Intermediate Jacobians and linearizability

TL;DR

This paper develops an equivariant version of the intermediate Jacobian torsor (IJT) obstruction theory and applies it to conic bundles over $\\mathbb{P}^2$, quadric surface bundles over $\\mathbb{P}^1$, and Fano threefolds. It constructs $G$-equivariant intermediate Jacobians $\\mathrm{IJ}(X)$ and Prym-type torsors, establishing obstructions to $G$-equivariant projective linearizability and deriving criteria for when such actions are linearizable. It shows that the $G$-equivariant Abel-Jacobi map $\\mathrm{AJ}: \\mathrm{CH}^2(X)_{\\mathrm{alg}} \\to \\mathrm{IJ}(X)(\\mathbb{C})$ induces identifications $(\\mathbf{CH}^2_{X/\\mathbb{C}})^\\gamma \\simeq \\mathrm{Pic}^m(C)$ for suitable $G$-curves $C$ and $\\gamma \\in \\mathrm{NS}^2(X)^G$, leading to obstructions and corollaries for conic bundles and quadric surface bundles. It then applies these obstructions to show (i) linearizability criteria in cyclic cases via the triviality of torsors $\\widetilde{P}$, $\\widetilde{P}^{(1)}$, (ii) identifications of CH^2-torsors with Prym varieties, and (iii) partial linearization results for selected Fano families via Abel-Jacobi isomorphisms $\\mathcal{F}_d(X)^G \\cong (\\mathbf{CH}^2_{X/\\mathbb{C}})^{\\ell}$; these yield new distinctions among $G$-equivariant birational types, including conjugacy classes of involutions in $\\mathrm{Cr}_3(\\mathbb{C})$.

Abstract

We develop an equivariant version of the formalism of intermediate Jacobian torsor obstructions, and apply it to conic bundles over rational surfaces, quadric surface bundles over $\mathbb P^1$, and Fano threefolds.

Theorems & Definitions (66)

• Theorem 1.1: Theorem \ref{['theorem:IJTobstructions']}
• Theorem 1.2: Corollary \ref{['thm:distinguish']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Lemma 3.1
• proof