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András Némethi

Publications and source records attributed to András Némethi.

At least 19 recordsLinked to original sources
arXiv2026

Splice formulae for Poincaré series of integral homology spheres

Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph $Γ$. One of its most important invariants is its multivariable Poincaré series (or zeta function) $f_Γ(\mathbf {t})$. Several numerical invariants can be read from $f_Γ(\mathbf{t})$, or even from its `polynomial part' $ \operatorname{Pol}_Γ(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_Γ(1)$. In this note we provide several splice (surgery) formulae for $f_Γ$ and $\operatorname{Pol}_Γ$ (reduced to the node variables, or to the node variables of connected sub-graphs). In this way, these global invariants can be recovered from a collection of certain smaller graphs. Recall that some (integral homology sphere) 3-manifold invariants are additive with respect to the splice decomposition (like the Casson's invariant). However, some invariants need some `splice correction terms'. In our formulae the correction terms are easily computable one variable Alexander polynomials.

math.GT↗
arXiv2026

Lattice homology of integrally closed submodules and Artin algebras

The general construction of lattice (co)homology assigns to a lattice $\mathbb{Z}^r$ and a weight function $w:\mathbb{Z}^r \to \mathbb{Z}$ a bigraded $\mathbb{Z}[U]$-module $\mathbb{H}_*$. The weight function $w$ is often obtained from some geometric data as the difference of two `height functions'. In this paper we consider the case when these height functions are Hilbert functions of valuative multifiltrations on a Noetherian $k$-algebra $\mathcal{O}$ and a finitely generated $\mathcal{O}$-module $M$. We introduce the notion of `realizable submodules' in $M$, the prime example of which are finite codimensional integrally closed submodules in the sense of Rees (or integrally closed ideals when $M=\mathcal{O}$). We prove, that whenever two sets of `extended' discrete valuations `realize' the same submodule $N \leq M$, then, although the corresponding lattices and weight functions might be different, the resulting lattice homology modules are isomorphic and have Euler characteristic $\dim_k(M/N)$. In this way, we associate a well-defined lattice homology to any quotient of type $M/N$, where $N$ is a realizable submodule of $M$. We also present some structural and computational results: e.g., we geometrically characterize the (lattice) homological dimension of integrally closed monomial ideals of $k[x,y]$. The main upshot of the paper, however, is the possibility of categorifying numerical invariants defined as codimensions of realizable submodules or integrally closed ideals. The geometric applications include: the delta invariant $δ(C, o)$ of a reduced curve singularity; the geometric genus $p_g(X, o)$, the irregularity $q(X, o)$ and the various plurigenera of higher dimensional isolated normal singularities. The corresponding categorifications generalize the analytic lattice homologies of Ágoston and the first author.

math.AG↗
arXiv2025

Delta invariant of $\mathbb{Q}$-Cartier curve germs and the genus of representable numerical semigroups

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider representable numerical semigroups, they are semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links (via the degrees of the homogeneous functions). We then prove that such a semigroup can be interpreted as the value semigroup of a generic orbit (as a curve singularity) given by the $\mathbb{C}^*$-action on the weighted homogeneous germ. Furthermore, we use the delta invariant formula to derive a combinatorially computable formula for the genus of representable semigroups. Finally, we characterize topologically those representable semigroups which are symmetric.

math.AG↗
arXiv2025

$Δ$ Invariants of Plumbed Manifolds

We study the minimal $q$-exponent $Δ$ in the BPS $q$-series $\widehat{Z}$ of negative definite plumbed 3-manifolds equipped with a spin$^{\rm c}$-structure. We express $Δ$ of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of $Δ$ for non-Seifert manifolds. Finally, we compare $Δ$ invariants with correction terms in Heegaard-Floer homology.

math.GT↗
arXiv2025

Cohen-Macaulay Type via Lattice Homology and the Motivic Poincaré Series

We give results on reduced complex-analytic curve germs which relate their indecomposable maximal Cohen-Macaulay (MCM) modules to their lattice homology groups and related invariants, thereby providing a connection between the algebraic theory of MCM modules and techniques arising from low-dimensional topology. In particular, we characterize the germs $(C, o)$ of finite Cohen-Macaulay type in terms of the lattice homology $\mathbb{H}_*(C, o)$, and those of tame type in terms of the lattice homologies and associated spectral sequences of $(C, o)$ and its subcurves, including the distinction between germs of finite and infinite growth. As a consequence of these results, we obtain corresponding characterizations of a germ's Cohen-Macaulay type in terms of the motivic Poincaré series.

