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.