Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie
Given a smooth projective variety \(X\) over a field of characteristic 0, according to the Hodge theory, the Hodge-to-de Rham spectral sequence \(H^p (X, \Omega^q_X) \implies H^{p+q}_{DR} (X)\) degenerates at the first term, so that there is a noncanonical decomposition of the form \(H^{p+q}_{DR} (X) = \bigoplus H^p (X, \Omega^q_X)\). The terms in the Hodge-to-de Rham spectral sequence can be generalized to the case of a site \(X\) equipped with a sheaf of noncommutative associative algebras \(\mathcal A\) by replacing the differential forms with the Hochschile cohomology sheave \(HH_\bullet (\mathcal A)\) and the de Rham complex with the cyclic homology complex \(HC_\bullet (\mathcal A)\). Thus the Hodge-to-de Rham spectral sequence is well-defined for noncommutative algebras, and it would be natural to ask when it degenerates. A recent conjecture of Kontsevich and Soibelman provides a condition under which the Hodge-to-de Rham spectral sequence degenerates in the noncommutative case. In this paper, the author proves the Kontsevich and Soibelman conjecture under some additional assumptions.
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