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The first one is based on the conjecture that Weil cohomology theories should yield exact and conservative functors on the category of mixed motives. In the paper Hanamura M. … The second approach was proposed by Voevodsky himeslf (in his well-known letter to Beilinson); it is based on the idea that the 'mixed motivic cohomology' of a (smooth) affine variety should satisfy the …

Is it true that all pure Hodge structures 'that come from geometry' (for example, the graded pieces of the weight filtration of the singular cohomology of varieties and motives) are polarized? …

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with eff …

There are plenty of examples of varieties whose singular cohomology with rational coefficients considered as a mixed Hodge structure does not decompose as the direct sum of its pure (weight) factors. … Yet if we consider the cohomology just as filtered vector spaces over rationals, such a decomposition certainly exist (for any variety). …

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 …

For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_-$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$-th singular cohomology of $X$ …

Certain examples and http://ncatlab.org/nlab/show/fiber+sequence#LongSequCoh seem to suggest that the cohomology of $D$ should be something like $H^*(B)\otimes_{H^*(A)}H^\ast(C)$ (in lower cohomological …

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. … Is it true that one doesn't have any similar results for etale cohomology since it is not known whether the corresponding bilinear forms are positive definite? …

Note that one can compute the cohomology of $U$ as the hypercohomology $H^*(Z,Rpr'_\ast\mathbb{Z}/l^n\mathbb{Z}_U)$, where $pr': U\to Z$ is the corresponding prinicple $G_m$-bundle. … Cf. https://mathoverflow.net/questions/89171/on-the-cohomology-of-g-m-bundles-and-purity-for-singular-varieties …

Also, the proof of the latter statements uses algebraic cobordism and motivic cohomology operations, which do not work integrally. …

algebraic varieties over a subfield $k$ of the field of complex numbers, can one define certain mixed Hodge modules with some $k$-structure that would be related with the $k$-structure on the De Rham cohomology …

As far as I understood, there are two way to assign weights to a mixed Hodge complex (a way that does not depend on shifts, and another one that does). There is a clever way to relate these to ways us …

Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of http://www.math.uiuc.edu … It seems sufficient to prove the latter for $A^\ast$ being the algebraic cobordism (as defined by Levine and Morel), since this is the universal algebraic oriented cohomology theory. …

Indeed, the cohomology groups in question would be exactly the weight zero part of the singular cohomology of $H$ (considered as a sequence of mixed Hodge structures), whereas for $i<\operatorname{dim} … Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties? …

Can one formulate those version of Weak Lefschetz that uses tubular neighbourhoods purely in terms of cohomology of (some) algebraic varieties? … I would be completely satisfied with cohomology with $Z/l^n Z$-coefficients i.e. etale cohomology. …