The Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of lines

Abstract

Given a central arrangement of lines \mathcal{A} in a 2 -dimensional vector space V over a field of characteristic zero, we study the algebra \mathscr{D}(\mathcal{A}) of differential operators on V which are logarithmic along \mathcal{A} . Among other things we determine the Hochschild cohomology of \mathscr{D}(\mathcal{A}) as a Gerstenhaber algebra, establish a connection between that cohomology and the de Rham cohomology of the complement M(\mathcal{A}) of the arrangement, determine the isomorphism group of \mathscr{D}(\mathcal{A}) and classify the algebras of that form up to isomorphism.