math.AG↗
arXiv2024

Multiplier ideals of normal surface singularities

We study the multiplier ideals and the corresponding jumping numbers and multiplicities $\{m(c)\}_{c\in \mathbb{R}}$ in the following context: $(X,o)$ is a complex analytic normal surface singularity, ${\mathfrak a}\subset \mathcal{O}_{X,o}$ is an ${\mathfrak m}_{X,o}$--primary ideal, $ϕ:\widetilde{X}\to X$ is a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$, for some nonzero effective divisor $F$ supported on $ϕ^{-1}(0)$. We show that $\{m(c)\}_{c>0}$ is combinatorially computable from $F$ and the resolution graph $Γ$ of $ϕ$, and we provide several formulae. We also extend Budur's result (valid for $(X,o)=(\mathbb{C}^2,0)$), which makes an identification of $\sum_{c\in[0,1]}m(c)t^c$ with a certain Hodge spectrum. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that $\{m(c)\}_{c\leq 0}$ usually depends on the analytic type of $(X,o)$. However, for some distinguished analytic types we determine it concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated with the central vertex), we recover $\sum_cm(c)t^c$ from the Poincaré series of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover $\sum_cm(c)t^c$ from the multivariable topological Poincaré (zeta) function of $Γ$.

math.AG↗
arXiv2024

Structural properties of the lattice cohomology of curve singularities

The lattice cohomology of a reduced curve singularity is a bigraded ${\mathbb Z}[U]$-module ${\mathbb H}^*=\oplus_{q,n}{\mathbb H}^q_{2n}$, that categorifies the $δ$-invariant and extract key geometric information from the semigroup of values. In the present paper we prove three structure theorems for this new invariant: (a) the weight-grading of the reduced cohomology is (just as in the case of the topological lattice cohomology of normal surface singularities) nonpositive; (b) the graded ${\mathbb Z}[U]$-module structure of ${\mathbb H}^0$ determines whether or not a given curve is Gorenstein; and finally (c) the lattice cohomology module ${\mathbb H}^0$ of any plane curve singularity determines its multiplicity.

math.AG↗
arXiv2024

Filtrations associated with singularities

We fix a complex analytic normal singularity germ $(X,o)$ of dimension $\geq 2$ and a (not necessarily irreducible) reduced Weil divisor $(S,o)\subset (X,o)$. The embedded resolution of the pair determines a multi-index filtration of the local ring $\mathcal{O}_{X,o}$, which measures the embedded geometry of the pair. Furthermore, from the (induced) resolution of $(S,o)$ we also consider a multi-index filtration associated with $(S,o)$. This latter one can be lifted to a filtration of $\mathcal{O}_{X,o}$ too. The main result proves that the second filtration of $\mathcal{O}_{X,o}$ can be realized as a `limit' filtration of the first one (if we blow up certain centers sufficiently many times).

math.AG↗
arXiv2023

Filtered lattice homology of curve singularities

Let $(C,o)$ be a complex analytic isolated curve singularity of arbitrary large embedded dimension. Its lattice cohomology ${\mathbb H}^*=\oplus_{q\geq 0}{\mathbb H}^q$ was introduced by Ágoston and the author, each ${\mathbb H}^q$ is a graded ${\mathbb Z}[U]$--module. Here we study its homological version ${\mathbb H}_*(C,o)=\oplus_{q\geq 0}{\mathbb H}_q$. 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 $\{S_n\}_{n\in {\mathbb Z}}$ such that ${\mathbb H}_q=\oplus_n H_q(S_n,{\mathbb Z})$. In this article for every $n$ we consider a natural filtration of the space $S_n$, which provides a homological spectral sequence converging to the homogeneous summand $H_q(S_n,{\mathbb Z})$ 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.

math.AG↗
arXiv2023

Filtered lattice homology of surface singularities

Let $(X,o)$ be a complex analytic normal surface singularity with rational homology sphere link $M$. The `topological' lattice cohomology ${\mathbb H}^*=\oplus_{q\geq 0} {\mathbb H}^q$ associated with $M$ and with any of its spin$^c$ structures was introduced by the author. Each ${\mathbb H}^q$ is a graded ${\mathbb Z}[U]$--module. Here we consider its homological version ${\mathbb H}_*=\oplus_{q\geq 0}{\mathbb H}_q$. The construction uses a Riemann-Roch type weight function. A key intermediate product is a tower of spaces $\{S_n\}_{n\in {\mathbb Z}}$ such that ${\mathbb H}_q=\oplus_n H_q(S_n,{\mathbb Z})$. In this article we fix the embedded topological type of a reduced curve singularity $(C,o)$ embedded into $(X,o)$, that is, a 1-dimensional link $L_C\subset M$. Each component of $L_C$ will also carry a non-negative integral decoration. For any fixed $n$, the embedded link $L_C$ provides a natural filtration of the space $S_n$, which induces a homological spectral sequence converging to the homogeneous summand $H_q(S_n,{\mathbb Z})$ of the lattice homology. All the entries of all the pages of the spectral sequences are new invariants of the decorated $(M,L_C)$. Each page provides a triple graded ${\mathbb Z}[U]$-module. We provide several concrete computations of these pages and structure theorems for the corresponding multivariable Poincaré series associated with the entries of the spectral sequences. Connections with Jacobi theta series are also discussed.

