Golod-Shafarevich type theorems and potential algebras

TL;DR

This paper develops new techniques using Gr"obner bases and Golod-Shafarevich theorems to analyze the finiteness and growth of potential algebras, providing classifications, bounds, and confirming conjectures in the field.

Contribution

It introduces novel methods to determine finiteness conditions of potential algebras, classifies degree 3 potentials, and proves growth properties for homogeneous potentials.

Findings

Two-generated potential algebras cannot have dimension smaller than 8.

Potential algebras with potentials of degree ≥5 are infinite dimensional.

Potential algebras with homogeneous potential of degree n≥3 are infinite dimensional.

Abstract

Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gr\"obner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gr\"obner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss \cite{Wemyss}, related to the geometric argument of Toda \cite{T}. We derive from the improved version of the Golod-Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree $n\geq 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class ${\cal P}_n$ of potential algebras with homogeneous potential of degree $n+1\geq 4$, the minimal Hilbert series is $H_n=\frac{1}{1-2t+2t^n-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but non-linear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invariants.