The equivariant Atiyah class

Andrea T. Ricolfi

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arXiv · 2003.05440

The equivariant Atiyah class

Andrea T. Ricolfi

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Abstract

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.

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Andrea T. Ricolfi. 2020-03-11. The equivariant Atiyah class. https://arxiv.org/abs/2003.05440

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