Deformations of Abelian categories in algebra, geometry & physics

W. Lowen and M. Van den Bergh developed a deformation theory for Abelian categories, such that deformations of the category of (quasi)coherent sheaves on an affine variety Spec A correspond to deformations of the algebra A, which in turn are controlled by the Hochschild cochain complex equipped with the Gerstenhaber bracket. For categories of coherent sheaves on a (non-affine) quasi-projective variety X, one should replace the single commutative algebra A by a "diagram" of commutative algebras, obtained as the restriction of the structure sheaf of the variety to an affine open cover which is closed under intersections.

Deformations of QCoh(X) then correspond to deformations of this diagram of algebras, which are controlled by the so-called Gerstenhaber-Schack complex, whose cohomology is isomorphic to the Hochschild cohomology of X. First-order deformations of QCoh(X) are parametrized by HH²(X), which for smooth complex varieties includes deformations of the complex structure as well as (complex) deformation quantizations.

I will explain how to obtain an explicit L-infinity algebra structure on this complex, controlling the higher deformation theory of QCoh(X), in case X can be covered by two affine open sets and explain how this point of view may be used to study the effect of deformations of QCoh(X) on moduli spaces of vector bundles or instanton moduli. This latter application can be viewed as an analogue of work by Nekrasov-Schwarz on non-commutative instantons on R⁴, for example to study non-commutative instantons on the blow-up of the complex plane or on the resolution of the A₁ surface singularity.

This talk is based on joint work with Yaël Frégier and work in progress joint with Zhengfang Wang.