Comptes Rendus
no. 15-16
Algebraic Geometry
The module structure of Hochschild homology in some examples
[La structure module de l'homologie de Hochschild dans quelques exemples]

Présenté par : Maxim Kontsevitch

Emanuele Macri`1 ; Marc Nieper-Wißkirchen2 ; Paolo Stellari3
1 Hausdorff Center for Mathematics, Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany
2 Institut für Mathematik, Johannes-Gutenberg-Universität, Staudinger Weg 9, 55128 Mainz, Germany
3 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 863-866.

Dans cette Note nous donnons une démonstration simple d'une conjecture d'A. Căldăraru énoncant la compatibilité entre l'isomorphisme modifié de Hochschild–Kostant–Rosenberg et l'action de la cohomologie de Hochschild sur l'homologie de Hochschild dans le cas des variétés de Calabi–Yau et des courbes projectives régulières.

In this Note we give a simple proof of a conjecture by A. Căldăraru stating the compatibility between the modified Hochschild–Kostant–Rosenberg isomorphism and the action of Hochschild cohomology on Hochschild homology in the case of Calabi–Yau manifolds and smooth projective curves.

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Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.05.017
Emanuele Macri` 1 ; Marc Nieper-Wißkirchen 2 ; Paolo Stellari 3

1 Hausdorff Center for Mathematics, Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany
2 Institut für Mathematik, Johannes-Gutenberg-Universität, Staudinger Weg 9, 55128 Mainz, Germany
3 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy 
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Emanuele Macri`; Marc Nieper-Wißkirchen; Paolo Stellari. The module structure of Hochschild homology in some examples. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 863-866. doi : 10.1016/j.crma.2008.05.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.017/

[1] D. Calaque; M. Van den Bergh Hochschild cohomology and Atiyah classes | arXiv

[2] A. Căldăraru The Mukai pairing II: The Hochschild–Kostant–Rosenberg isomorphism, Adv. Math., Volume 194 (2005), pp. 34-66

[3] D. Huybrechts; M. Nieper-Wisskirchen Remarks on derived equivalences of Ricci-flat manifolds | arXiv

[4] M. Kontsevich Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003), pp. 157-216

[5] N. Markarian The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem | arXiv

[6] A. Ramadoss The relative Riemann–Roch theorem from Hochschild homology | arXiv

[7] J. Roberts; S. Willerton On the Rozansky–Witten weight systems | arXiv

[8] A. Yekutieli The continuous Hochschild cochain complex of a scheme, Canadian J. Math., Volume 54 (2002), pp. 1319-1337

Cité par Sources :

⁎ This work was supported by the SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG (German Research Foundation).

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