Smooth compactifications in derived non-commutative geometry

Abstract

This is a short overview of the author’s results related to the notion of a smooth categorical compactification. We cover the construction of a categorical smooth compactification of the derived categories of coherent sheaves, using the categorical resolution of Kuznetsov and Lunts. We also mention examples of homotopically finitely presented DG categories which do not admit a smooth compactification. This is closely related to Kontsevich’s conjectures on the generalized versions of categorical Hodge-to-de Rham degeneration, which we disproved. Finally, we mention our new result on the DG categorical analogue of Wall’s finiteness obstruction, which in particular gives a criterion for existence of a smooth compactification of a homotopically finite DG category.