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A remark on a result of Huber and Kahn

Somayeh Habibi, Farhad Rahmati

TL;DR

The paper addresses the inverse implication of Huber–Kahn's result for principal bundles: whether the mixed Tate property of the base forces the total space to be mixed Tate for a principal $G$-bundle with split reductive $G$. It employs the slice filtration framework and motivic fundamental invariants $c_n(M)$ to characterize mixed Tate motives and uses a torus-bundle filtration as a stepping stone to handle $G$-bundles, including a reduction to a sequence of $\mathbb{G}_m$-bundles and projective bundle arguments. The main contributions are (i) establishing that if the base $Y$ is (stratified) mixed Tate then the total space $X$ is mixed Tate, and (ii) the converse in the smooth base case: if $M(X)$ is mixed Tate then $M(Y)$ is mixed Tate, under either Zariski-local triviality of $X$ or $k=\mathbb{C}$; with corollaries on the finiteness of Chow groups over finite fields. These results facilitate computing motives of quotient varieties under reductive group actions and deepen understanding of how mixed Tate properties propagate through principal bundles.

Abstract

A. Huber and B. Kahn construct a relative slice filtration on the motive M(X) associated to a principal T-bundle X over a smooth scheme Y. As a consequence of their result, one can observe that the mixed Tateness of the motive M(Y) implies that the motive M(X) is mixed Tate. In this note we prove the inverse implication for a principal G-bundle, for a split reductive group G.

A remark on a result of Huber and Kahn

TL;DR

The paper addresses the inverse implication of Huber–Kahn's result for principal bundles: whether the mixed Tate property of the base forces the total space to be mixed Tate for a principal -bundle with split reductive . It employs the slice filtration framework and motivic fundamental invariants to characterize mixed Tate motives and uses a torus-bundle filtration as a stepping stone to handle -bundles, including a reduction to a sequence of -bundles and projective bundle arguments. The main contributions are (i) establishing that if the base is (stratified) mixed Tate then the total space is mixed Tate, and (ii) the converse in the smooth base case: if is mixed Tate then is mixed Tate, under either Zariski-local triviality of or ; with corollaries on the finiteness of Chow groups over finite fields. These results facilitate computing motives of quotient varieties under reductive group actions and deepen understanding of how mixed Tate properties propagate through principal bundles.

Abstract

A. Huber and B. Kahn construct a relative slice filtration on the motive M(X) associated to a principal T-bundle X over a smooth scheme Y. As a consequence of their result, one can observe that the mixed Tateness of the motive M(Y) implies that the motive M(X) is mixed Tate. In this note we prove the inverse implication for a principal G-bundle, for a split reductive group G.
Paper Structure (3 sections, 7 theorems, 19 equations)

This paper contains 3 sections, 7 theorems, 19 equations.

Table of Contents

  1. Notation and Conventions
  2. Slice filtration and motive of $G$-bundles
  3. Acknowledgment:

Key Result

Theorem 1

Let $G$ be a split reductive group over a perfect field $k$. Let $X$ be a principal $G$-bundle over $Y$. Furthermore, assume either $X$ is locally trivial for the Zariski topology on $Y$ or $k={\mathbb{C}}$. Then we have the following statements In particular, when $k$ is a finite field and $M(Y)$ is stratified mixed Tate (resp. $M(X)$ is mixed Tate and $Y$ is smooth), the ${\mathbb{Q}}$-vector s

Theorems & Definitions (19)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • ...and 9 more