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Let $D$ be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an $n$-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of $D$ as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology $H{H}^{2n}({𝒟}_n,{𝒟}_n^{*})$ of the algebra of differential operators on a formal neighbourhood of a...