2004/01/09 by Ania Otwinowska, Otwinowska, Ania
Arts and Humanities · Mathematics · #14C25 #14C30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Meromorphic and Entire Functions #math.AG #msc:14C25 #msc:14C30
paper · pdf · doi:10.48550/arxiv.math/0401092
16 pages
arxiv created 2004/01/09 · openalex publication_date 2004/01/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Y be a smooth complex projective variety of dimension N+1, L an invertible sufficiently ample sheaf, X∈ |L| a smooth hypersurface and λ∈ FkHN(X,C) a vanishing cohomology class, where F* is the Hodge filtration and k∈\1,...,[N/2]\. Assume that L is sufficiently ample and that the codimension in |L| of the Hodge variety associated to λ (locally defined as the locus where the image of λ by flat transport over |L| remains in Fk) is sufficiently small. I show that this forces N to be even and k=[N/2], and that the class λ 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.