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.