Algebraic Versus Homological Equivalence for Singular Varieties. Issue 6 (2nd June 2016)

• Record Type: Journal Article
• Authors: Di Gennaro, Vincenzo; Franco, Davide; Marini, Giambattista
• Is Part Of: Communications in algebra. Volume 44:Issue 6(2016)
• Page Start: 2547
• Page End: 2560
• Publication Date: 2016-06-02
• Subjects: Algebraic cycle -- Algebraic equivalence -- Chow variety -- Connectivity Theorem -- Hilbert scheme -- Homological equivalence -- Projective variety -- Singularity; 14C05 -- 14C15 -- 14C25 -- 14F17 -- 14F43 -- 14F45 -- 14J17 -- 14M10; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2015.1053904
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract : Let Y ⊆ ℙ N be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dim Y = n + h and dim Y sing ≤ min { d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H 2( d + h ) ( Y ; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.