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Smooth compactifications in derived non-commutative geometry
P. 535–551.
Efimov A.
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.
Keywords: resolution of singularitiesdifferential graded categorieshomotopy finitenessderived categories Verdier localization
Publication based on the results of:
Kuznetsov A., Algebraic Geometry 2025 Vol. 12 No. 4 P. 519–574
We construct S-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations X/S with relative Picard rank 1 and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of X/S. ...
Added: November 29, 2025
Kuznetsov A., Shinder E., Inventiones Mathematicae 2025 Vol. 239 P. 377–430
Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree d∈{2,3,4,5}d∈{2,3,4,5} and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree d=1d=1 can be smoothly deformed to the nontrivial components of the derived categories of prime ...
Added: November 29, 2025
Guseva L., Journal de Mathématiques Pures et Appliquées 2023 Vol. 180 P. 1–46
We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian Gr(3,7) parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from two operations ...
Added: November 16, 2023
Shinder E., Kuznetsov A., Karmazyn J., Journal of the European Mathematical Society 2022 Vol. 24 No. 2 P. 461–526
We develop an approach that allows one to construct semiorthogonal decompositions of
derived categories of surfaces with cyclic quotient singularities whose components are equivalent
to derived categories of local finite-dimensional algebras.
We first explain how to induce a semiorthogonal decomposition of a surface X with rational
singularities from a semiorthogonal decomposition of its resolution. In the case when X ...
Added: November 24, 2022
Lunts V., Moscow Mathematical Journal 2022 Vol. 22 No. 4 P. 705–739
Let X be a smooth complex projective variety. The group of autoequivalences of the derived category of X acts naturally on its singular cohomology H(X; Q) and we denote by Geq(X) GL(H(X; Q)) its image. Let Geq(X) GL(H(X; Q) be its Zariski closure. We study the relation of the Lie algebra LieGeq(X) and ...
Added: November 15, 2022
Kalck M., Pavic N., Shinder E., Moscow Mathematical Journal 2021 Vol. 21 No. 3 P. 567–592
We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound A(n), singularities. We introduce obstructions coming from Algebraic K-theory and translate them into the concept of maximal nonfactoriality.
Using these obstructions we show that many classes of nodal three-folds do not admit Kawamata ...
Added: December 1, 2021
Fonarev A., International Mathematics Research Notices 2020
We show fullness of the exceptional collections of maximal length constructed by Kuznetsov and Polishchuk in the bounded derived categories of coherent sheaves on Lagrangian Grassmannians. ...
Added: October 10, 2021
Belmans P., Kuznetsov A., Smirnov M., Transformation Groups 2021
For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type E6, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A general hyperplane section of the Cayley plane is the coadjoint Grassmannian of Dynkin type F4. We show that the restriction of the Faenzi–Manivel collection to such ...
Added: September 3, 2021
Efimov A., Journal of the European Mathematical Society 2020 Vol. 22 No. 9 P. 2879–2942
In this paper, we prove that the bounded derived category D-coh(b) (Y) of coherent sheaves on a separated scheme Y of finite type over a field k of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: D-coh(b) (Y) is equivalent to a DG quotient D-coh(b) ((Y) over tilde)/T, where (Y) over tilde is some smooth and proper ...
Added: September 24, 2020
Kuznetsov A., Smirnov M., Belmans P., / Series arXiv "math". 2020.
For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type E_6, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A general hyperplane section of the Cayley plane is the coadjoint Grassmannian of Dynkin type F_4. We show that the restriction of the Faenzi-Manivel collection to the hyperplane section ...
Added: August 19, 2020
Brav C. I., Compositio Mathematica 2019 Vol. 155 No. 2 P. 372–412
We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya ...
Added: October 18, 2019
Kuznetsov A., / Series arXiv "math". 2018.
We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond. ...
Added: December 3, 2018
Kuznetsov A., Shinder E., Karmazyn J., / Series arXiv "math". 2018.
We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with rational singularities with components equivalent to derived categories of local finite dimensional algebras. First, we discuss how a semiorthogonal decomposition of a resolution of singularities of a surface X may induce a semiorthogonal decomposition of X. In the case when Xhas cyclic quotient ...
Added: December 3, 2018
Katzarkov L. V., Mathematische Annalen 2018 Vol. 371 No. 1-2 P. 337–370
We provide descriptions of the derived categories of degree d hypersurface
fibrations which generalize a result of Kuznetsov for quadric fibrations and give
a relative version of a well-known theorem of Orlov. Using a local generator and
Morita theory, we re-interpret the resulting matrix factorization category as a derivedequivalent
sheaf of dg-algebras on the base. Then, applying homological perturbation
methods, ...
Added: November 29, 2018
Burban I., Drozd Y., Gavran V., European Journal of Mathematics 2017 Vol. 3 No. 2 P. 311–341
We develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra. ...
Added: May 10, 2018
Katzarkov L. V., Deliu D., Favero D. et al., Advances in Mathematics 2016 Vol. 295 P. 195–249
Building upon ideas of Eisenbud, Buchweitz, Positselski, and others, we introduce the notion of a factorization category. We then develop some essential tools for working with factorization categories, including constructions of resolutions of factorizations from resolutions of their components and derived functors. Using these resolutions, we lift fully-faithfulness and equivalence statements from derived categories of ...
Added: October 23, 2017
Alexander I. Efimov, Journal of London Mathematical Society 2014 Vol. 90 No. 2 P. 350–372
In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac 34$ there exist infinitely many toric Fano varieties $Y$ with ...
Added: January 28, 2015
Bodzenta-Skibinska A., / Series math "arxiv.org". 2013.
Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces. ...
Added: November 5, 2014
Kuznetsov A., Lunts V., / Series math "arxiv.org". 2012. No. 1212.6170.
We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity. ...
Added: October 4, 2013