# Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

1. Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in terms of mixed Hodge structures)? I only found a partial statement of this sort in section 7.1. of 'Mixed Hodge Theory' by Peters and Steenbrink, as well as is the book of Jannsen (cited there).

2. It seems that I can prove a statement of this sort using the ('usual') generalized Hodge cobnjecture. Could this be true (i.e. could it be possible to describe the coniveau filtration using Hodge structures at all)? What possible applications could be found for such a (conjectural) description?

3. An easy question: it seems easy to prove that the coniveau filtration does not depend on the choice of an algebraically closed base field (statements of this sort are usually called 'rigidity' ones). Could I say that this result is well-known?

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