Algebraic Versus Homological Equivalence for Singular Varieties

Abstract

Let Y ⊆ ℙN 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 dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(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.

Locations

• arXiv (Cornell University) - View - PDF
• Communications in Algebra - View