RC-positivity, rational connectedness and Yau's conjecture

Abstract

In this paper, we give a differential geometric interpretation of Mumford's conjecture on rational connectedness and outline a differential geometric approach. To this end, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles over compact complex manifolds. We prove that if \(E\) is an RC-positive vector bundle over a compact complex manifold \(X\), then any line subbundle of the dual vector bundle \(E^*\) can not be pseudo-effective and 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 projective manifold \(X\), if the anticanonical bundle \(\Lambda^{\dim X}T_X\) is RC-positive, then \(X\) is uniruled; if \(\Lambda^p T_X\) is RC-positive for every \(1\leq p\leq \dim X\), then \(X\) is 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 \(1\leq p\leq \dim X\); in particular, we establish that \(X\) is a projective and rationally connected manifold, which confirms a conjecture of Yau.