Geodesic
Căldăraru’s conjecture and Tsygan’s formality
Damien Calaque1 ; Carlo A. Rossi2 ; Michel Van den Bergh3
1 ETH Zürich, 8092 Zürich, Switzerland, Institut Camille Jordan, CNRS and Université Lyon 1, France
2 MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
3 Departement WNI, Hasselt University, Agoralaan, 3590 Diepenbeek, Belgium
Annals of mathematics, Tome 176 (2012) no. 2, pp. 865-923

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In this paper we complete the proof of Căldăraru’s conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects.
Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured by Tsygan.) We also rely on the fact — recently proved by the first two authors — that Shoikhet’s quasi-isomorphism is compatible with cap products after twisting with a Maurer-Cartan element.

DOI : 10.4007/annals.2012.176.2.4

Damien Calaque 1 ; Carlo A. Rossi 2 ; Michel Van den Bergh 3

1 ETH Zürich, 8092 Zürich, Switzerland, Institut Camille Jordan, CNRS and Université Lyon 1, France
2 MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
3 Departement WNI, Hasselt University, Agoralaan, 3590 Diepenbeek, Belgium 
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Damien Calaque; Carlo A. Rossi; Michel Van den Bergh. Căldăraru’s conjecture and Tsygan’s formality. Annals of mathematics, Tome 176 (2012) no. 2, pp. 865-923. doi: 10.4007/annals.2012.176.2.4

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