2023/06/24 by András Némethi, Némethi, András · 1 citation
Computer Science · Mathematics · #32S05 #32S10 #32S25 #57K10 #57K14 (Secondary) #57K18 (Primary) 14Bxx #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2306.13889
openalex publication_date 2023/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (C,o) be a complex analytic isolated curve singularity of arbitrary large embedded dimension. Its lattice cohomology \mathbb H^*=⊕q≥ 0\mathbb Hq was introduced by Ágoston and the author, each \mathbb Hq is a graded \mathbb Z[U]--module. Here we study its homological version \mathbb H_*(C,o)=⊕q≥ 0\mathbb Hq. The construction uses the multivariable Hilbert function associated with the valuations provided by the normalization of the curve. A key intermediate product is a tower of spaces \Sn\_n∈ \mathbb Z such that \mathbb Hq=⊕n Hq(Sn,\mathbb Z). In this article for every n we consider a natural filtration of the space Sn, which provides a homological spectral sequence converging to the homogeneous summand Hq(Sn,\mathbb Z) of the lattice homology. All the entries of all the pages of the spectral sequences are new invariants of (C,o). We show how the collection of the first pages is equivalent with the motivic Poincaré series of (C,o).We provide several concrete computations of the corresponding multivariable Poincaré series associated with the entries of the spectral sequences. In the case of plane curve singularities, the first page can also be identified with the Heegaard Floer Link homology of the link of the singularity. In this way, the new invariants provide for an arbitrary (non necessarily plane) singularity a homological theory which is the analogue of the Heegaard Floer Link theory for links of plane curve singularities.