[Paper Review] On a rationality question in the Grothendieck ring of varieties

Ene Esnault, Eckart Viehweg|ArXiv.org|Aug 16, 2009

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper investigates rationality questions in the Grothendieck ring of varieties, focusing on quotients of vector spaces by finite group actions. It establishes that for a finite abelian group $ G $ acting linearly on a $ K $-vector space $ V $, where $ K/k $ is a Galois extension with group $ \Gamma $, and $ k $ contains all $ |G| $-th roots of unity, the class $[V/G]$ equals $ \mathbb{L}^{\dim_K V} $ in $ K_0(\mathrm{Var}_k) $, generalizing Looijenga's result and highlighting the necessity of roots of unity for descent.

ABSTRACT

This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group. Part of it reproduces the content of a letter dated September 27, 2008 addressed to Johannes Nicaise

Motivation & Objective

• To understand when the class of a quotient variety $ V/G $ in the Grothendieck ring $ K_0(\mathrm{Var}_k) $ is rational, i.e., equal to a power of the Lefschetz class $ \mathbb{L} $.
• To generalize Looijenga's result on abelian group actions to non-trivial Galois extensions of the base field.
• To investigate the failure of descent in the Grothendieck ring when the base field lacks sufficient roots of unity.
• To clarify the role of group actions on fields and vector spaces in determining classes in $ K_0(\mathrm{Var}_k) $.

Proposed method

• Uses the Grothendieck ring $ K_0(\mathrm{Var}_k) $, defined via scissor relations and product structure, to study classes of quotient varieties.
• Applies the Weak Factorization Theorem and the isomorphism $ K_0(\mathrm{Var}_k)/\langle \mathbb{L} \rangle \cong \mathbb{Z}[SB] $ to analyze relations.
• Employs descent techniques and étale cohomology to study $ G $-torsors and Zariski locally trivial fibrations.
• Uses flat descent to show that certain fiber bundles are Zariski locally trivial, enabling class decomposition in $ K_0(\mathrm{Var}_k) $.
• Applies induction on the number of irreducible factors in the representation of $ V $ as a $ G $-module.
• Analyzes the action of $ G $ on $ \mathrm{Spec}\,K $ and proves triviality of the action when $ G $ acts via a cyclic matrix preserving $ T = Y/X $.

Experimental results

Research questions

• RQ1 Under what conditions does $[V/G] = \mathbb{L}^{\dim_K V}$ hold in $ K_0(\mathrm{Var}_k) $ for a finite abelian group $ G $ acting on a $ K $-vector space $ V $?
• RQ2 Is the rationality of $[V/G]$ compatible with base change to smaller fields, particularly when roots of unity are missing?
• RQ3 What is the role of the Galois action on the base field in determining the class of $ V/G $ in the Grothendieck ring?
• RQ4 How does the failure of descent manifest when $ k $ does not contain all $ |G| $-th roots of unity?
• RQ5 Can the class $[V/G]$ be expressed as a combination of simpler classes when the action on the base field is non-trivial?

Key findings

• For a finite abelian group $ G $ with quotient $ G \to \Gamma $, and a $ G $-action on a $ K $-vector space $ V $, where $ K/k $ is a Galois extension with group $ \Gamma $, and $ k $ contains all $ |G| $-th roots of unity, one has $[V/G] = \mathbb{L}^{\dim_K V} $ in $ K_0(\mathrm{Var}_k) $.
• The condition that $ k $ contains all $ |G| $-th roots of unity is necessary; without it, the formula fails, as shown in Example 1.2 where $ \mathbb{L}^2 \neq [V/G] $ in $ K_0(\mathrm{Var}_{\mathbb{Q}}) $.
• The action of $ G $ on $ \mathrm{Spec}\,K $ is trivial when $ G $ acts via a matrix preserving $ T = Y/X $, which is crucial for the decomposition of classes.
• The difference $[V] - [V/G]$ decomposes as $ (1 + [\mathbb{G}_m] \cdot [\mathrm{Spec}\,K]) \cdot ([W] - [W/G]) $, enabling induction on the number of irreducible factors of $ V $.
• The quotient $ (D^\times \times_k W)/G \to (\mathrm{Spec}\,K \times_k W)/G $ is a Zariski locally trivial $ \mathbb{G}_m $-fibration, so $ [(D^\times \times_k W)/G] = [\mathbb{G}_m] \cdot [\mathrm{Spec}\,K] \cdot [W/G] $.
• The class $[U^{(2)}/G] $ equals $[U^{(2)}] $ in $ K_0(\mathrm{Var}_k) $, since $ \mathbb{P}(U)/G \cong \mathbb{P}^1_k $ and $ G $ acts trivially on $ \mathrm{Spec}\,K $.