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HHNCDMIR Report Summary

Project ID: 257004
Funded under: FP7-IDEAS-ERC
Country: Belgium

Final Report Summary - HHNCDMIR (Hochschild cohomology, non-commutative deformations and mirror symmetry)

The research project involves the development of deformation theory in the realm of non-commutative geometry, a rapidly growing field with deep ties to both quantum mechanics and string theory in theoretical physics.

Deformation theory concerns the study of “nearby structures”, and in non-commutative geometry we are particularly interested in ways of deforming classical spaces (schemes) into non-commutative (nc) spaces. To this end, we study models which are sufficiently geometric in nature and at the same time allow for a natural intrinsic deformation theory governed by a notion of Hochschild cohomology.

Our first group of results involves global algebraic models for nc spaces. Using tails topologies on Z-algebras, we identify a broad class of schemes with “projective” deformations. More general models are given by differential graded (dg) algebras. In general, in their deformation theory one encounters the “curvature problem”, placing arbitrary deformations outside classical homological algebra. Inspired by Fukaya categories in symplectic geometry, joint with Olivier De Deken we developed parts of a general curved theory, leading to a notion of “curvature compensating deformation”. Joint with Michel Van den Bergh, we obtained important specific results. Firstly, we managed to prove that all infinitesimal deformations of reasonable schemes have accompanying dg algebras. Secondly, for formal deformations we arrived at a complete solution of the curvature problem for bounded above dg algebras.

Our second group of results involves local models for nc spaces, namely prestacks, inspired by structure sheaves of schemes. Joint with Hoang Dinh Van, we introduced a Gerstenhaber-Schack (GS) complex for arbitrary prestacks, involving higher components in the differential, and we developed a technique for endowing the complex with the higher structure necessary for deformation theory.
One of the main technical tools is the map-graded Grothendieck construction of a prestack. For map-graded categories, we developed a notion of Hochschild cohomology with support, leading for instance to the existence of Mayer-Vietoris triangles. Joint with Liyu Liu, we applied the GS complex to the study of deformations of singular hypersurfaces.

At the crossroads of algebraic geometry and algebraic topology, joint with Dmitry Kaledin we introduced a notion of Mac Lane cohomology of exact categories, leading to a new type of deformations which is currently being investigated. Likewise, our results in various directions of the project have given rise to promising new research questions and projects which will be pursued further in the future.

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