Let be an algebraic variety over an algebraically closed field of characteristic and let 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 as an Abelian category can be seen to be controlled by the Gerstenhaber–Schack complex associated to the restriction of the structure sheaf to a cover of affine open sets. We construct an explicit L algebra structure on the Gerstenhaber–Schack complex controlling the higher deformation theory of or in case can be covered by two acyclic open sets, giving an explicit deformation-obstruction calculus for such deformations. For smooth , such deformations recover the deformation of complex structures and deformation quantizations of as degenerate cases, as we show by means of concrete examples.
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Severin Barmeier, Yaël Frégier, Deformation-obstruction theory for diagrams of algebras and applications to geometry. J. Noncommut. Geom. 14 (2020), no. 3, pp. 1019–1047DOI 10.4171/JNCG/385