Dec 2019
- RESULTS MATH

In this paper, we prove that any complete shrinking gradient Kähler–Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler–Ricci solitons with nonnegative orthogonal bisectional curvature. - Preprint
- Oct 2018

In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $$\mathscr Y=g_{ij}f^i_\alpha f^j_\beta \frac{W^\alpha W^\beta}{\sum h_{\gamma\delta} W^\gamma W^\delta}.$$ We get new Hessian estimates to this energy density and obtain various new Liouville type theorems for holomorphic maps, harmonic maps and pluri-harmonic maps. For instance, we show that there is no non-constant holomorphic map from a compact \emph{Hermitian manifold} with positive (resp. non-negative) holomorphic sectional curvature to a \emph{Hermitian manifold} with non-positive (resp. negative) holomorphic sectional curvature.

- Complex analytic and algebraic geometry, book online https
- J.-P Demailly

J.-P. Demailly, Complex analytic and algebraic geometry, book online https://www-fourier.ujf-grenoble.fr/ ~ demailly/books.html. - Article
- Aug 1993
- T AM MATH SOC

We prove that a compact orientable 2n-dimensional Riemannian manifold, with second Betti number nonzero, nonnegative curvature on totally isotropic 2-planes, and satisfying a positivity-type condition at one point, is necessarily Kähler, with second Betti number 1. Using the methods of Siu and Yau, we prove that if the positivity condition is satisfied at every point, then the manifold is biholomorphic to complex projective space. - Article
- Dec 2012

In this lecture notes, we aim at giving an introduction to the K\"ahler-Ricci flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in its first twenty years (1984-2003), especially an essentially self-contained exposition of Perelman's uniform estimates on the scalar curvature, the diameter, and the Ricci potential function for the normalized K\"ahler-Ricci flow (NKRF), including the monotonicity of Perelman's \mu-entropy and \kappa-noncollapsing theorems for the Ricci flow on compact manifolds. The Notes is based on a mini-course on KRF delivered at University of Toulouse III in February 2010, a talk on Perelman's uniform estimates for NKRF at Columbia University's Geometry and Analysis Seminar in Fall 2005, and several conference talks, including "Einstein Manifolds and Beyond" at CIRM (Marseille - Luminy, fall 2007), "Program on Extremal K\"ahler Metrics and K\"ahler-Ricci Flow" at the De Giorgi Center (Pisa, spring 2008), and "Analytic Aspects of Algebraic and Complex Geometry" at CIRM (Marseille - Luminy, spring 2011). - Article
- May 2010

In this paper, we will give an extension of Mok’s theorem on the generalized Frankel conjecture under the condition of the orthogonal holomorphic bisectional curvature. KeywordsKähler Ricci flow-orthogonal holomorphic bisectional curvature-first Chern class MSC(2000)53C55 - Article
- Nov 2007
- ADV MATH

For any irreducible Kähler manifold which admits positive orthogonal bisectional curvature and C1>0, if this positivity condition is preserved under the flow, then the underlying manifold is biholomorphic to CPn. - ArticleFull-text available
- Jan 1981