SPECTRUM OF PLANE CURVES VIA KNOT THEORY

MACIEJ BORODZIK AND ANDRÁS NÉMETHI

Abstract. In this paper we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram– Levine signatures of links, and the associated Murasugi type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink and Varchenko on semicontinuity of the spectrum of singular points under deformations and result of Némethi and Sabbah on semicontinuity of the spectrum at infinity regarding families of polynomial maps. We also relate the spectrum at infinity of a polynomial map with the collection of the spectra of singular points of a chosen fiber.

1. Introduction

The Hodge spectrum of a local isolated hypersurface singularity f : (ℂ^{n+1}, 0) → (ℂ, 0) can be derived from the mixed Hodge structure of the vanishing cohomology of the singular germ [A, St1, St2, Var, Var2]. Usually, it is not topological, it is one of the finest analytic invariants of the germ. Although, it does not characterize the singularity completely, it gives extremely strong information about it. As it was conjectured by Arnold [A], and proved by Varchenko [Var, Var2] and Steenbrink [St2], the spectrum behaves semicontinuously under deformations, which makes it, for example, a very strong tool in attempts to solve the adjacency problem (i.e., to determine, which singularities can specialize to a given one).

A more precise picture is the following: the algebraic monodromy acts on the vanishing cohomology, which supports the Seifert form (which can be identified with the variation map) and the mixed Hodge structure polarized by the intersection form. The equivariant Hodge numbers were codified by Steenbrink in the spectral pairs; if one deletes the information about the weight filtration one gets the spectrum/spectral numbers Sp(f). They are (in some normalization) rational numbers in the interval (0, n + 1). In the presence of a deformation f_{t}, where t is the deformation parameter t ∈ (ℂ, 0), the semicontinuity guarantees that |Sp(f_{0}) ∩ I| ≥ |Sp(f_{t≠0}) ∩ I| for certain semicontinuity domains I. Arnold conjectured that I = (−∞, x] is a semicontinuity domain for any x ∈ ℝ, Steenbrink and Varchenko proved the statement for I = (x, x + 1], which implies Arnold’s conjecture. Additionally, for some cases, Varchenko verified the stronger version, namely semicontinuity for I = (x, x + 1) [Var].

The semicontinuity property (with any domain) cannot be extended to the spectral pairs. Therefore, in studies targeting these kind of applications one usually works with the spectrum only. This is what we will do in the present article as well.

On the other hand, one of the strongest topological invariants of f is its Seifert form, for terminology see e.g. [AGV]. The relation between the Hodge invariants and the Seifert form was established by the second author in [Nem2], proving that the collection of mod 2 spectral pairs are equivalent with the real Seifert form. In this way, the real Seifert form is in strong relationship with the mod 2 spectrum, that is with the collection of numbers x mod 2 in (0, 2], where x run over Sp(f). Clearly, for plane curve singularities, i.e. when n = 1, by taking mod 2 reduction we loose no information.

Date: December 31, 2011.

2010 Mathematics Subject Classification. primary: 32S55, secondary: 14B07, 14D07, 14H50, 32G20.

Key words and phrases. semicontinuity of the spectrum, Tristram–Levine signature, Seifert forms, variation structures, Mixed Hodge Structure, link at infinity, plane algebraic curve.

The first author is supported by Polish MNiSzW Grant No N N201 397937 and also by a Foundation for Polish Science FNP. The second author is partially supported by OTKA Grant K67928.

Our primary goal is to extend the above correspondence for an arbitrary link (S_{R}^{3}, L), where S_{R}^{3} is the boundary of some ball with radius R in ℂ^{2}, and L is the intersection of S_{R}^{3} with some affine algebraic curve C in ℂ^{2}. The primary interest is the link at infinity of such affine curve (hence R ≫ 0), but we also wish to develop a method to study any general (S_{R}^{3}, L), for which the available methods in the literature are rather sparse.

Let us consider a complex polynomial map F : ℂ^{2} → ℂ. For its topology at infinity see Neumann’s article [Neu3]. Our first main result recovers the spectrum at infinity associated with the limit mixed Hodge structure at infinity (supported by the cohomology of the generic fiber) from the real Seifert form of the regular link at infinity associated with F. In particular, we reobtain the spectrum at infinity topologically in a pure link–theoretical language.

