On integral Hodge classes on uniruled or Calabi-Yau threefolds

By Claire Voisin. 2004

76 citations recorded by the source corpus.

Let X be a smooth complex projective variety of dimension n. The Hodge conjecture is then true for rational Hodge classes of degree 2n−2, that is, degree 2n−2 rational cohomology classes of Hodge type (n − 1, n − 1) are algebraic, which means that they are the cohomology classes of algebraic cycles with Q-coefficients. Indeed, this follows from the hard Lefschetz theorem, which provides an isomorphism:

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c14e1140d6024f49f0b40c029f6c19f6cdb650c8

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, unknown): “We even have the following result [103].”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “(a) Our proof used the Hodge Conjecture over Z for Calabi– Yau 3-folds, proved by Voisin [103].”
Some aspects of the Hodge conjecture (2007, methodology): “This is essentially taken from [41], but the discussion concerning the 4-dimensional case here is new.”
Evidence for a sublattice weak gravity conjecture (2016, background): “This latter statement is known to be true for M-theory compactifications on smooth Calabi-Yau three-folds: the integral Hodge conjecture, proven by Voisin for Calabi-Yau three-folds [60], shows that holomorphic curve classes generate the full homology lattice H2.”
On the universal $CH_0$ group of cubic hypersurfaces (2014, background): “4, it is satisfied by uniruled threefolds by [31], but it is not clear that it is satisfied by Fano complete intersections in any dimension.”
Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal (2010, methodology): “Concerning the vanishing of the group Z(4)(X), the following is proved in [28].”