Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration
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Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration Alexander I. Efimov1,2
Received: 26 December 2018 / Accepted: 2 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin (in: Algebra, geometry, and physics in the 21st century. Birkhäuser/Springer, Cham, pp 99–129, 2017). In particular, we show that there exists a minimal 10-dimensional A∞ -algebra over a field of characteristic zero, for which the supertrace of μ3 on the second argument is non-zero. As a byproduct, we obtain an example of a homotopically finitely presented DG category (over a field of characteristic zero) that does not have a smooth categorical compactification, giving a negative answer to a question of Toën. This can be interpreted as a lack of resolution of singularities in the noncommutative setup. We also obtain an example of a proper DG category which does not admit a categorical resolution of singularities in the terminology of Kuznetsov and Lunts (Int Math Res Not 2015(13):4536–4625, 2015) (that is, it cannot be embedded into a smooth and proper DG category).
The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. N 14.641.31.0001.
B Alexander I. Efimov [email protected]
1
Steklov Mathematical Institute of RAS, Gubkin str. 8, GSP-1, Moscow, Russian Federation 119991
2
National Research University Higher School of Economics, Moscow, Russian Federation
123
A. I. Efimov
Contents 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminaries on DG categories and A∞ -algebras . . . . . . . . . 1.1 DG categories . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A∞ -algebras and A∞ -(bi)modules . . . . . . . . . . . . . . 2 Preliminaries on the Hochschild complex, pairings and copairings 3 A counterexample to the generalized degeneration conjecture . . 4 A counterexample to Conjecture 0.3 . . . . . . . . . . . . . . . . 5 A counterexample to Conjecture 0.2 . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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0 Introduction Given a smooth algebraic variety X over a field of characteristic zero, we have p,q p p+q the Hodge-to-de Rham spectral sequence E 1 = H q (X, X ) ⇒ H D R (X ). It is classically known that when X is additionally proper, this spectral sequence degenerates at E 1 , that is, all differentials vanish. This follows from the classical Hodge theory for compact Kähler manifolds, and can be also proved algebraically [3]. We recall the following fundamental re
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