The equivariant Atiyah class

Ricolfi, Andrea T.

Comptes Rendus. Mathématique, Tome 359 (2021) no. 3, pp. 257-282.

Soit $X$ un schéma complexe sur lequel agit un groupe algébrique affine $G$. Nous démontrons que la classe d’Atiyah d’un complexe parfait $G$-équivariant au dessus de $X$, construite par Huybrechts et Thomas, est $G$-équivariante dans un sense précis. Comme application, nous démontrons que, si $G$ est réductif, la théorie d’obstruction sur l’espace de modules relatif fin $M\to B$ des complexes parfaits simples sur une famille lisse projective $Y\to B$ est $G$-équivariante. Les résultats contenus ici vont suggérer comment vérifier l’équivariance de la théorie d’obstruction naturelle sur un nombre d’espaces de modules munis de l’action d’un tore, notamment ceux qui sont construits en théorie de Donaldson–Thomas et en théorie de Vafa–Witten.

Let $X$ be a complex scheme acted on by an affine algebraic group $G$. We prove that the Atiyah class of a $G$-equivariant perfect complex on $X$, as constructed by Huybrechts and Thomas, is $G$-equivariant in a precise sense. As an application, we show that, if $G$ is reductive, the obstruction theory on the fine relative moduli space $M\to B$ of simple perfect complexes on a $G$-invariant smooth projective family $Y\to B$ is $G$-equivariant. The results contained here are meant to suggest how to check the equivariance of the natural obstruction theories on a wide variety of moduli spaces equipped with a torus action, arising for instance in Donaldson–Thomas theory and Vafa–Witten theory.

Classification : 14F05, 14D20, 14N10

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