Motivic decomposition of a compactification of a Merkurjev-Suslin variety

Motivic spheres and the image of the Suslin--Hurewicz map

We show that an old conjecture of A.A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree $n$ to Milnor K-theory in degree $n$ admits an interpretation in terms of unstable ${\mathbb A}^1$-homotopy sheaves of the general linear group. Using this identification, we establish Suslin's conjecture in degree $5$ for infinite fields having characteristic unequal to $2$ or $3$. We do this by linking the relevant unstable ${\mathbb A}^1$-homotopy sheaf of the general linear group to the stable ${\mathbb A}^1$-homotopy of motivic spheres.

On motivic decompositions arising from the method of Bialynicki-Birula

Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of Bialynicki-Birula, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko.

局部化与椭圆动机关系

Localization and elliptic motivic relations

该研究利用局部化等推导动机框架，将苏斯林互律推广到任意基上光滑概形，还细化了模单位杯积关系，推广到整系数、整基及非满水平结构等场景。

Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives

I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneou…

On the Albanese map for smooth quasi-projective varieties

Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its Albanese variety induces an isomorphism on torsion prime to the characteristic of k. In the present paper we prove a generalisation to quasi-projective varieties admitting a smooth compactification. As was first observed by Ramachandran, for such a generalisation one should replace the Chow group of zero cycles by Suslin's 0-th algebraic singular homology group …