RC-positivity, rational connectedness and Yau's conjecture

Abstract

In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if $E$ is an RC-positive vector bundle over a compact complex manifold $X$, then for any vector bundle $A$, there exists a positive integer $c_A=c(A,E)$ such that $$H^0(X,{\mathrm{Sym}}^{\otimes \ell }{E}^{\ast }\otimes A^{\otimes k})=0$$ for $\ell \ge c_A(k+1)$ and $k\ge 0$. Moreover, we obtain that, on a compact Kähler manifold $X$, if ${\Lambda }^p{T}_X$ is RC-positive for every $1\le p\le \dim X$, then $X$ is projective and rationally connected. As applications, we show that if a compact Kähler manifold $(X,\omega )$ has positive holomorphic sectional curvature, then ${\Lambda }^p{T}_X$ is RC-positive and $H_{\bar{\partial }}^{p,0}(X)=0$ for every $1\le p\le \dim X$, and in particular, we establish that $X$ is a projective and rationally connected manifold, which confirms a conjecture of Yau([57, Problem 47]).

Accepted by Cambridge Journal of Mathematics

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• A partial converse to the Andreotti-Grauert theorem
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• Scalar curvature on compact complex manifolds
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Cited by in corpus (2)

• On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature
• Comparison and vanishing theorems for Kähler manifolds