Hochschild Cohomology of Cubic Surfaces

TL;DR

This paper computes the Hochschild homology and cohomology of cubic surfaces defined by a specific polynomial, using Kontsevich's method, Groebner bases, and computer algebra tools.

Contribution

It develops a method combining Kontsevich's approach with Groebner bases for explicit Hochschild (co)homology calculations of cubic surfaces.

Findings

Explicit Hochschild homology and cohomology for the cubic surface.

Application of Groebner bases and Maple for computations.

Extension of Kontsevich's method to this class of algebraic varieties.

Abstract

We consider the polynomial algebra $\mathbb{C}[\mathbf{z}]:=\mathbb{C}[z_1,\,z_2,\,z_3]$ and the polynomial $f:=z_1^3+z_2^3+z_3^3+3qz_1z_2z_3$, where $q\in \mathbb{C}$. Our aim is to compute the Hochschild homology and cohomology of the cubic surface $$\mathcal{X}_f:=\{\mathbf{z}\in\mathbb{C}^3\ /\ f(\mathbf{z})=0\}.$$ For explicit computations, we shall make use of a method suggested by M. Kontsevich. Then, we shall develop it in order to determine the Hochschild homology and cohomology by means of multivariate division and Groebner bases. Some formal computations with Maple are also used.