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

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We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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. Birkhauser/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 Toen. 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).