Lattice cohomology and the embedded topological type of plane curve singularities
Alexander A. Kubasch, Gergő Schefler
TL;DR
This work develops analytic lattice cohomology for curve singularities and proves that, for irreducible plane curve singularities, the invariant $\mathbb{H}^0(C,o)$ fully encodes the embedded topological type and enables reconstruction of the semigroup $\mathcal{S}_{C,o}$ from the ${\mathbb Z}[U]$-module structure; it also clarifies the limitations of lattice cohomology as a complete invariant in multi-branch settings and contrasts it with Seifert forms and the multivariate Poincaré series. The authors introduce graded roots and local minima as crucial tools, outline an algorithm to extract initial semigroup data from $\mathbb{H}^0(C,o)$, and provide examples demonstrating both the strengths (planar irreducibles) and limits (multi-branch or higher codimension) of the invariant. They also compare lattice cohomology to the Seifert form, showing non-determinacy in general, and connect the invariant to the multivariate Poincaré series, illustrating its sufficiency in planar cases but insufficiency in higher codimensions. Overall, the paper advances understanding of how analytic invariants of curve singularities reflect embedded topology and identifies precise boundaries of the invariant’s reconstruction power.
Abstract
Analytic lattice cohomology is a new invariant of reduced curve singularities. In the case of plane curves, it is an algebro-geometric analogue of Heegaard Floer Link homology. However, by the rigidity of the analytic structure, lattice cohomology can be naturally defined in higher codimensions as well. In this paper we show that in the case of irreducible plane curve singularities the lattice cohomology is a complete embedded topological invariant. We also compare it to the integral Seifert form in the case of multiple branches.