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Generalized Jacobian rings for open complete intersections
Author(s) -
Asakura Masanori,
Saito Shuji
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310343
Subject(s) - mathematics , hypersurface , complete intersection , jacobian matrix and determinant , pure mathematics , cohomology , intersection theory , duality (order theory) , mathematical analysis , differential algebraic equation , ordinary differential equation , differential equation
In this paper, we develop the theory of Jacobian rings of open complete intersections, which are pairs ( X, Z ) where X is a smooth complete intersection in the projective space 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 Macaulay's duality theorem and 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. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)