Some recent transcendental techniques in algebraic and complex geometry

TL;DR

This paper reviews recent transcendental methods applied in algebraic and complex geometry to prove key conjectures related to deformation invariance, the nonexistence of certain hypersurfaces, and hyperbolicity of high-degree hypersurfaces.

Contribution

It highlights the application of transcendental techniques to solve longstanding conjectures in algebraic and complex geometry.

Findings

Proof of invariance of plurigenera under deformation

Nonexistence of smooth Levi-flat hypersurfaces in the projective plane

High-degree hypersurfaces in projective space are hyperbolic

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.