Some recent transcendental techniques in algebraic and complex geometry

Abstract

This article discusses the recent transcendental techniques used in the proofs of the following three conjectures. (1)~The plurigenera of a compact projective algebraic manifold are invariant under holomorphic deformation. (2)~There exists no smooth Leviflat hypersurface in the complex projective plane. (3)~A generic hypersurface of sufficiently high degree in the complex projective space is hyperbolic in the sense that there is no nonconstant holomorphic map from the complex Euclidean line to it.