On automorphisms of $p$-torsion $\mathbf{G}_m$-gerbes

TL;DR

The paper investigates when the identity-component map $Aut^0_{\mathcal{X}} \to Aut^0_X$ is surjective for a $\mathbf{G}_m$-gerbe $\mathcal{X}\to X$ over a smooth projective variety $X$ in positive characteristic. Building on Olsson's result for Brauer classes of order prime to $p$, it constructs an Igusa-like counterexample in characteristic $p$ where the Brauer class has order $p$ and the map is not surjective, using a quotient $X=(A\times B)/\mathbf{Z}/2$ with $A,B$ ordinary abelian varieties. The argument intertwines deformation theory of fppf Artin–Mazur formal groups, flat cohomology of ordinary varieties, and recent representability results to analyze the obstruction map $a: Aut_X \to R^2\pi_*G_m$, showing a nonzero differential of $a$ at the identity. Additionally, the paper proves several sufficient conditions for surjectivity, including when $H^0(X,T_X)=0$, $H^2(X,O_X)=0$, or $Aut^0_X$ is abelian, and discusses when these conditions hold for K3 surfaces or abelian varieties, providing a nuanced picture of how the Brauer class and deformation-theoretic obstructions govern automorphism groups of gerbes. Overall, it advances our understanding of the interaction between twisted derived equivalences, gerbes, and automorphism group structures in positive characteristic.

Abstract

Olsson showed in [Ols25] that if $\mathcal{X} \to X$ is a $\mathbf{G}_m$-gerbe over a smooth projective variety over an algebraically closed field $k$ such that the Brauer class of $\mathcal{X}$ has order prime to the characteristic of $k$, then the homomorphism of $k$-group algebraic spaces $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ is surjective. We provide an example to show that this need not be the case when the Brauer class of $\mathcal{X}$ has order equal to the characteristic. Our main tools are deformation theory of the fppf sheafified Artin--Mazur formal groups and nice properties of the flat cohomology of ordinary varieties in positive characteristic which are presumably well-known, but which we collect and give an exposition of here. We additionally prove some sufficient conditions for surjectivity of $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ using representability results of Bragg and Olsson.

Theorems & Definitions (50)

• Theorem 1.1: See Theorem \ref{['thm-mainpaperversion']}
• Lemma 2.1
• proof
• Theorem 2.2: Raynaud, Ekedahl
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 3.1: Dwyer, Greenlees, May
• Example 3.2