A remark on the Generalized Hodge Conjecture

Abstract

Let X be a smooth, projective variety over the field of complex numbers. On the space H of its rational cohomology of degree i we have the arithmetic filtration F^p. On the other hand, on the space of cohomology of degree i of X with complex coefficients we have the Hodge filtration G^p. It is well known that F^p is contained in the intersection of G^p with H, and that, in general, this inclusion is strict. In this paper we propose a natural substitute S^p for the Hodge filtration space G^p such that the intersection of S^p with H is the space F^p of the arithmetic filtration. In particular, S^p is a complex subspace of G^p. This result leaves untouched Grothendieck's Generalized Hodge Conjecture. But the method used here to construct algebraic supports for suitable cohomology classes seems to me of some interest. The main technical tool is the use of semi-algebraic sets, which are available by the triangulation of complex projective algebraic varieties.