Some aspects of the Hodge conjecture

By Claire Voisin. 2007

98 citations recorded by the source corpus.

Abstract.I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general Kähler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e8f66c4d9959743eb7b3e2d91f6d34a4fa38577

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Cubic Fourfolds, K3 Surfaces, and Rationality Questions (2016, background): “ated by the classes of smooth rational surfaces? Voisin has shown that the cohomology can be generated by smooth surfaces (see the proof of [Voi14, Th. 5.6]) or by possibly singular rational surfaces [Voi07]. Low discriminant examples suggest we might be able to achieve both. CUBIC FOURFOLDS, K3 SUR”
Unirational threefolds with no universal codimension $$2$$2 cycle (2013, background): “Indeed, by [30], they satisfy the integral Hodge conjecture in degree 4 and the other assumptions are easy to check.”
Hodge theory and derived categories of cubic fourfolds (2012, unknown): “…) ⊗ C = p + q =2 (cid:101) H p,q ( A X ), where Since X satisfies the Hodge conjecture [45] and K top ( X ) is torsion-free, we can identify the numerical K-theory In Proposition 2.4 we will see that this inclusion is an equality, essentially because X satisfies the integral Hodge conjecture [44].”
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, background): “The groups Z(X) defined as Z(X) := Hdg(X,Z)/H B (X,Z)alg themselves are birationally invariant for i = 2, i = n−1, n = dimX, as remarked in [93], [105].”
Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal (2010, background): “11 thus provides Z(4)(X) = 0 for cubic fourfolds, a result first proved in [29].”
On the universal $CH_0$ group of cubic hypersurfaces (2014, background): “All classes in Hdg4(X,Z) are algebraic by the Hodge conjecture for integral Hodge classes on cubic fourfolds proved in [32], and all classes in H ∗6=4(X,Z) are algebraic.”