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 cA=c (A, E) such that H⁰ (X, Sym^ E^* A^ k) =0 for cA (k+1) and k 0. Moreover, we obtain that, on a compact Kähler manifold X, if Λᵖ TX is RC-positive for every 1 p X, then X is projective and rationally connected. As applications, we show that if a compact Kähler manifold (X, ω) has positive holomorphic sectional curvature, then Λᵖ TX is RC-positive and H_^p, 0 (X) =0 for every 1 p X, and in particular, we establish that X is a projective and rationally connected manifold, which confirms a conjecture of Yau (57, Problem 47).