1999/04/16 by J. Biswas, Jayabrata Biswas, V. Srinivas +2 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #math.AG
paper · pdf · doi:10.48550/arxiv.math/9904086
AMSLaTeX, 29 pages
arxiv created 1999/04/16 · openalex publication_date 1999/04/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an analogue of the Lefschetz (1,1) Theorem characterizing cohomology classes of Cartier divisors (or equivalently first Chern classes of line bundles) in the second integral cohomology. Let X be a normal complex projective variety. We show that the classes of Cartier divisors in H2(X,Z) are precisely the classes x such that (i) the image of x in H2(X,C) (cohomology with complex coefficients) lies in F1 H2(X,C) (first level of the Hodge filtration for Deligne's mixed Hodge structure), and (ii) x is Zariski-locally trivial, i.e., there is a covering of X by Zariski open sets U such that x has zero image in H2(U,Z). For normal quasi-projective varieties, this positively answers a question of Barbieri-Viale and Srinivas (J. Reine Ang. Math. 450 (1994)), where examples are given to show that divisor classes are not characterized by either one of the above conditions (i), (ii), taken by itself, unlike in the case of non-singular varieties. The present paper also contains an example of a non-normal projective variety for which (i) and (ii) do not suffice to characterize divisor classes.