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Maybe it is helpful to look at these lectures by Yekutieli: arXiv:0801.3233. There he discusses twisted deformations of algebraic varieties $X$ suggested by Kontsevich, i.e. deformations of the category of coherent sheaves on $X$. These deformations are classified by the second Hochschild cohomology of $X$, which by the Hochschild-Kostant-Rosenberg theorem is $H^0(\wedge^2T)\oplus H^1(T)\oplus H^2({\mathcal O})$, where ${\mathcal O}$ is the structure sheaf and $T$ is the tangent sheaf of $X$.
Maybe it is helpful to look at these lectures by Yekutieli: arXiv:0801.3233. There he discusses twisted deformations of algebraic varieties $X$ suggested by Kontsevich, i.e. deformations of the category of coherent sheaves on $X$. These deformations are classified by the second Hochschild cohomology of $X$, which by the Hochschild-Kostant-Rosenberg theorem is $H^0(\wedge^2T)\oplus H^1(T)\oplus H^2({\mathcal O})$, where ${\mathcal O}$ is the structure sheaf and $T$ is the tangent sheaf of $X$.