Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

Abstract

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.