The key bridge which connects the link–theoretical language and invariants with the Hodge theoretical spectrum are the Tristram–Levine signatures [Tri, Le]. For example, for the weighted homogeneous singularity given by {x^{p} − y^{q} = 0} with p and q relative prime integers, the spectrum is Sp_{p,q} = \{\frac{i}{p} + \frac{j}{q} , 1 ≤ i ≤ p − 1, 1 ≤ j ≤ q − 1\}, while the Tristram–Levine signature function of the (p, q)−torus knot, evaluated at e^{2πix} with x ∈ (0, 1), pqx ∉ ℤ, is equal to 2 · |Sp_{p,q} ∩ (x, x + 1)| − (p − 1)(q − 1), see e.g. [Li]. In [BN] we made this relation rigorous, showing a direct translation between the spectrum of singularities and Tristram–Levine signatures of their links.

In this correspondence, what is really surprising — and this is the seconds main message of the article — is the fact that the semicontinuity of the mod 2 spectrum is topological: it can be recovered independently of analytic (Hodge theoretical) tools, it follows from pure link theory. More precisely, we prove that length one ‘intervals’ intersected by the mod 2 spectrum, namely sets of type Sp∩(x, x+1) and (Sp∩(0, x))∪(Sp∩(x+1, 2]), for x ∈ [0, 1], satisfy semicontinuty properties, whenever this question is well–posed.

In this article we exemplify this by three cases: we recover the semicontinuity (in the above form, with certain weak assumptions) for deformations of local plane curve singularities, corresponding to the above mentioned results of Varchenko and Steenbrink, and we also prove the semicontinuity of the spectrum at infinity associated with a family of polynomials in two variables, in the spirit of [NS]. The third case targets a new phenomenon: in the context of an affine curve C ⊂ ℂ^{2} we show a semicontinuity property which connects the local spectrum of the singularities of C with the spectrum at infinity of C.

In all these cases, the key link–theoretical ingredient is a Murasugi type inequality, which controls the modification of the Tristram–Levine signatures under those type of surgeries which appears when we pass from C ∩ S_{r}^{3} to C ∩ S_{R}^{3} via Morse theory (r < R). This was studied by the first author in [Bo].

The article is organized as follows. In section 2 we review the theory of hermitian variation structures from [Nem2], their relation with the spectrum, and the methods how one associates such a structure to a link [BN]; furthermore, we connect the spectrum with the Tristram–Levine signatures [BN]. We also recall some of the main results of [Bo] about surgery inequalities of links of type S_{R}^{3} ∩ C. Section 3 contains the study of the spectrum at infinity of a polynomial map in terms of the Seifert form at infinity. In section 4 we prove semicontinuity results regarding the spectrum.

For a finite set A, we denote by |A| the cardinality of A.

Acknowledgements. The main part of the paper was accomplished during the stay of the first author at Alfréd Rényi Institute in Budapest. The first author expresses his thanks for the hospitality. His stay was supported by Foundation for Polish Science (program KOLUMB) and by Polish MNiSzW Grant No N N201 397937.

2. Hermitian variations structures of links

In subsection 2.1 we recall the definition of an abstract hermitian variation structure and its spectrum, while in 2.2 the definition of the hermitian variation structure and spectrum associated with links in a three-sphere. Subsection 2.3 reviews the definition of mixed Hodge structures and their ‘Hodge’ spectrum. Finally, in 2.4 we draw a relationship observed in [BN] between the spectrum and Tristram–Levine signatures of links. In 2.5 we recall some results from [Bo] which are crucial ingredients in the proof of the semicontinuity results of the last section.

2.1. Hermitian variation structures were introduced in [Nem2], they generalize the ε–symmetric isometric structures. Here we review the minimal basics, for more detail see [Nem2, Nem3].

