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Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration

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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We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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 \(A_{\infty }\)-algebra over a field of characteristic zero, for which the supertrace of \(\mu _3\) 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).

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Acknowledgements

I am grateful to Dmitry Kaledin, Maxim Kontsevich and Bertrand Toën for useful discussions. I am also grateful to an anonymous referee for a number of useful comments and suggestions, and for pointing out a number of inaccuracies in the previous version of the paper.

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  1. Steklov Mathematical Institute of RAS, Gubkin str. 8, GSP-1, Moscow, Russian Federation, 119991

    Alexander I. Efimov

  2. National Research University Higher School of Economics, Moscow, Russian Federation

    Alexander I. Efimov

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Correspondence to Alexander I. Efimov.

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The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. N 14.641.31.0001.

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Efimov, A.I. Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. math. 222, 667–694 (2020). https://doi.org/10.1007/s00222-020-00980-9

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