This lecture is an introduction to the analytic approach to the study of positivity of direct images. I will first explain Berndtsson's famous curvature formula of the direct images, and several important generalisations. In the second part, I will explain some applications in complex geometry: transcendental weak sub-adjunction formula, Iitaka conjecture etc. If the time is permitted, I will explain Berndtsson, Paun and Wang's curvature formula of higher direct images and its applications which is closely related to Ya Deng's previous lectures about the hyperbolicity of moduli spaces.

曹俊彦,法国Universite Côte D'Azur教授.

Junyan Cao, Professor of Universite Côte D'Azur.