Recall that a structure (U = ℂ^{n}; b, h), where b is an ε–symmetric hermitian form on U preserved by the automorphism h of U, is called an isometric structure (for ε = ±1). The classification of isometric structures when b is non–degenerate was established by Milnor [Mil] (see also [Neu1, Neu2]). Any ε–hermitian variation structure (in short ε–HVS) can be regarded as an isometric structure together with an operator V : U^{∗} → U such that

(2.1.1)  \overline{V^{∗}} = −εV \overline{h^{∗}} and V ∘ b̃ = h − Id,

where b̃ is the form b regarded as map from U to U^{∗}. We denote it by V = (U; b, h, V). Here ·^{∗} denotes passing to the dual space, while \overline{·} the complex conjugation.

Definition 2.1.2. We say that the isometric structure (U; b, h) can be completed to a hermitian variation structure, if there exists V : U^{∗} → U such that (2.1.1) is satisfied.

If b is non–degenerate, then the isometric structure can be uniquely completed to a HVS: V = (h − Id) ∘ b̃^{−1}. In general, not every isometric structures can be completed (see e.g. (3.2.8)(c) below). Moreover, if a completion exists, in general, it is not unique (even if we restrict ourselves to non–degenerate matrices V, see e.g. [Nem2, (2.7.7)]).

A HVS is called simple if V is an isomorphism. The classification of simple HVS’s is established in [Nem2]. Each simple variation structure is a direct sum of indecomposable simple variation structures. Indecomposable structures can be listed: for each positive integer k, and for each λ ∈ ℂ such that 0 < |λ| ≤ 1 we have

• a unique simple indecomposable variation structure V_{λ}^{2k} if |λ| < 1;

• two simple indecomposable structures, denoted by W_{λ}^{k}(+1) and W_{λ}^{k}(−1), if |λ| = 1.

This classification is a refinement of the Jordan block decomposition of the matrix h (or of Milnor’s classification of non–degenerate isometric structures). More precisely, the matrix h corresponding to W_{λ}^{k}(±1) is a single Jordan block of size k and eigenvalue λ, while the one corresponding to V_{λ}^{2k} has two Jordan blocks of size k: one with eigenvalue λ, the other with eigenvalue 1/\overline{λ}. For their precise form see [Nem2].

Let us write a simple variation structure V as the unique sum of the indecomposable ones:

(2.1.3)  V = ⨁_{\begin{matrix}0<|λ|<1 \\ k≥1 \end{matrix}} q_{λ}^{k} · V_{λ}^{2k} ⊕ ⨁_{\begin{matrix}|λ|=1 \\ k≥1, u=±1 \end{matrix}} p_{λ}^{k}(u) · W_{λ}^{k}(u)

for certain non–negative integers q_{λ}^{k} and p_{λ}^{k}(u). Here we write m·V for V ⊕⋯⊕V (m-times). The numbers {q_{λ}^{k}}_{|λ|<1} and {p_{λ}^{k}(±1)}_{λ∈S^{1}} are called the H–numbers of the HVS V. Using H–numbers we can define the spectrum of V. Sometimes, in order to emphasize

the source of the definition, we call it HVS–spectrum.

Definition 2.1.4. ([Nem1] or [BN, (2.3.1)-(2.3.3)]) Consider the H–numbers {q_{λ}^{k}}_{|λ|<1} and {p_{λ}^{k}(±1)}_{λ∈S^{1}} of V. The extended spectrum ESp is the union ESp = Sp ∪ ISp, where

(a) Sp, the spectrum, is a finite set of real numbers from the interval (0, 2] such that any real number α occurs in Sp precisely s(α) times, where

s(α) = ∑_{n=1}^{∞} ∑_{u=±1} (\frac{2n − 1 − u(−1)^{⌊α⌋}}{2} p_{λ}^{2n−1}(u) + np_{λ}^{2n}(u)) , (e^{2πiα} = λ).

