  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.
