Progr\`es r\'ecents sur les fonctions normales (d'apr\`es Green-Griffiths, Brosnan-Pearlstein, M. Saito, Schnell...)

TL;DR

This paper discusses recent progress on the algebraicity of loci of Hodge classes and normal functions in complex algebraic geometry, highlighting recent proofs by Brosnan-Pearlstein, Schnell, and others.

Contribution

It reviews recent proofs of the algebraicity of zero loci of normal functions, extending classical results to mixed Hodge structures.

Findings

Proof of algebraicity of zero loci of normal functions

Extension of classical Hodge conjecture results

Integration of work by Brosnan-Pearlstein, Schnell, and Saito

Abstract

Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around this theorem.