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.
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.
DI GENNARO, V., Franco, D., Marini, G. (2016). Algebraic versus homological equivalence for singular varieties. COMMUNICATIONS IN ALGEBRA, 44(6), 2547-2560 [10.1080/00927872.2015.1053904].
Algebraic versus homological equivalence for singular varieties
Di Gennaro Vincenzo;Marini Giambattista
2016-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
