Angelo Vistoli: Neutral representations of finite groups, arithmetic of quotient singularities, and fields of moduli

Abstract:

Fix a base field k₀ (for example, \(k_0 = \mathbb{C}\) is a good choice), which for simplicity we will assume to be of characteristic 0. Fix an algebraic variety X over an algebraically closed extension K of k₀, possibly with an additional structure, such as a polarization, or a finite set of marked points; we will always assume that \(\operatorname{Aut}_K X\) is finite. 

Under very general general condition one can define a field of moduli k for X. There is a well defined moduli problem of twisted forms of X, giving rise to the so called residual gerbe \(\mathcal{G}\) of X, which is an étale gerbe over \(\operatorname{Spec}k\). A natural question to ask is: when is X defined over its field of moduli k? This is equivalent to the gerbe \(\mathcal{G}\) being neutral. We are searching for criteria that depend on the geometry of X over K, and not on arithmetic information.

I am going to discuss two techniques that were introduced by Giulio Bresciani and myself to answer this question.

One, that works particularly well for varieties with a smooth marked point, is based on the concept of R-singularity.

The other, which applies much more generally, that of neutral representations. This gives criteria to show that X is defined on its field of moduli, by studying the action of the automorphism group of X on the intrinsically defined cohomology groups of X (for example, the cohomology of the structure sheaf, or the cotangent sheaf).