Moduli of vector bundles on $\mu_n$-gerbes over genus 2 curves and the period-index problem

TL;DR

This paper develops a framework for vector bundles on $bc_n$-gerbes over genus 2 curves, demonstrating that certain Brauer classes have equal period and index, and extends these results to higher-dimensional varieties over $C_1$-fields.

Contribution

It introduces a new geometric approach to the period-index problem for Brauer classes on genus 2 curves and constructs higher-dimensional examples satisfying the conjecture.

Findings

Existence of Brauer classes with equal period and index on genus 2 curves.

Every 2-torsion Brauer class on genus 2 curves over $C_1$-fields satisfies the period-index relation.

Construction of higher-dimensional varieties with 2-torsion Brauer classes satisfying the period-index problem.

Abstract

We develop a framework for describing vector bundles on $\mu_n$-gerbes over curves and illustrate the construction through two detailed examples. Using the interpretation of Brauer classes as obstructions to descending determinantal line bundles from the algebraic closure, together with a geometric analysis of the moduli space of twisted sheaves, we prove that for genus $2$ curves there exist Brauer classes over the base field whose period equals their index. Over $C_1$-fields, we further show that every $2$-torsion class in the Brauer group of a genus $2$ curve satisfies the period-index problem. As an application, we construct higher-dimensional varieties obtained as fibre products of genus $2$ curves over $C_1$-fields whose $2$-torsion algebraic Brauer classes also satisfy the period-index problem, providing new evidence toward the period-index conjecture.