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Article

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

Inventiones Mathematicae. 2020. Vol. 222. No. 2. P. 667-694.

We disprove two (unpublished) conjectures of Kontsevich which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).