(b) ISp is a subset of complex numbers from (0, 2]×iℝ, ISp∩ℝ = ∅, such that z = α+iβ occurs in ISp precisely s(z) times, where

s(z) = \begin{cases}∑ k · q_{λ}^{k} & if α ≤ 1, β > 0 and e^{2πiz} = λ \\ ∑ k · q_{λ}^{k} & if α > 1, β < 0 and e^{2πiz} = 1/λ̄ \\ 0 & if α ≤ 1 and β < 0, or α > 1 and β > 0. \end{cases}

Since the dimension of the vector space supporting V_{λ}^{2k} is 2k and of W_{λ}^{k}(±1) is k, one gets

(2.1.5)  |ESp| = dim U = deg det(h − t Id).

2.2. The HVS and the spectrum of a link. The variation structure and H–numbers of a link in S^{3} were defined in [BN]. Let us review shortly how the construction is performed.

Let S be a Seifert matrix of a link L. (For the conventions used in its definition see 3.2.) By Keef’s result [Keef] S is S–equivalent either to an empty matrix, or to a matrix S′, which can be decomposed into a direct sum

(2.2.1)  S′ = S_{0} ⊕ S_{ndeg},

where S_{0} is a zero matrix and S_{ndeg} is non–degenerate, that is det S_{ndeg} ≠ 0. Moreover, any two such non–degenerate models S_{ndeg} of the same link are congruent over ℚ. The size of S_{0} is also determined by L (it is equal to dim(ker S ∩ker S^{T})), we will call it the irregularity of L, and we will denote it by

(2.2.2)  Irr = Irr(L) := size(S_{0}).

Let n be the size of S_{ndeg}. The quadruple V = (U, b, h, V), where U = ℂ^{n}, V = (S_{ndeg}^{T})^{−1}, h = V S, b = S − S^{T}, constitutes a HVS with the sign choice ε = −1. (Here ·^{T} denotes the transposition.) As changing a Seifert matrix results in congruency of S_{ndeg}, which leads to an isomorphism of variations structures, the structure V does not depend on the choice of a Seifert matrix, so it is a well-defined link invariant, called V_{L}. Additionally, V_{L} is simple. Note that V_{L} is defined over the rational numbers ℚ. The characteristic polynomial Δ^{h} = det(h−t Id) of h will be called the characteristic polynomial of the link. Its connection with Alexander polynomial is as follows (see e.g. [BN, §4]):

Lemma 2.2.3. Let V_{L} be as above. If the Alexander polynomial Δ of L is non–zero then Δ = Δ^{h} up to multiplication by an invertible element of ℚ[t, t^{−1}]. If the Alexander polynomial is zero, then Δ^{h} is proportional to the first higher Alexander polynomial Δ_{k}, which is not identically zero: Δ_{k} = 0 for 0 ≤ k < Irr and Δ_{Irr} = Δ^{h} (up to an invertible element).

Definition 2.2.4. Consider the integers {q_{λ}^{k}}_{|λ|<1} and {p_{λ}^{k}(±1)}_{λ∈S^{1}} provided by the direct sum decomposition (2.1.3) of V_{L}. They are called the H–numbers of the link L. The associated (extended) spectrum is called the (extended) spectrum of the link.

¿From (2.1.5) one has |ESp| = deg Δ^{h}. Moreover, Sp \ ℤ is symmetric with respect to 1.

2.3. Mixed Hodge structures and their spectrum. The name of spectrum in Definition 2.1.4 is motivated by the fact that if L is an algebraic link, i.e. the link of (local) isolated plane curve singularity, then ISp is empty and Sp is the ‘classical’ spectrum associated with the mixed Hodge structure of the vanishing cohomology (for this see e.g. [St1, St2, Var]).

More generally, let f: (ℂ^{n+1}, 0) → (ℂ, 0) be the germ of an analytic function with an isolated singularity at 0, and let Y be the Milnor fiber and U = H̃_{n}(Y, ℝ). (For details regarding the Milnor fibration, see e.g. [Mi2, AGV, Nem2].) One takes the monodromy operator h: U → U, the intersection form b: U × U → ℝ and the variation operator V : U^{∗} → U. One checks (see e.g. [AGV] or [Nem2, § 5]) that the complexification of (U; b, h, V) constitutes a (−1)^{n}–HVS. If S is the Seifert matrix of the Milnor fibration, then at the level of matrices V = (S^{T})^{−1}. Since S is unimodular, V is an isomorphism, hence the variation structure is simple. For plane curves one has ε = −1, hence h = (S^{T})^{−1}S and b = S − S^{T}. The structure (U; b, h, V) ⊗ ℂ is called the ‘homological HVS’ of the germ.

