Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differential forms and Brauer classes in positive characteristic

Domenico Valloni

TL;DR

This work develops a differential-forms-based approach to $p$-torsion Brauer classes in positive characteristic, introducing a filtration $\mathrm{Br}^{\delta}_{n}(X)$ built from global sections of $\Omega^1_X/B_{n,X}$ and relating it to the Brauer group of supersingular K3 surfaces. It provides explicit descriptions of the Brauer-Manin obstruction in terms of Cartier-theoretic data, shows strong obstruction phenomena for supersingular K3s that descend along Frobenius, and produces density and purity results for adelic points under global-generation and jet-ampleness conditions. The paper also constructs and analyzes a differential Brauer group, proves representability over perfect fields, and offers concrete examples for curves, abelian surfaces, and K3 surfaces, including pairings from infinitesimal torsors. Altogether, the results illuminate how differential forms in positive characteristic control adelic obstructions, yielding new avenues to study rational points and descent phenomena in this setting.

Abstract

We study $p$-torsion Brauer classes in positive characteristic arising from differential forms. We relate this construction to the Brauer group of supersingular K3 surfaces and analyze the contribution of these classes to the Brauer-Manin obstruction. As an application, we examine the Brauer-Manin set of supersingular K3 surfaces and of varieties admitting many differential forms.

Differential forms and Brauer classes in positive characteristic

TL;DR

This work develops a differential-forms-based approach to -torsion Brauer classes in positive characteristic, introducing a filtration built from global sections of and relating it to the Brauer group of supersingular K3 surfaces. It provides explicit descriptions of the Brauer-Manin obstruction in terms of Cartier-theoretic data, shows strong obstruction phenomena for supersingular K3s that descend along Frobenius, and produces density and purity results for adelic points under global-generation and jet-ampleness conditions. The paper also constructs and analyzes a differential Brauer group, proves representability over perfect fields, and offers concrete examples for curves, abelian surfaces, and K3 surfaces, including pairings from infinitesimal torsors. Altogether, the results illuminate how differential forms in positive characteristic control adelic obstructions, yielding new avenues to study rational points and descent phenomena in this setting.

Abstract

We study -torsion Brauer classes in positive characteristic arising from differential forms. We relate this construction to the Brauer group of supersingular K3 surfaces and analyze the contribution of these classes to the Brauer-Manin obstruction. As an application, we examine the Brauer-Manin set of supersingular K3 surfaces and of varieties admitting many differential forms.

Paper Structure

This paper contains 21 sections, 44 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Consequences and applications
  3. Some literature
  4. Differential Brauer classes
  5. Preliminaries
  6. The p-torsion Brauer group
  7. Example: local fields
  8. Differential Brauer group
  9. Representability over perfect fields.
  10. Examples
  11. Curves
  12. Abelian surfaces
  13. K3 surfaces
  14. Pairing infinitesimal torsors
  15. Absolute vs relative differentials
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(x_v) \in X(\mathbf{A}_K)$ and $\omega \in H^0(X, \Omega^1_X / B_{n,X})$. Then $(x_v)$ is unobstructed for all Brauer classes of the form $[t^{p^n}\omega] \in \mathop{\mathrm{Br}}\nolimits(X)[p]$ with $t \in K$ if and only if

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5: Cartier, Serre
  • ...and 75 more