Hochschild cohomology of algebras of differential operators tangent to a central arrangement of lines

TL;DR

This paper investigates the Hochschild cohomology of differential operator algebras associated with line arrangements, revealing their structure, symmetries, and connections to geometric invariants of the arrangement complement.

Contribution

It computes the Hochschild cohomology of these algebras, links it to de Rham cohomology, and classifies the algebras up to isomorphism, providing new insights into their structure.

Findings

Hochschild cohomology as a Gerstenhaber algebra is explicitly determined.

A connection between Hochschild cohomology and the de Rham cohomology of the arrangement complement is established.

The automorphism group of the algebra is characterized and classification results are obtained.

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 $\mathcal D(\mathcal A)$ of differential operators on $V$ which are logarithmic along $\mathcal A$. Among other things we determine the Hochschild cohomology of $\mathcal D(\mathcal A)$ 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 $\mathcal D(\mathcal A)$ and classify the algebras of that form up to isomorphism.