Derived equivalent varieties with differing integral Mukai-Hodge structures?

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

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

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

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?

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

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

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

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

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

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