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Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poissonmanifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it isimportant to be able to quantize it, in order to obtain more precise resultsthan through classical mechanics. An available method is the deformationquantization, which consists in constructing a star-product on the algebra offormal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study ofstar-products is the calculation of Hochschild cohomology of $\mathcal{F}(M)$.The aim of this article is to determine this Hochschild cohomology in the caseof singular curves of the plane -- so we rediscover, by a different way, aresult proved by Fronsdal and make it more precise -- and in the case of Kleinsurfaces. The use of a complex suggested by Kontsevich and the help ofGr\"obner bases allow us to solve the problem.

Cohomologie De Hochschild Des Surfaces De Klein