On The Motivic Leray-Hirsch Theorem For Pure Tate Fibre Bundles

TL;DR

The paper proves a motivic Leray–Hirsch theorem for smooth, proper fibrations with pure Tate fibres, establishing a canonical decomposition of the motive $M_{CH}(X)$ as $M_{CH}(Y)\otimes M_{CH}(F)$ when the fibre is pure Tate and Poincaré duality holds. It provides explicit correspondences yielding isomorphisms of Chow groups and higher Chow groups, and extends to a broad class of fibrations, including Grassmann and projective-bundle cases. The results give concrete motivic decompositions, enabling computations of $CH^p(X,q)$ and enabling applications such as Chow–Künneth decompositions and connections to the Motivic Decomposition Theorem for semi-small resolutions. Concrete families, like projective homogeneous bundles and $GL_n$-Grassmann bundles, are worked out to illustrate the decompositions in terms of Schubert cell data. Overall, the work generalizes classical topological Leray–Hirsch ideas to the motivic setting, yielding practical tools for motivic and higher Chow computations in fibre bundle constructions.

Abstract

In this note we prove a motivic version of Leray-Hirsch theorem for pure Tate fibre bundles in the Grothendieck category of Chow motives. We then discuss some of its applications.

Theorems & Definitions (14)

• Remark 1
• Proposition 2
• proof
• Theorem 2.1
• proof
• Remark 2.2
• Remark 2.3
• Theorem 2.4: Motivic Leray--Hirsch for pure Tate fibre bundles
• proof
• Corollary 3.1