math.AG↗
arXiv2023

The classification of reflexive modules of rank one over rational and minimally elliptic singularities

We classify the reflexive modules of rank one over rational and minimally elliptic singularities. Equivalently, we classify full line bundles on the resolutions of rational and minimally elliptic singularities. As an application, we determine among such reflexive modules of rank one all the special ones (in the sense of Wunram) and all the flat ones. In this way, we also classify the non-flat reflexive modules as well (as a generalization of a construction of Dan and Romano). In particular, we prove (in the rank one case) a conjecture of Behnke, namely that in the case of a cusp singularity any reflexive module admits a flat connection. The results generalize the classical Mckay correspondence, and results of Artin, Verdier, Esnault, Khan and Wunram valid for different particular families of singularities.

math.AG↗
arXiv2023

Analytic lattice cohomology of isolated curve singularities

We construct a lattice cohomology ${\mathbb H}^*(C,o)=\oplus_{q\geq 0}{\mathbb H}^q(C,o)$ and a graded root ${\mathfrak R}(C,o)$ to any complex isolated curve singularity $(C,o)$. Each ${\mathbb H}^q(C,o)$ is a ${\mathbb Z}$-graded ${\mathbb Z}[U]$-module. The Euler characteristic of ${\mathbb H}^*(C,o)$ is the delta-invariant of $(C,o)$. 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 $(C_t,o)_{t\in ({\mathbb C},0)}$ of isolated curve singularities induces an explicit degree zero graded ${\mathbb Z}[U]$-module morphism ${\mathbb H}^0(C_{t=0},o)\to {\mathbb H}^0(C_{t\not=0},o)$, and a graded (graph) map of degree zero at the level of graded roots ${\mathfrak R}(C_{t\not=0},o)\to{\mathfrak R}(C_{t=0},o)$. In the treatment of the deformation functor we need a second construction of the lattice cohomology in terms of the system of linear subspace arrangements associated with the above filtration.

math.AG↗
arXiv2021

The analytic lattice cohomology of isolated singularities

We associate (under a minor assumption) to any analytic isolated singularity of dimension $n\geq 2$ the `analytic lattice cohomology' ${\mathbb H}^*_{an}=\oplus_{q\geq 0}{\mathbb H}^q_{an}$. Each ${\mathbb H}^q_{an}$ is a graded ${\mathbb Z}[U]$--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 $\widetilde{X}$ of the singularity $(X,o)$. Then we prove the independence of the choice of the resolution, and we show that the Euler characteristic of ${\mathbb H}^*_{an}$ is $h^{n-1}({\mathcal O}_{\widetilde{X}})$. In the case of a hypersurface weighted homogeneous singularity we relate it to the Hodge spectral numbers of the first interval.

math.AG↗
arXiv2021

Normal reduction number of normal surface singularities

Let $(X,o)$ be a complex analytic normal surface singularity and let ${\mathcal O}_{X,o}$ be its local ring. We investigate the normal reduction number of ${\mathcal O}_{X,o}$ and related numerical analytical invariants via resolutions $\widetilde{X}\to X$ of $(X,o)$ and cohomology groups of different line bundles ${\mathcal L}\in {\rm Pic}(\widetilde{X})$. The normal reduction number is the universal optimal bound from which powers of certain ideals have stabilization properties. Here we combine this with stability properties of the iterated Abel maps. Some of the main results provide topological upper bounds for both stabilization properties.

math.AG↗
arXiv2021

Analytic lattice cohomology of surface singularities

We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it is a rational homology sphere. This topological lattice cohomology is the categorification of the Seiberg--Witten invariant, and conjecturally it is isomorphic with the Heegaard Floer cohomology. We compare the two lattice cohomologies: in some simple cases they coincide, but in general, the analytic cohomology is sensitive to the analytic structure. We expect a deep connection with deformation theory. We provide several basic properties and key examples, and we formulate several conjectures and problems. This is the initial article of a series, in which we develop the analytic lattice cohomology of singularities.

math.AG↗
arXiv2021

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

We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric genus of the germ. This is the analytic analogue of the topological lattice cohomology, associated with the link of the germ, and indexed by the spin$^c$--structures of the link (which is a categorification of the Seiberg--Witten invariant and conjecturally it is isomorphic with the Heegaard Floer cohomology).

math.AG↗
arXiv2021

Polar exploration of complex surface germs

We prove that the topological type of a normal surface singularity $(X,0)$ provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of $(X,0)$. A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of $(X,0)$, which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of $(X,0)$ through its Nash transform. This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted.

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