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

TL;DR

This paper extends Kontsevich's graph complex to arbitrary smooth algebraic varieties, linking it to the deformation complex of polyvector fields and revealing connections with the Grothendieck-Teichmueller Lie algebra and characteristic classes.

Contribution

It constructs a homotopy-theoretic map from Kontsevich's graph complex to the deformation complex of polyvector fields on any smooth algebraic variety, generalizing previous affine space results.

Findings

Action of Deligne-Drinfeld elements matches odd Chern character components.

The A-hat genus in Calaque-Van den Bergh formula can be replaced by a generalized A-hat genus.

Established a link between graph complex actions and characteristic classes.

Abstract

We generalize Kontsevich's construction of L-infinity 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-Teichmueller 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 A-hat genus in the Calaque-Van den Bergh formula arXiv:0708.2725 for the isomorphism between harmonic and Hochschild structures can be replaced by a generalized A-hat genus.