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]).