The main question can be summarized in the following form:
For a smooth projective complex variety $X$, is the cohomology $H^{2p}(X, \tau^{\leq p}\Omega_{alg}^{\bullet})$ supposed to surject onto $(H^{p,p}(X)\cap H^{2p}_{sing}(X, \mathbb{Q}))\otimes \mathbb{C}$?
The longer version:
A conjecture of Suslin also known as integral Bloch-Kato expects that there to be a quasi-isomorphism of complexes of Zariski sheaves on the category of smooth quasi-projective complex varieties, which is in the following form:
$$\mathbb{Z}^{sst}(t)\rightarrow tr^{\leq t}\mathbb{R}\epsilon_*\mathbb{Z}$$
Here the cohomology groups of the complex $\mathbb{Z}^{sst}(t)$ gives us the morphic cohomology groups i.e. $L^tH^n(X, \mathbb{Z})=\mathbb{H}^n_{Zar}(X, \mathbb{Z}^{sst}(t))$. The functor $\epsilon$ is the forgetful functor from the category of topological spaces homeomorphic to a finite CW complex (as every complex quasi-projective variety is with the analytic topology), to the schemes over the complex number with the Zariski site.
It is known that this conjecture implies the standard conjectures. I have a question with the assumption that this conjecture is true.
- I was wondering about the relation of this conjecture to the Hodge conjecture. If the conjecture above is true then it follows that codimension $p$ Chow groups mod the algebraic relations is given by $L^pH^{2p}(X, \mathbb{Q})\cong \mathbb{H}^{2p}_{Zar}(X, tr^{\leq p}\mathbb{R}\epsilon_*\mathbb{Q})$. There is a natural morphism from this to the singular cohomology which should land in the $(p,p)$ part of the cohomology and since standard conjectures holds then algebraic and numerical adequate equivalence relations are the same. This implies that the morphism from $\mathbb{H}^{2p}_{Zar}(X, tr^{\leq p}\mathbb{R}\epsilon_*\mathbb{Q})$ to $H^{p,p}(X, \mathbb{Q})$ should be surjective (Hodge conjecture). Is there a way to verify this?
