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# The Hodge conjecture is one of the seven “Millenium problems” for which the Clay Institute offers a prize of one million dollars It was formulated by Hodge in H1 much before H2 often quoted as the original source as part of a more general problem now called the general Hodge conjecture We will give the precise formulation below roughly Hodge proves that the fact that a topological cycle of dimension p on a projective manifold is contained in an algebraic subvariety of complex dimension r p forces the vanishing of the integrals along of certain differential forms he then asks whether the converse is true

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Niveau: Supérieur, Doctorat, Bac+8
The Hodge conjecture Arnaud BEAUVILLE Introduction The Hodge conjecture is one of the seven “Millenium problems” for which the Clay Institute offers a prize of one million dollars. It was formulated by Hodge in [H1] (much before [H2], often quoted as the original source) as part of a more general problem, now called the general Hodge conjecture. We will give the precise formulation below; roughly, Hodge proves that the fact that a topological cycle ? of dimension p on a projective manifold is contained in an algebraic subvariety of (complex) dimension r < p forces the vanishing of the integrals along ? of certain differential forms; he then asks whether the converse is true: “The question whether [these necessary conditions] are sufficient is of great importance in the application of the theory of harmonic integrals to algebraic geometry, but as yet it can only be answered in special cases.” The particular case where p = 2r , that is, the characterization of those topo- logical cycles which are algebraic (in a sense that we will explain below) has become known as the Hodge conjecture – though Hodge did not seem to consider it as par- ticularly important, and never formulated it as a conjecture. It is probably time to say that Hodge's formulation was actually incorrect – in a minor way for the usual Hodge conjecture, more seriously for the general one (see the discussion in 4 and 7 below).

• hodge

• though hodge

• tangent space

• space has

• pi-periodic function over

• canon- ical isomorphism

• manifold

• invariant called

• forms

• torsion subgroups

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