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arXiv · 2410.00551

Structural properties of the lattice cohomology of curve singularities

Alexander A. Kubasch·András Némethi·Gergő Schefler
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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.

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Alexander A. Kubasch, András Némethi, Gergő Schefler. 2024-10-01. Structural properties of the lattice cohomology of curve singularities. https://arxiv.org/abs/2410.00551

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