

Iskovskikh Seminar
April 5, 2018 14:00, Moscow, Steklov Mathematical Institute, room 540






On smooth categorical compactifications and noncommutative Hodge theory
A. I. Efimov^{} ^{} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

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Abstract:
In this talk we disprove two conjectures of Kontsevich which state generalized versions of categorical Hodgetode Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin). In particular, we show that there exists a minimal 8dimensional $A_{\infty}$algebra, for which the supertrace of $m_3$ on the second argument is nonzero.
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 (i.e. it cannot be represented as a quotient of a smooth and proper DG category). This gives a negative answer to a question of Toen. We also obtain an example of a proper DG category, which does not have a categorical resolution of singularities (i.e. it cannot be embedded into a smooth and proper DG category).

