Algebraic Versus Homological Equivalence for Singular Varieties

作者：Di Gennaro Vincenzo^*; Franco Davide; Marini Giambattista

来源：Communications in Algebra, 2016, 44(6): 2547-2560.

DOI：10.1080/00927872.2015.1053904

• Algebraic cycle
• Algebraic equivalence
• Chow variety
• Connectivity Theorem
• Hilbert scheme
• Homological equivalence
• Projective variety
• Singularity

Let Y subset of 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 dimY=n+h and dimY(sing)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; ) 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.

• 出版日期2016