@article {Fronsdal:February 2007:0377-9017:109,
	author = "Fronsdal, Christian",
	author = "Kontsevich, Maxim",
	title = "Quantization on Curves: With an Appendix by Maxim Kontsevich",
	journal = "Letters in Mathematical Physics",
	volume = "79",
	year = "February 2007",
	abstract = "Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over &#60;EquationSource Format="TEX"&#62;&#60;![CDATA[&#36;&#36;mathbb{C}&#36;&#36;]]&#62;&#60;/EquationSource&#62; , being algebraic varieties of the form &#60;EquationSource Format="TEX"&#62;&#60;![CDATA[&#36;&#36;{mathbb{C}}^2/R&#36;&#36;]]&#62;&#60;/EquationSource&#62; , where <I>R</I> is a polynomial in two variables; that is, abelian deformations of the coordinate algebra &#60;EquationSource Format="TEX"&#62;&#60;![CDATA[&#36;&#36;mathbb{C}[x,y]/(R&#36;&#36;]]&#62;&#60;/EquationSource&#62; ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane curves &#60;EquationSource Format="TEX"&#62;&#60;![CDATA[&#36;&#36;mathbb{C}[x,y]/R&#36;&#36;]]&#62;&#60;/EquationSource&#62; , but the cohomology depends on the local algebra of the singularity of <I>R</I> at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.",
	pages = "109-129(21)",
	url = "http://www.ingentaconnect.com/content/klu/math/2007/00000079/00000002/00000137"
	doi = "doi:10.1007/s11005-006-0137-8"
}