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

TL;DR

This paper disproves two conjectures related to categorical Hodge-to-de Rham degeneration, providing counterexamples in the noncommutative setting, and explores the limitations of resolutions and compactifications of DG categories.

Contribution

It presents explicit counterexamples to conjectures on categorical Hodge-to-de Rham degeneration and shows the non-existence of certain resolutions in noncommutative geometry.

Findings

Existence of a 10-dimensional A-infinity algebra with non-zero supertrace of μ₃

A homotopically finitely presented DG category without a smooth categorical compactification

A proper DG category lacking a categorical resolution of singularities

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We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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 \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 $\mu_3$ 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 To\"en. 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).