Analytic lattice cohomology of surface singularities, II (the equivariant case)

Ágoston, Tamás; Némethi, András

doi:10.48550/arxiv.2108.12429

Cited by 3 publications

(6 citation statements)

References 27 publications

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“…Example 3.1.5. The 'Combinatorial Duality Property' for the analytic lattice cohomology, case n ≥ 2, was verified in [1,2,3]. The 'Combinatorial Duality Property' in the context of the topological lattice cohomology of normal surface singularities usually is not true.…”

Section: Combinatorial Lattice Cohomologymentioning

confidence: 99%

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Analytic lattice cohomology of isolated curve singularities

Ágoston¹,

Némethi²

2023

Preprint

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We construct a lattice cohomology H * (C, o) = ⊕ q≥0 H q (C, o) and a graded root R(C, o) to any complex isolated curve singularity (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 (Ct, o) t∈(C,0) of isolated curve singularities induces an explicit degree zero graded Z[U ]-module morphism H 0 (Ct=0, o) → H 0 (C t =0 , o), and a graded (graph) map of degree zero at the level of graded roots R(C t =0 , o) → R(Ct=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 arrangement associated with the above filtration.

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Section: Combinatorial Lattice Cohomologymentioning

confidence: 99%

“…[30,28]), and in agreement with the analytic lattice cohomologies for n ≥ 2 (cf. [1,2,3]), in this case we will have the coincidence of the Euler characteristic of any path lattice cohomology and of the lattice cohomology, cf. Corollary 4.2.2.…”

mentioning

confidence: 99%

Analytic lattice cohomology of isolated curve singularities

Ágoston¹,

Némethi²

2023

Preprint

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“…Indeed, e.g. for the Brieskorn (normal, minimally elliptic surface) singularities of type (2, 3, 7) and (2,3,11) have the same graded root, but their spectrum in (0, 1) are different. Both have only one spectral number in (0, 1), they are 41/42 and 61/66 respectively.…”

Section: H N−1 (O Ementioning

confidence: 99%

“…1.1.2. Recently, in [1,2] we introduced their analytic analogues, the analytic lattice cohomology H * an = ⊕ q≥0 H q an , associated with a normal surface singularity with a rational homology sphere link. It is constructed from analytic invariants of a good resolution, however it turns out that it is independent of the choice of the resolution.…”

Section: Introductionmentioning

confidence: 99%

The analytic lattice cohomology of isolated singularities

Ágoston¹,

Némethi²

2021

Preprint

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We associate (under a minor assumption) to any analytic isolated singularity of dimension n ≥ 2 the 'analytic lattice cohomology' H * an = ⊕ q≥0 H q an . Each H q an is a graded Z[U]-module. It is the extension to higher dimension of the 'analytic lattice cohomology' defined for a normal surface singularity with a rational homology sphere link. This latest one is the analytic analogue of the 'topological lattice cohomology' of the link of the normal surface singularity, which conjecturally is isomorphic to the Heegaard Floer cohomology of the link.The definition uses a good resolution X of the singularity (X,o). Then we prove the independence of the choice of the resolution, and we show that the Euler characteristic of H * an is h n−1 (O X ). In the case of a hypersurface weighted homogeneous singularity we relate it to the Hodge spectral numbers of the first interval.

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“…The analytic version H * an (X , o) of the lattice cohomology associated with a normal surface singularity was defined in [1,2], and even a graded Z[U ]-module morphism H * an (X , o) → H * top (X , o) was provided. In this analytic case the Euler characteristic equals the geometric genus of the germ.…”

Section: Introductionmentioning

confidence: 99%

Lattice cohomology and subspace arrangements: the topological and analytic cases

Ágoston

2024

Period Math Hung

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In this paper we consider the (topological) lattice cohomology $$\mathbb {H}^*$$ H ∗ of a surface singularity with rational homology sphere link. In particular, we will be studying two sets of (topological) invariants related to it: the weight function $$\upchi $$ χ that induces the cohomology and the topological subspace arrangement $$T(\ell ,I)$$ T ( ℓ , I ) at each lattice point $$\ell $$ ℓ — the latter of which is the weaker of the two. We shall prove that the two are in fact equivalent by establishing an algorithm to compute $$\upchi $$ χ from the subspace arrangement. Replacing the topological arrangements with the analytic, we get another formula — one that connects them with the analytic lattice cohomology introduced and studied in our earlier papers. In fact, this connection served as the original motivation for the definition of the latter. Aside from the historical interest, this parallel also provides us with tools to study more easily the connection between the two cohomologies.

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Analytic lattice cohomology of surface singularities, II (the equivariant case)

Cited by 3 publications

References 27 publications

Analytic lattice cohomology of isolated curve singularities

Analytic lattice cohomology of isolated curve singularities

The analytic lattice cohomology of isolated singularities

Lattice cohomology and subspace arrangements: the topological and analytic cases

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