Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves
type: Item
instance of: scholarly article
title: Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves
zbMATH Open document ID: 1241.13015
DOI: 10.4171/JEMS/310
author: Domenico Fiorenza
author: Donatella Iacono
author: E. Martinengo
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2012-04-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.1301
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 13D10
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 6023340
zbMATH Keywords: differential graded Lie algebras
zbMATH Keywords: DGLA
zbMATH Keywords: deformation
zbMATH Keywords: deligne deformation
MaRDI profile type: Publication
arXiv ID: 0904.1301
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OpenAlex ID: W1975741552
review text: In the particular case of a locally free sheaf \(\mathcal E\) of \(\mathcal O_X\)-modules on a complex manifold \(X\), the Kodaira-Spencer description of deformations of \(\mathcal E\) is given in terms of the Čech functor \(H^1(X;\, \text{exp} \, \mathcal End(\mathcal E))\). A deformation of \(\mathcal E\) is reduced to a deformation of the gluing data of its local charts, and the compatibility condition these gluing data have to satisfy is precisely expressed by the cocycle condition in the Čech functor. Also, deformations of \(\mathcal E\) are controlled by the Differential Graded Lie Algebra (DGLA) \(A^{0,\ast}_X(\mathcal End(\mathcal E))\) of \((0,\ast)\)-forms on \(X\) with values in the sheaf of endomorphisms of \(\mathcal E\). The equivalence of these two descriptions can be understood by moving to groupoid-valued deformation functors: The groupoid \(\text{Def}_{\mathcal E|_U}\) of \(\mathcal E\) over \(U\) defines a stack over \(\text\mathbf{Top}_X\). That is, the global deformations of \(\mathcal E\) are the same thing as the descent data for its local deformations: \(\text{Def}_{\mathcal E}\simeq\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\text{Def}_{\mathcal E|_U},\) where \(\Delta_{\mathcal U}\) is the semisimplicial object in \(\text\mathbf{Top}_X\) associated with an open cover \(\mathcal U\) of \(X\). One sees that locally the groupoid of deformations of \(\mathcal E|_U\) is equivalent to the Deligne groupoid of \(\mathcal End(\mathcal E)(U)\), and since these equivalences are compatible with restriction maps, one has an equivalence of semicosimplicial groupoids. Finally, the Deligne groupoid commutes with homotopy limits of DGLA concentrated in nonnegative degree so that NEWLINE\[CARRIAGE_RETURNNEWLINE\text{Def}_{\mathcal E}\simeq\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\text{Del}_{\mathcal End(\mathcal E)(U)}\simeq\text{Del}_{\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathcal End(\mathcal E)(U)}CARRIAGE_RETURNNEWLINE\]NEWLINE pr…
description: scientific article; zbMATH DE number 6023340