Geodesic
Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields
V. A. Dolgushev1 ; C. L. Rogers2 ; T. H. Willwacher3
1 Department of Mathematics, Temple University, Philadelphia, PA
2 Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Germany
3 University of Zürich, Institute of Mathematics, Zürich, Switzerland
Annals of mathematics, Tome 182 (2015) no. 3, pp. 855-943.

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We generalize Kontsevich’s construction of $L_{\infty}$-derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich’s graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmüller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that the $\hat{A}$-genus in the Calaque-Van den Bergh formula for the isomorphism between harmonic and Hochschild structures can be replaced by a generalized $\hat{A}$-genus.
DOI : 10.4007/annals.2015.182.3.2

V. A. Dolgushev 1 ; C. L. Rogers 2 ; T. H. Willwacher 3

1 Department of Mathematics, Temple University, Philadelphia, PA
2 Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Germany
3 University of Zürich, Institute of Mathematics, Zürich, Switzerland 
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     title = {Kontsevich{\textquoteright}s graph complex, {GRT,} and the deformation complex of the sheaf of polyvector fields},
     journal = {Annals of mathematics},
     pages = {855--943},
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V. A. Dolgushev; C. L. Rogers; T. H. Willwacher. Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields. Annals of mathematics, Tome 182 (2015) no. 3, pp. 855-943. doi : 10.4007/annals.2015.182.3.2. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.182.3.2/

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