[Paper Review] Equivariant birational types and Burnside volume

Andrew Kresch, Yuri Tschinkel|arXiv (Cornell University)|Jul 24, 2020

Algebraic Geometry and Number Theory27 references11 citations

TL;DR

This paper introduces equivariant birational invariants using a new construction called the equivariant Burnside group, generalizing Kontsevich's birational symbols to actions of finite groups. It establishes a specialization map for equivariant birational types, proving that if two smooth projective $G$-varieties are $G$-equivariantly birational over a discrete valuation ring, then their special fibers are also $G$-equivariantly birational—extending classical specialization techniques to the equivariant setting.

ABSTRACT

We introduce equivariant Burnside groups, new invariants in equivariant birational geometry, generalizing birational symbols groups for actions of finite abelian groups, due to Kontsevich, Pestun, and the second author, and study their properties. We establish a specialization map for the equivariant birational type of a smooth algebraic variety with an action of a finite group.

Motivation & Objective

• To develop new invariants in $G$-equivariant birational geometry for finite group actions on algebraic varieties.
• To generalize Kontsevich's birational symbols to non-trivial group actions via equivariant Burnside groups.
• To establish a specialization map for equivariant birational types, extending classical specialization techniques to the equivariant setting.
• To compare the new invariants with existing invariants such as ${\mathcal{B}}_n(G,k)$ for abelian groups.
• To prove that $G$-equivariant birationality of generic fibers over a DVR implies $G$-equivariant birationality of special fibers.

Proposed method

• Define the equivariant Burnside group $\operatorname{Burn}_n(G)$ as a quotient of $\mathbb{Z}$-span of symbols $ (H, N_G(H)/H \curvearrowright K, \beta) $, where $H$ is an abelian stabilizer subgroup.
• Use the divisorialification algorithm of Bergh and Bergh-Rydh to resolve $[X/G]$ into a stack with abelian stabilizers.
• Construct the equivariant Burnside volume map $ \rho^G_\pi: \operatorname{Burn}_{n,K}(G) \to \operatorname{Burn}_{n,k}(G) $ for a complete DVR $\mathfrak{o}$ with residue field $k$ and uniformizer $\pi$.
• Apply weak factorization in the $G$-equivariant setting to relate birational types across different strata of the action.
• Establish a homomorphism from $\operatorname{Burn}_n(G)$ to the orbifold Burnside group $\overline{\operatorname{Burn}}_n$, sending $[X \curvearrowleft G]$ to the class of $[X/G]$.
• For abelian $G$, show that $\operatorname{Burn}_n^G(G) \to \mathcal{B}_n(G,k)$ is surjective, and that $[X \curvearrowleft G]$ maps to $\beta_k(X)$.

Experimental results

Research questions

• RQ1 Can a new invariant be constructed in $G$-equivariant birational geometry that captures the birational type of a smooth projective $G$-variety?
• RQ2 Does the specialization of generic fibers over a DVR preserve $G$-equivariant birationality?
• RQ3 How do the new equivariant Burnside groups relate to existing invariants like $\mathcal{B}_n(G,k)$ for abelian $G$?
• RQ4 Can the equivariant Burnside volume map be used to relate birational types over function fields and their special fibers?
• RQ5 Is there a canonical way to associate a symbol in $\operatorname{Burn}_n(G)$ to each orbit type of a $G$-action with abelian stabilizers?

Key findings

• The equivariant Burnside group $\operatorname{Burn}_n(G)$ is defined as a quotient of symbols encoding stabilizer type, normalizer action, and normal bundle representation, with relations ensuring invariance under $G$-equivariant birational maps.
• The specialization map $\rho^G_\pi$ is constructed, and it induces a homomorphism that respects the $G$-equivariant birational type across fibers of a family over a DVR.
• Corollary 6.8 establishes that if two smooth projective $G$-varieties are $G$-equivariantly birational over the generic fiber of a DVR, then their special fibers are also $G$-equivariantly birational.
• For abelian $G$, the map $\operatorname{Burn}_n^G(G) \to \mathcal{B}_n(G,k)$ is surjective, and the class $[X \curvearrowleft G]$ maps to $\beta_k(X)$, the invariant defined in [16].
• The construction of the equivariant Burnside volume provides an equivariant analogue of the specialization map in [17], extending it to the non-trivial group action setting.
• The group $\operatorname{Burn}_n(G)$ maps surjectively to the orbifold Burnside group $\overline{\operatorname{Burn}}_n$, sending $[X \curvearrowleft G]$ to the class of the quotient stack $[X/G]$.