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            <title xml:lang="en">Hochschild Cohomology of Cubic Surfaces</title>
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                <forename type="first">Frédéric</forename>
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            <idno type="stamp" n="UNIV-LYON1">Université Claude Bernard - Lyon I</idno>
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                <term xml:lang="en">Hochschild cohomology</term>
                <term xml:lang="en">Hochschild homology</term>
                <term xml:lang="en">cubic surfaces</term>
                <term xml:lang="en">Groebner bases</term>
                <term xml:lang="en">algebraic resolution</term>
                <term xml:lang="en">quantization</term>
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              <p>We consider the polynomial algebra $\mathbb{C}[\mathbf{z}]:=\mathbb{C}[z_1,\,z_2,\,z_3]$ and the polynomial $f:=z_1^3+z_2^3+z_3^3+3qz_1z_2z_3$, where~$q\in \mathbb{C}$. Our aim is to compute the Hochschild homology and cohomology of the cubic surface $$\mathcal{X}_f:=\{\mathbf{z}\in\mathbb{C}^3\ /\ f(\mathbf{z})=0\}.$$ For explicit computations, we shall make use of a method suggested by M. Kontsevich. Then, we shall develop it in order to determine the Hochschild homology and cohomology by means of multivariate division and Groebner bases. Some formal computations with Maple are also used.</p>
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