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I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, …

triangulated subcategory of Artin's motives (generated by motives of varieties of dimension $0$). … Hanamura considers his own category of motives; yet I proved that it is (anti)-equivalent to Voevodsky's motives. 2. There is also the category of Nori's motives. …

Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with integral coefficients. Furthermore, motives with transfers(! … Now let's consider motives with integral coefficients. …

If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram commutative). … For this purpose one may recall that Chow motives embed into Voevodsky one and note that both the cycle classes and the Hodge classes are determined by Chow motives of the corresponding varieties. …

In particular, could there exist some 'Chow motives for compact Kähler manifolds'? …

I know of two conjectural descriptions of the motivic $t$-structure for Voevodsky's motives. … Mixed motives and algebraic cycles, III// Math. Res. …

Here for motivic cohomology I would like to take $Hom_{DM}(M(X),\mathbb{Z}(p)[q]$; $DM$ is the category of Voevodsky's motives, and $M(X)$ is the motif of $X$ (I don't want to take the motif with compact …

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 effective motives. …

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one. For the algebraic cobordism theory …

The basic property of the Lefschetz motif (in this aspect) is the Cancellation Theorem: $Hom(X,Y)\cong Hom(X\otimes L,Y\otimes L)$ for any effective motives X and Y. … If it does, then its image in the ('usual') Chow motives is invertible (with respect to the tensor product). …

In the paper Smirnov, Oleg N., Graded associative algebras and Grothendieck standard conjectures// Invent. Math. 128 (1997), no. 1, 201–206 it is proved that Standard Conjecture D (numerical equivalen …

Any characteristic zero field is an inductive limit of fields that can be embedded into complex numbers (i.e., those of characteristic 0 and of cardinality at most continuum). Hence the assumption t …

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively? This stateme …

The idea is to rewrite the $L$-function as the sum of $\sharp [Sym^nM](F_p)t^n$; here "the number of $F_p$-points" of a motif is a natural homomorphism from the Grothendieck group of motives to abelian …

As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. …