Let A be a differential graded algebra. Its cohomological Hochschild complex C(A,A) is a double complex, one differential being induced by the differential in A and the other one is the Hochschild differential. Any double complex admits two totalizations, one given by direct sums and the other – by direct products. The ordinary Hochschild complex is formed by taking the direct product totalization. The Hochschild complex obtained by the direct sum totalization was considered by Positselski and Polischchuk and they called it the Hochschild complex of the second kind. Their goal was to prove that the Hochschild cohomology of A of the second kind is isomorphic to the ordinary Hochschild cohomology (of the first kind) of the category of cofibrant perfect A-modules; they proved that this is true under certain technical assumptions on A.

In this talk I will consider another version of the Hochschild cohomology of the second kind, which is lies between the Positselski-Polischchuk Hochschild cohomology and the ordinary Hochschild cohomology. Its definition is not as elementary as that of Positselski-Polishchuk but it has better formal properties. We consider two examples giving geometrically meaningful results: one when A is the Dolbeault algebra of a smooth complex projective variety and the other when A is the de Rham algebra of a smooth manifold. In the first case the Hochschild cohomology is isomorphic to the Hodge cohomology of the variety, and in the second it leads to the string homology of the manifold.