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On the Problem of Mixed-Tateness of the Motives of G-Varieties

Somayeh Habibi

TL;DR

This work investigates when motives of G-varieties are mixed Tate, motivated by connections to arithmetic, geometry, and representation theory. It introduces rationally special pairs (G,H) and proves that, under stabilizer-based conditions, the motives of the corresponding G-varieties are mixed Tate, with the homogeneous spaces G/H themselves enjoying mixed Tate motives. The paper applies the main result to a range of classical constructions: irreducible SL_n representations, symmetric spaces of Sp(2n), moduli of Lagrangian splittings, and the space of nondegenerate alternating forms, using semi-small maps and compactifications to establish explicit motivic decompositions. Collectively, the results provide a broad toolkit for identifying mixed-Tate motives in G-equivariant settings and connect these motives to geometric structures like Pfaffian hypersurfaces and Higgs branches.

Abstract

Building on earlier work concerning the motives of $G$-bundles, we study the structure of motives associated with certain classes of $G$-varieties. In particular, we show that the corresponding motives lie within the category of mixed-Tate motives, under certain condition on the stabilizers. We further discuss some applications and provide some examples to illustrate the limitations.

On the Problem of Mixed-Tateness of the Motives of G-Varieties

TL;DR

This work investigates when motives of G-varieties are mixed Tate, motivated by connections to arithmetic, geometry, and representation theory. It introduces rationally special pairs (G,H) and proves that, under stabilizer-based conditions, the motives of the corresponding G-varieties are mixed Tate, with the homogeneous spaces G/H themselves enjoying mixed Tate motives. The paper applies the main result to a range of classical constructions: irreducible SL_n representations, symmetric spaces of Sp(2n), moduli of Lagrangian splittings, and the space of nondegenerate alternating forms, using semi-small maps and compactifications to establish explicit motivic decompositions. Collectively, the results provide a broad toolkit for identifying mixed-Tate motives in G-equivariant settings and connect these motives to geometric structures like Pfaffian hypersurfaces and Higgs branches.

Abstract

Building on earlier work concerning the motives of -bundles, we study the structure of motives associated with certain classes of -varieties. In particular, we show that the corresponding motives lie within the category of mixed-Tate motives, under certain condition on the stabilizers. We further discuss some applications and provide some examples to illustrate the limitations.

Paper Structure

This paper contains 10 sections, 12 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Motive of Rationally Special $G$-varieties
  4. Consequences
  5. Homogeneous space of an irreducible representation of $\mathrm{SL}_n$
  6. The symmetric homogeneous space $X_{Sp}(p,n)$
  7. Moduli of Lagrangian Splittings
  8. The space of nondegenerate alternating forms and Pfaffian Hypersurface
  9. Projective compactification and the Pfaffian hypersurface
  10. Orbit stratification of the Pfaffian

Key Result

Theorem 3.2

Let $X$ be a principal $G$-bundle over $Y$. Furthermore, assume $X$ is locally trivial for the Zariski topology on $Y$. If $Y$ is smooth and $M(X)$ is mixed Tate, then $M(Y)$ is mixed Tate. Moreover if $M(Y)$ is mixed Tate, then $M(X)$ is mixed Tate.

Theorems & Definitions (36)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 26 more