Sur les varietes de Hodge

TL;DR

This paper investigates the structure of Hodge varieties on smooth projective varieties, showing that certain small codimension loci are generated by algebraic subvarieties, supporting aspects of the Hodge conjecture.

Contribution

It establishes conditions under which Hodge classes are algebraic and describes the infinitesimal structure of the Noether-Lefschetz locus, advancing understanding of the Hodge conjecture.

Findings

N is even and k equals N/2 under small codimension assumptions

Hodge classes of small codimension are linear combinations of algebraic subvarieties

Components of smallest codimension in the Noether-Lefschetz locus are spanned by algebraic classes

Abstract

Let $Y$ be a smooth complex projective variety of dimension $N+1$, $L$ an invertible sufficiently ample sheaf, $X\in |L|$ a smooth hypersurface and $\lambda\in F^kH^N(X,C)$ a vanishing cohomology class, where $F^{*}$ is the Hodge filtration and $k\in\{1,...,[N/2]\}$. Assume that $L$ is sufficiently ample and that the codimension in $|L|$ of the Hodge variety associated to $\lambda$ (locally defined as the locus where the image of $\lambda$ by flat transport over $|L|$ remains in $F^k$) is sufficiently small. I show that this forces $N$ to be even and $k=[N/2]$, and that the class $\lambda$ is a linear combination with complex coefficients of classes of algebraic subvarieties of $X$ of small degree. As a corollary, I obtain that the components of smallest codimensions of the Noether-Lefschetz locus are spanned by classes of algebraic subvarieties as predicted by Hodge conjecture. The proof relies on an algebraic description of the infinitesimal neighboorghood of the Noether-Lefschetz locus at any order and on a (global) monodromy result.