Lattice and Heegaard Floer Homologies of Algebraic Links

We compute the Heegaard Floer link homology of algebraic links in terms of the multi-variate 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, (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 generalized version of the Orlik–Solomon algebra of these local arrangements. In particular, the Poincaré polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic Poincaré series of the singularity.