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Peer-reviewedCC BYAdvances in Mathematics

Filtered lattice homology of curve singularities

  • András Némethi1,2,3,4
4 affiliations
  1. 1Alfréd Rényi Institute of Math., Reáltanoda utca 13-15, H-1053, Budapest, Hungary
  2. 2BBU - Babeş-Bolyai Univ., Str. M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania
  3. 3BCAM - Basque Center for Applied Math., Mazarredo, 14, E48009 Bilbao, Basque Country, Spain
  4. 4ELTE - Univ. of Budapest, Dept. of Geo., Pázmány Péter sétány 1/A, 1117, Budapest, Hungary
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Abstract

Let be a complex analytic reduced curve singularity of arbitrary large embedded dimension. Its lattice cohomology ⁎ was introduced in [2] , each is a graded –module. Here we study its homological version ⁎ . 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 topological spaces such that . In this article for every n we consider a natural filtration of the space , which provides a homological spectral sequence converging to the homogeneous summand of the lattice homology. All the entries of all the pages of the spectral sequences are new invariants of . We show that the collection of the ranks of the entries of the first pages is equivalent with the motivic Poincaré series of . 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 (formulated for links in the 3–sphere).

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Cite

BibTeXRIS
BibTeX
@article{nemethi2026filtered,
  title = {{Filtered lattice homology of curve singularities}},
  author = {András Némethi},
  journal = {Advances in Mathematics},
  year = {2026},
  month = aug,
  doi = {10.1016/j.aim.2026.111215},
  url = {https://doi.org/10.1016/j.aim.2026.111215},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Filtered lattice homology of curve singularities
AU  - András Némethi
PY  - 2026
DA  - 2026/08/25
JO  - Advances in Mathematics
DO  - 10.1016/j.aim.2026.111215
UR  - https://doi.org/10.1016/j.aim.2026.111215
AB  - Let be a complex analytic reduced curve singularity of arbitrary large embedded dimension. Its lattice cohomology ⁎ was introduced in [2] , each is a graded –module. Here we study its homological version ⁎ . 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 topological spaces such that . In this article for every n we consider a natural filtration of the space , which provides a homological spectral sequence converging to the homogeneous summand of the lattice homology. All the entries of all the pages of the spectral sequences are new invariants of . We show that the collection of the ranks of the entries of the first pages is equivalent with the motivic Poincaré series of . 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 (formulated for links in the 3–sphere).
SN  - 0001-8708
SN  - 1090-2082
ER  -