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BIMSA AG Seminar
Neutral representations of finite groups, arithmetic of quotient singularities, and fields of moduli
Neutral representations of finite groups, arithmetic of quotient singularities, and fields of moduli
演讲者
Angelo Vistoli
时间
2024年10月24日 15:00 至 16:00
地点
A6-101
线上
Zoom 638 227 8222
(BIMSA)
摘要
Fix a base field k, which for simplicity we will assume to be of characteristic 0. Fix an algebraic variety X over the algebraic closure k' of k, possibly with an additional structure, such as a polarization, or a finite set of marked points; we will always assume that the automorphism group of X over k' is finite.
Under very general general conditions there exists a well defined moduli problem of twisted form of X, giving rise to the so called residual gerbe G_X of X, which is an étale gerbe over a subfield E of k' containing k, which is known as the field of moduli of X. A natural question to ask is: when is X defined over its field of moduli E? This is equivalent to the residual gerbe G_X 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).
Under very general general conditions there exists a well defined moduli problem of twisted form of X, giving rise to the so called residual gerbe G_X of X, which is an étale gerbe over a subfield E of k' containing k, which is known as the field of moduli of X. A natural question to ask is: when is X defined over its field of moduli E? This is equivalent to the residual gerbe G_X 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).