Analytic lattice cohomology of isolated curve singularities

TL;DR

This paper introduces a new lattice cohomology framework for complex isolated curve singularities, linking algebraic invariants to topological structures and analyzing their behavior under deformations.

Contribution

It constructs a novel lattice cohomology and graded root for isolated curve singularities, connecting algebraic invariants with topological and combinatorial structures.

Findings

Lattice cohomology's Euler characteristic equals the delta-invariant.

Deformation induces explicit morphisms between cohomologies.

Examples include Gorenstein and Newton non-degenerate curves.

Abstract

We construct a lattice cohomology ${\mathbb H}^*(C,o)=\oplus_{q\geq 0}{\mathbb H}^q(C,o)$ and a graded root ${\mathfrak R}(C,o)$ to any complex isolated curve singularity $(C,o)$. Each ${\mathbb H}^q(C,o)$ is a ${\mathbb Z}$-graded ${\mathbb Z}[U]$-module. The Euler characteristic of ${\mathbb H}^*(C,o)$ is the delta-invariant of $(C,o)$. The construction is based on the multivariable Hilbert series of the multifiltration provided by valuations of the normalization. Several examples are discussed, e.g. Gorenstein curves (where an additional symmetry is established), plane curves (in particular, Newton non-degenerate ones), ordinary $r$-tuples. We also prove that a flat deformation $(C_t,o)_{t\in ({\mathbb C},0)}$ of isolated curve singularities induces an explicit degree zero graded ${\mathbb Z}[U]$-module morphism ${\mathbb H}^0(C_{t=0},o)\to {\mathbb H}^0(C_{t\not=0},o)$, and a graded (graph) map of degree zero at the level of graded roots ${\mathfrak R}(C_{t\not=0},o)\to{\mathfrak R}(C_{t=0},o)$. In the treatment of the deformation functor we need a second construction of the lattice cohomology in terms of the system of linear subspace arrangements associated with the above filtration.