Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866909480974811136 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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 \cite{Ka}). 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 $μ_3$ 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 \cite{KL} (that is, it cannot be embedded into a smooth and proper DG category). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_09283 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration Efimov, Alexander I. Algebraic Geometry Category Theory K-Theory and Homology We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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 \cite{Ka}). 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 $μ_3$ 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 \cite{KL} (that is, it cannot be embedded into a smooth and proper DG category). |
| title | Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration |
| topic | Algebraic Geometry Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/1805.09283 |