The equivariant Atiyah class

2020/03/11 by Ricolfi, Andrea T. · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.05440

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→ B of simple perfect complexes on a G-invariant smooth projective family Y→ 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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• Virtual invariants of Quot schemes of points on threefolds
• The refined local Donaldson-Thomas theory of curves

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