RC-positivity, rational connectedness and Yau’s conjecture

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\[H_0 (X, \mathrm{Sym}^{\otimes \ell} E^{*} \otimes A^{\otimes k}) = 0\]for $\ell \geq c_A (k + 1)$ and $k \geq 0$. Moreover, we obtain that, on a compact Kähler manifold $X$, if $\Lambda^p T_X$ is RC-positive for every $1 \leq p \leq \dim X$, 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 $H^{p,0}_{\overline{\partial}} (X) = 0$ for every $1 \leq p \leq \dim X$, and in particular, we establish that $X$ is a projective and rationally connected manifold, which confirms a conjecture of Yau.

Keywords

RC-positivity, vanishing theorem, holomorphic sectional curvature, rationally connected

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This work was partially supported by China’s Recruitment Program of Global Experts and NSFC 11688101.

Received 27 February 2018

Published 18 May 2018