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 K\"ahler 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 K\"ahler 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 $1\leq p\leq \dim X$, and in particular, we establish that $X$ is a projective and rationally connected manifold, which confirms a conjecture of Yau([57, Problem 47]).