  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.
