The period-index conjecture for abelian threefolds and Donaldson-Thomas theory

2024/05/06 by James Hotchkiss, Hotchkiss, James, Perry Alexander +1 · 4 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2405.03315

openalex publication_date 2024/05/06 · openalex created_date 2024/05/09 · openalex updated_date 2026/07/28

Abstract

We prove the period-index conjecture for unramified Brauer classes on abelian threefolds. To do so, we develop a theory of reduced Donaldson-Thomas invariants for 3-dimensional Calabi-Yau categories, with the feature that the noncommutative variational integral Hodge conjecture holds for classes with nonvanishing invariant. The period-index result is then proved by interpreting it as the algebraicity of a Hodge class on the twisted derived category, and specializing within the Hodge locus to an untwisted abelian threefold with nonvanishing invariant. As a consequence, we also deduce the integral Hodge conjecture for generically twisted abelian threefolds.

Cited by

• Stability conditions on products of curves and Hilbert schemes of surfaces
• Moduli of vector bundles on μ_n-gerbes over genus 2 curves and the period-index problem
• Secant sheaves on abelian n-folds with real multiplication and Weil classes on abelian 2n-folds with complex multiplication
• Specialization and rigidity

Related

• Hodge theory of twisted derived categories and the period-index problem
• The period-index problem and Hodge theory
• Splitting unramified Brauer classes by abelian torsors and the period-index problem
• Generalized Donaldson-Thomas theory over fields K ≠ C
• A theory of generalized Donaldson-Thomas invariants