motives

Derived equivalent varieties with differing integral Mukai-Hodge structures?

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For a smooth projective complex variety X of dimension n, let Hi(X) denote its integral Hodge structure of weight i. Define ~H0(X)=⨁H2i(X)⊗Z(i) and ~H1(X)=⨁H2i+1⊗Z(i), respectively. It is known that any derived equivalence Φ:Db(X)→Db(Y) induces isomorphisms of rational Hodge structures ~H0Q(X)≅~H0Q(Y) and ~H1Q(X)≅~H1Q(Y), however these isomorphisms are defined by characteristic classes whose coefficients aren’t necessarily integral. … Read more

Passing motivic decompositions from rational to algebraic equivalence

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It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions). The category of motives Mk over a field k, known as Grothendieck-Chow-Motives is based on choosing rational equivalence for ∼. But one can also choose other equivalence relations for ∼, and thus get a … Read more

Quadrics contained in the (complex) Cayley plane

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In the paper Ilev, Manivel – The Chow ring of the Cayley plane we can learn, that CH8(X), with X:=E6/P1, denoting the Cayley plane, has three generators with one of them being the class of an 8-dimensional quadric [Q]. We consider CH(−) mod rational equivalence. Now I would like to know more about these quadrics. … Read more

Effectivity and Lower Shriek for Voevodsky Motives

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I am in a situation where I need a result of the following form. Suppose $X$ is a smooth $k$-variety, $U$ is a dense open subvariety with complement $Z$ a smooth divisor. Let $\pi^X:X\rightarrow\text{Spec }k$ and $\pi^U:U\rightarrow\text{Spec }k$ be the structure morphisms. Let us work with Voevodsky motives $\text{DM}(k)$. Is it true that if $\pi^X_!\mathbf{1}_X(n)$ … Read more

Why do motivic stacks make sense?

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In the paper “Motivic model categories and motivic derived algebraic geometry”, Yuki Kato, whose email-address I sadly couldn’t find out, describes a procedure to “motivy” the objects of any (∞,1)-category by doing the construction used to get motivic spaces over an arbitrary base. He then applies this construction to (∞,1)-Cat itself to get motivic categories … Read more

A1\mathbf{A}^1- contractibility

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Suppose U is an A1-contractible smooth scheme over a field k, that is, it is isomorphic to a point in the A1-homotopy category of smooth schemes over k. Does motivic cohomology of U satisfy some cohomological vanishing property? That is, HiM(U,Z(n))=0 for what (i,n)? Answer AttributionSource : Link , Question Author : Community , Answer … Read more

Locus of Hodge classes

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Let π:X→S be a proper smooth morphism of complex analytic spaces, with connected smooth X and S over C, projective fibers, and HpX/S:=Rpπ∗Ω∙X/S. Suppose π is algebraic and defined over Q. By Deligne-Cattani-Kaplan, the locus of Hodge classes Hdgp,pX/S in HpX/S is a countable union of closed algebraic subsets. Is it known that Hdgp,pX/S is … Read more

holomorphic continuation of motivic LL-functions

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The question is rather easy to formulate: when is the L-function of a pure motive over Q expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex plane? The Dedekind zeta function of a number field, which is a pure motive of weight 0, has a pole at s=1. Is this … Read more

Etale cohomology of projective spaces in the rigid analytic setting

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Take K a complete non-archimedean field (maybe algebraically closed, to simplify the question), and PdK the rigid projective space over K. Can we compute the étale cohomology with coefficients in Gm ? Can we also describe the cohomology of the complementary of an arrangement of tubular neighborhood of hyperplans ? I know that the Voevodsky … Read more

Explicit linear object underlying ll-adic cohomology for almost all ll

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If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with Z-coefficients. If you are studying the cohomology of varieties you often have to choose a random prime number. Is there a reasonably explicit linear object from which one can … Read more