arXiv subjects

Nikita Karpenko

Publications and source records attributed to Nikita Karpenko.

4 recordsLinked to original sources

arXiv2025

Pullback and Weil transfer on Chow groups

In the paper ``Weil transfer of algebraic cycles'', published by the second author in Indagationes Mathematicae about 25 years ago, a Weil transfer map for Chow groups of smooth algebraic varieties has been constructed and its basic properties have been established. The proof of commutativity with the pullback homomorphisms given there used a variant of Moving Lemma suffering a lack of reference. Here we are providing an alternative proof based on a more contemporary construction of the pullback via a deformation to the normal cone.

math.AG↗

arXiv2025

A-upper motives of reductive groups

Given a prime number $p$, we perform the study of Chow motives and motivic decompositions, with coefficients in $\mathbb{Z}/p\mathbb{Z}$, of projective homogeneous varieties for $p'$-inner $p$-consistent reductive algebraic groups. Assorted with the known case of $p$-inner reductive groups, our results cover all absolutely simple groups of type not $^3\!D_4$ or $^6\!D_4$, among other examples. First, we define the A-upper motives of such a reductive group $G$; they are indecomposable motives, naturally related to Artin motives built out of spectra of subextensions of a minimal extension over which $G$ become of inner type. With this in hand, we carry on the qualitative study of motivic decompositions for projective $G$-homogeneous varieties. Providing geometric isomorphism criteria for A-upper motives, we obtain a classification of motives of projective $G$-homogeneous varieties, by means of their higher Artin-Tate traces. We also show that the higher Tits $p$-indexes of the group $G$ determine its motivic equivalence class.

math.AG↗

arXiv2024

Artin shapes

We introduce and study on examples a notion of the Artin shape for a motive related to a projective homogenous variety. We apply it to the problem of finding the complete motivic decomposition of the variety. Our examples cover unitary involution varieties as well as some varieties given by a quadratic Weil transfer. Some of the decompositions obtained dispel prior expectations on how motivic decompositions of projective homogeneous varieties can look like.

math.AG↗

arXiv2012

Isotropy of unitary involutions

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.

math.AG↗