Orateur
Prof.
Michel Van den Bergh
(Hasselt et NFWO)
Description
It is well known that the Hochschild (co)homology of a smooth
algebraic variety (or real/complex manifold) may be additively
identified with its tangent (co)homology through the
Hochschild-Kostant-Rosenberg isomorphism.
In 2003 Caldararu published an intriguing conjecture (which he
attributes to Kontsevich) that in order to make the HKR morphism
compatible with the multiplicative structures one has to twist it with
the square root of the Todd class, yielding an unexpected connection
with the Riemann-Roch theorem. Surprisingly this conjecture has so
far resisted all attempts to prove it by elementary means.
Calderaru's conjecture was proved by Damien Calaque, Carlo Rossi and
myself in 2009 (see arXiv:0904.4890) using techniques from modern
deformation theory. The aim of the course is to give an introduction
to this theory and to explain the proof of the conjecture. A rough outline
will be as follows.
(1) Kontsevich and Tsygan formality results.
(2) Globalization methods.
(3) Application to the proof of Caldararu's conjectury.
If time permits then I will discuss some more applications to deformation
theory.
| Mots Clés / Keywords | Hochschild (co)homology; deformation theory; formality results |
|---|
Auteur principal
Prof.
Michel Van den Bergh
(Hasselt et NFWO)
