Formality of the homotopy calculus algebra of Hochschild (co)chains

TL;DR

This paper proves that the sheaf of homotopy calculi on Hochschild (co)chains of a smooth algebraic variety is formal, extending previous results to a broad geometric setting.

Contribution

It establishes the formality of the homotopy calculus structure on Hochschild (co)chains for any smooth algebraic variety, generalizing prior work.

Findings

The sheaf (C^*(O_X), C_*(O_X)) is formal for smooth algebraic varieties.

The result applies to a broad class of algebraic geometrical objects.

Supports the conjectural framework connecting homotopy calculus and algebraic geometry.

Abstract

The Kontsevich-Soibelman solution of the cyclic version of Deligne's conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^*(A), C_*(A)) ``Hochschild cochains + Hochschild chains'' of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^*(O_X), C_*(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author.