Deformation-obstruction theory for diagrams of algebras and applications to geometry

  • Severin Barmeier

    Albert-Ludwigs-Universität Freiburg, Germany
  • Yaël Frégier

    Université d'Artois, France
Deformation-obstruction theory for diagrams of algebras and applications to geometry cover
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Abstract

Let XX be an algebraic variety over an algebraically closed field of characteristic 00 and let \Coh(X)\Coh (X) denote its Abelian category of coherent sheaves. By the work of W. Lowen and M. Van den Bergh, it is known that the deformation theory of Coh(XX) as an Abelian category can be seen to be controlled by the Gerstenhaber–Schack complex associated to the restriction of the structure sheaf OXU\mathcal O_X \vert_{\mathfrak U} to a cover of affine open sets. We construct an explicit L_\infty algebra structure on the Gerstenhaber–Schack complex controlling the higher deformation theory of OXU\mathcal O_X \vert_{\mathfrak U} or Coh(XX) in case XX can be covered by two acyclic open sets, giving an explicit deformation-obstruction calculus for such deformations. For smooth XX, such deformations recover the deformation of complex structures and deformation quantizations of XX as degenerate cases, as we show by means of concrete examples.

Cite this article

Severin Barmeier, Yaël Frégier, Deformation-obstruction theory for diagrams of algebras and applications to geometry. J. Noncommut. Geom. 14 (2020), no. 3, pp. 1019–1047

DOI 10.4171/JNCG/385