Jie Liu

This thesis is devoted to the study of complex Fano varieties via the properties of subsheaves of the tangent bundle and the geometry of the fundamental divisor. The main results contained in this text are:
(1) a generalization of Hartshorne's conjecture: a projective manifold is isomorphic to a projective space if and only if its tangent bundle contains an ample subsheaf
(2) vanishing theorem on irreducible Hermitian symmetric spaces of compact type \(M\) and its applications to the stability of the tangent bundles of smooth complete intersections in \(M\) and effective restriction problem;
(3) Bogomolov-type inequality for Fano manifolds of Picard number one and its applications to effective nonvanishing problem, anticanonical geometry of Fano manifolds and the calculation of Seshadri constants of the anticanonical divisors of Fano manifolds with large index;
(4) geometry of smooth Moishezon manifolds with Picard number one.

The main aim of this thesis is to review the proof of Oshawa-Takegoshi extension theorem and its applications in complex geometry, including Siu's invariance of plurigenera, semi-continuity theorem of Demailly-Kollár and strong openness conjecture and so on.