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Intermediate Jacobians and Burnside invariants

Andrew Kresch, Sho Tanimoto, Yuri Tschinkel

TL;DR

The work tackles equivariant birational geometry for finite group actions on smooth projective rational threefolds by constructing curve-localized Burnside invariants that couple stabilizer data with the action on intermediate Jacobians. It introduces $Burn_3^C(G)$ and the class $[X\to G]^C$, develops blow-up invariance and a KT-struct filtration framework, and specializes to abelian $G$ to obtain an integer invariant $I=-I_1-2I_2+I_3$ that obstructs linearizability when nonzero. The authors provide concrete applications to involutions, conic and quadric surface bundles, and nodal cubic threefolds, recovering and strengthening existing obstructions (CKK, CTZ) and illustrating the practical impact of combining Burnside theory with equivariant Jacobians in birational classification.

Abstract

We propose new invariants in equivariant birational geometry, combining equivariant intermediate Jacobians and the Burnside formalism, for smooth rationally connected threefolds with actions of finite groups.

Intermediate Jacobians and Burnside invariants

TL;DR

The work tackles equivariant birational geometry for finite group actions on smooth projective rational threefolds by constructing curve-localized Burnside invariants that couple stabilizer data with the action on intermediate Jacobians. It introduces and the class , develops blow-up invariance and a KT-struct filtration framework, and specializes to abelian to obtain an integer invariant that obstructs linearizability when nonzero. The authors provide concrete applications to involutions, conic and quadric surface bundles, and nodal cubic threefolds, recovering and strengthening existing obstructions (CKK, CTZ) and illustrating the practical impact of combining Burnside theory with equivariant Jacobians in birational classification.

Abstract

We propose new invariants in equivariant birational geometry, combining equivariant intermediate Jacobians and the Burnside formalism, for smooth rationally connected threefolds with actions of finite groups.

Paper Structure

This paper contains 6 sections, 3 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Curve-localized Burnside groups
  4. Blow-up relations
  5. Structure
  6. Applications

Key Result

Theorem 4.1

Let $X$ and $X'$ be smooth projective rationally connected threefolds with a regular action of a finite group $G$, in divisorial form. Let $C$ be a curve of genus $\ge 1$. Then

Theorems & Definitions (13)

  • Example 3.1
  • Theorem 4.1
  • proof
  • Example 4.2
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Example 6.1
  • Example 6.2
  • ...and 3 more