  The solution of Deligne's conjecture on Hochschild cochains and the formality
of the operad of little disks provide us with a natural homotopy Gerstenhaber
algebra structure on the Hochschild cochains of an associative algebra. In this
paper we construct a natural chain of quasi-isomorphisms of homotopy
Gerstenhaber algebras between the Hochschild cochain complex C(A) of a regular
commutative algebra A over a field of characteristic zero and the Gerstenhaber
algebra of multiderivations of A. Unlike the original approach of the second
author based on the computation of obstructions our method allows us to avoid
the bulky Gelfand-Fuchs trick and prove the formality of the homotopy
Gerstenhaber algebra structure on the sheaf of polydifferential operators on a
smooth algebraic variety, a complex manifold, and a smooth real manifold.
