Lattice and Heegaard-Floer homologies of algebraic links

TL;DR

This paper establishes a deep connection between Heegaard-Floer link homology, lattice homology, and algebraic invariants of plane curve singularities, revealing that their Poincaré polynomials coincide and relate to motivic Poincaré series.

Contribution

It identifies four different homologies associated with algebraic links and proves their equivalence in terms of Poincaré polynomials and motivic series.

Findings

Homologies of algebraic links are shown to have identical Poincaré polynomials.

The homologies correspond to the motivic Poincaré series of the singularity.

The paper links topological, algebraic, and motivic invariants of plane curve singularities.

Abstract

We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a certain variant of the Orlik-Solomon algebra of these local arrangements. In particular, the Poincar\'e polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic Poincar\'e series of the singularity.