A counterexample to the Hodge conjecture for Kähler varieties

H(X,C) = ⊕p+q=kH (X). A class α ∈ H(X,Q) is said to be a rational Hodge class if its image in H(X,C) belongs to H(X). As is well known, the classes which are Poincaré dual to irreducible algebraic subvarieties of codimension p of X are degree 2p Hodge classes. The Hodge conjecture asserts that any rational Hodge class is a combination with rational coefficients of such classes. In the case of a general compact Kähler variety X, the conjecture above, where the algebraic subvarieties are replaced with closed analytic subsets, is known to be false (cf [13]). The simplest example is provided by a complex torus endowed with a holomorphic line bundle of indefinite curvature. If the torus is chosen general enough, it will not contain any analytic hypersurface, while the first Chern class of the line bundle will provide a Hodge class of degree 2. In fact, another general method to construct Hodge classes is to consider Chern classes of holomorphic vector bundles. In the projective case, the set of classes generated this way is the same as the set generated by classes of subvarieties. To see this, one looks at a still more general set of classes, which is the set generated by the Chern classes of coherent sheaves on X. Since any coherent sheaf has a finite resolution by locally free sheaves, one does not get more classes than with locally free sheaves. On the other hand, this later set obviously contains the classes of subvarieties (one computes for this the top Chern class of IZ for Z irreducible of codimension p, and one shows that it is proportional to the class of Z). Finally, to see that the classes of coherent sheaves can be generated by classes of subvarieties, one puts a filtration on any coherent sheaf, whose associated graded consists of rank 1 sheaves supported on subvarieties, which makes the result easy. In the general Kähler case, none of these equalities holds. The only obvious result is that the space generated by the Chern classes of analytic coherent sheaves contains both the classes which are Poincaré dual to irreducible closed analytic subspaces and the Chern classes of holomorphic vector bundles (or locally free analytic coherent sheaves). The example above shows that a Hodge class of degree 2 may be the Chern class of a holomorphic line bundle, even if X does not contain any complex analytic subset. On the other hand, it may be the case that coherent sheaves do not admit a resolution by locally free sheaves, (although it is true in dimension 2 [9]), and that