The geometry of projective algebraic varieties is intimately related to positivity or negativity properties of their curvature tensor. In the introduction, we will first recall some basic concepts of complex differential geometry: complex manifolds, Dolbeault cohomology, vector bundles, connections, Chern classes, holomorphic Morse inequalities. These fundamental concepts will then be used to investigate the existence of entire holomorphic curves drawn in projective algebraic varieties. This problem is related to deep conjectures concerning number theory (Diophantine equations) and, in a more algebraic setting, to curvature properties of jet bundles. The ultimate goal will be to prove a recent result of the lecturer in the direction of the Green-Griffiths-Lang conjecture, according to which every entire curve drawn in a variety of general type satisfies algebraic differential equations.

Written by: Zhang Jiang
Source: PKU Lecture Hall