2002/03/15 by Masanori Asakura, Asakura, Masanori, Shuji Saito +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #math.AG
paper · pdf · doi:10.48550/arxiv.math/0203147
33pages
arxiv created 2002/03/15 · openalex publication_date 2002/03/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair (X,Z) where X is a smooth complete intersection in the projective space and and Z is a simple normal crossing divisor in X whose irreducible components are smooth hypersurface sections on X. Our Jacobian rings give an algebraic description of the cohomology of the open complement X-Z and it is a natural generalization of the Poincaré residue representation of the cohomology of a hypersurface originally invented by Griffiths. The main results generalize the Macaulay's duality theorem and the Donagi's symmetrizer lemma for usual Jacobian rings for hypersurfaces. A feature that distinguishes our generalized Jacobian rings from usual ones is that there are instances where duality fails to be perfect while the defect can be controlled explicitly by using the defining equations of Z in X. Two applications of the main results are given: One is the infinitesimal Torelli problem for open complete intersections. Another is an explicit bound for Nori's connectivity in case of complete intersections. The results have been applied also to study of algebraic cycles in several other works.