The Hodge Conjecture

We recall that a pseudo complex structure on a C∞-manifold X of dimension 2N is a C-module structure on the tangent bundle TX . Such a module structure induces an action of the group C∗ on TX , with λ ∈ C∗ acting by multiplication by λ. By transport of structures, the group C∗ acts also on each exterior power ∧TX , as well as on the complexified dual Ω := Hom(∧TX ,C). For p+ q = n, a (p, q)-form is a section of Ω on which λ ∈ C∗ acts by multiplication by λ−pλ̄−q. From now on, we assumeX complex analytic. A (p, q)-form is then a form which, in local holomorphic coordinates, can be written as ∑ ai1,...,ip,j1...jqdzi1∧ · · · ∧dzip∧dz̄j1∧ · · · ∧dz̄jq , and the decomposition Ω = ⊕Ω induces a decomposition d = d′ + d′′ of the exterior differential, with d′ (resp. d′′) of degree (1, 0) (resp. (0, 1)). If X is compact and admits a Kähler metric, for instance if X is a projective non-singular algebraic variety, this action of C∗ on forms induces an action on cohomology. More precisely, H(X,C) is the space of closed n-forms modulo exact forms, and if we define H to be the space of closed (p, q)-forms modulo the d′d′′ of (p− 1, q − 1)-forms, the natural map