Cambridge Journal of Mathematics

Volume 6 (2018)

Number 2

RC-positivity, rational connectedness and Yau’s conjecture

Pages: 183 – 212

DOI: http://dx.doi.org/10.4310/CJM.2018.v6.n2.a2

Author

Xiaokui Yang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

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\[H_0 (X, \mathrm{Sym}^{\otimes \ell} E^{*} \otimes A^{\otimes k}) = 0\]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

Full Text (PDF format)

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

Received 27 February 2018

Published 18 May 2018