A remark on the Generalized Hodge Conjecture

TL;DR

The paper introduces a new complex subspace S^p within the Hodge filtration G^p that aligns with the arithmetic filtration F^p on rational cohomology, offering a novel approach using semi-algebraic sets without resolving the Generalized Hodge Conjecture.

Contribution

It proposes a natural substitute S^p for the Hodge filtration G^p, aligning it with the arithmetic filtration F^p through semi-algebraic set techniques, without resolving the conjecture.

Findings

S^p intersects with H to give F^p

S^p is a complex subspace of G^p

Method uses semi-algebraic sets and triangulation

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.