Torsion cohomology classes and algebraic cycles on complex projective manifolds

By Christophe Soulé, Claire Voisin. 2004

62 citations recorded by the source corpus.

Abstract Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincare dual to an algebraic cycle. We notice that the order of these classes must be small compared to the dimension of the manifold. However, building upon a construction of Kollar, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths group, and lie in a arbitrary high level of the H. Saito filtration on Chow groups.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8cd0c1559868f3036d9ff69639080e738cd65ead

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In Geometry and Topology

Citations in the corpus

Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal (2010, background): “The group Z 2 i ( X ) is a birational invariant of X for i = 2 , d − 1, where d = dim X (cf. [26]).”
The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, background): “The the Voisin group V3(X) is 2-torsion.”
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, background): “Theorem (Beauville and Voisin 2004; see Theorem 5.3).”
Some aspects of the Hodge conjecture (2007, background): “In [35], [41], we observed that if X is a smooth projective variety which is birational to Pn, then the Hodge conjecture holds for integral Hodge classes on X of degree 2n− 2 and 4.”