Structural properties of the lattice cohomology of curve singularities

TL;DR

This paper investigates the lattice cohomology of reduced curve singularities, revealing its structure, relation to Gorenstein properties, and how it determines multiplicity, thus providing new insights into the geometric and algebraic features of singularities.

Contribution

The paper establishes three key structure theorems for the lattice cohomology of curve singularities, linking it to Gorenstein conditions and multiplicity determination.

Findings

Weight-grading of reduced cohomology is nonpositive.

${ m H}^0$ module structure indicates Gorenstein property.

${ m H}^0$ module determines the singularity's multiplicity.

Abstract

The lattice cohomology of a reduced curve singularity is a bigraded ${\mathbb Z}[U]$-module ${\mathbb H}^*=\oplus_{q,n}{\mathbb H}^q_{2n}$, that categorifies the $\delta$-invariant and extract key geometric information from the semigroup of values. In the present paper we prove three structure theorems for this new invariant: (a) the weight-grading of the reduced cohomology is (just as in the case of the topological lattice cohomology of normal surface singularities) nonpositive; (b) the graded ${\mathbb Z}[U]$-module structure of ${\mathbb H}^0$ determines whether or not a given curve is Gorenstein; and finally (c) the lattice cohomology module ${\mathbb H}^0$ of any plane curve singularity determines its multiplicity.