Algebraic Versus Homological Equivalence for Singular Varieties. Issue 6 (2nd June 2016) - Di Gennaro, Vincenzo; Franco, Davide; Marini, Giambattista
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.