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Smooth compactifications in derived non-commutative geometry. (English) Zbl 1523.14034

Hujdurović, Ademir (ed.) et al., European congress of mathematics. Proceedings of the 8th congress, 8ECM, Portorož, Slovenia, June 20–26, 2021. Berlin: European Mathematical Society (EMS). 535-551 (2023).
Summary: 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.
For the entire collection see [Zbl 1519.00033].

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14A22 Noncommutative algebraic geometry
14A30 Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.)
18G35 Chain complexes (category-theoretic aspects), dg categories
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
Full Text: DOI

References:

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