Hochschild Homology and Cohomology of Klein Surfaces

TL;DR

This paper computes Hochschild homology and cohomology for algebraic varieties, specifically singular plane curves and Klein surfaces, using complex analysis and Groebner bases within deformation quantization.

Contribution

It provides new calculations of Hochschild (co)homology for Klein surfaces and singular curves, refining previous results with novel methods.

Findings

Hochschild homology and cohomology for Klein surfaces are explicitly determined.

A new approach using Kontsevich's complex and Groebner bases is developed.

Results refine and extend earlier work on singular plane curves.

Abstract

Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem.