Sur les varietes de Hodge

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.