Golod-Shafarevich-Vinberg type theorems and finiteness conditions for potential algebras

TL;DR

This paper extends Golod-Shafarevich-Vinberg theorems to potential algebras, providing lower bounds for their Hilbert series and characterizing conditions for finiteness or linear growth, especially in non-homogeneous cases.

Contribution

It introduces a new lower estimate for the Hilbert series of Jacobi algebras in the potential case, including non-homogeneous situations, and characterizes when such algebras are finite dimensional or of linear growth.

Findings

Lower bounds for Hilbert series of Jacobi algebras established.

Finite dimensionality or linear growth only occurs in two-variable, degree-three potential case.

Results have implications for noncommutative singularities and deformation theory.

Abstract

We obtain a lower estimate for the Hilbert series of Jacobi algebras and their completions by providing analogue of the Golog-Shafarevich-Vinberg theorem for potential case. We especially treat non-homogeneous situation. This estimate allows to answer number of questions arising in the work of Wemyss-Donovan-Brown on noncommutative singularities and deformation theory. In particular, we prove that the only case when a potential algebra or its completion could be finite dimensional or of linear growth, is the case of two variables and potential having terms of degree three.