“…As an application of this approach, we confirm a conjecture of Yau ([57, Problem 47]) that a compact Kähler manifold with positive holomorphic sectional curvature is a projective and rationally connected manifold. This project is motivated by a number of well-known conjectures proposed by Yau, Mumford, Demailly, Campana, Peternell and etc., and we refer to [55,57,10,33,32,16,17,38,27,7,12,14,28,6,49,11,15] and the references therein.…”