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.
The equivariant Atiyah class / Ricolfi, A. T.. - In: COMPTES RENDUS MATHÉMATIQUE. - ISSN 1631-073X. - 359:3(2021), pp. 257-282. [10.5802/crmath.166]
The equivariant Atiyah class
Ricolfi, A. T.
2021-01-01
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.File | Dimensione | Formato | |
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