There is a dual HVS as well, the ‘cohomological HVS’ associated with the germ, which is supported on H^{∗} := H̃^{n}(Y, ℂ). Additionally, H̃^{n}(Y, ℂ) carries a limit mixed Hodge structure with Hodge filtration F and weight filtration W such that the semisimple part h_{ss}^{∗} of the cohomological monodromy operator acts on (H^{∗}, F, W). They define spectral pairs. In order to eliminate any confusion about the existing different normalizations, we provide some details.

One considers the generalized λ–eigenspaces U_{λ}^{∗} for all the eigenvalues λ of the Gauss– Manin monodromy operator h_{GM} = (h_{ss}^{∗})^{−1} and the equivariant (Gauss–Manin) Hodge numbers h_{λ}^{p,q} := dim Gr_{F}^{p} Gr_{p+q}^{W}U_{λ}^{∗}.

Then these numbers can be codified in a different way in the collection of Hodge spectral pairs of (U^{∗}, F, W; h_{ss}^{∗}). This is a collection of pairs (α, w) from ℝ × ℕ defined by

(2.3.1)  Spp_{GM}(f) = ∑_{(α,w)} h_{exp(−2πiα)}^{n+[−α],w+s−n−[−α]}(α, w) ∈ ℕ[ℝ × ℕ],

where s = 1 if λ = exp(−2πiα) = 1 and s = 0 otherwise.

This can be transformed in several ways. If by some geometric reason, one wishes to emphasize more the cohomological monodromy operator h_{ss}^{∗} (instead of h_{GM}), one considers

(2.3.2)  Spp_{∗}(f) = ∑_{(α,w)} h_{exp(2πiα)}^{n+[−α],w+s−n−[−α]}(α, w) ∈ ℕ[ℝ × ℕ],

If one forgets the weight filtration, then from the equivariant Hodge filtration one can read the Hodge spectrum, namely

(2.3.3) Sp_{∗}(f) = ∑ α ∈ ℕ[ℝ] (the sum over the spectral pairs (α, w) of Spp_{∗}(f)).

Any such spectral number α is in the interval (−1, n). Another normalization of the spectrum defines the spectral numbers in the interval (0, n + 1): Sp_{MHS}(f) is the collection of numbers (α + 1) where α runs over the entries of Sp_{∗}(f).

The identification of the Hodge invariants with the associated hermitian variation structure goes through the crucial polarization property of the mixed Hodge structure. In this way, the cohomological hermitian variation structure of f can be obtained from (U^{∗}, F, W) by collapsing the Hodge filtration mod 2, having the collapsed spectral numbers in (−1, 1]. The corresponding H–numbers are, in fact, the equivariant primitive Hodge numbers of (U^{∗}, F, W) under this collapsing procedure. Usually, the homological and cohomological HVS’s do not agree, in the case ε = (−1)^{n} = −1 they differ by a sign: V_{coh} = −V_{hom}. This explains the two slightly different definition of the spectral numbers (2.1.4)(a) and (2.3.2). Nevertheless, one has the following identification:

Proposition 2.3.4 ([Nem2, (6.5)]). The HVS–spectrum Sp_{HVS} is a mod 2 reduction of the Hodge spectrum Sp_{MHS} considered in (0, 2]. In other words

Sp_{HVS} = {x mod 2: x ∈ Sp_{MHS}}.

Therefore, for a germ of an isolated plane curve singularity one gets Sp_{HVS} = Sp_{MHS}. That means, that the Hodge spectrum can be described completely in terms of the (real) Seifert form of the link. This is the model of our further investigation.

2.4. Spectrum of a link and the Tristram–Levine signature. The Tristram–Levine signatures (defined first in [Tri, Le]) turn out to be a knot–theoretic counterpart of the spectrum of singular points. We recall how can it be explicitly expressed from the spectrum of the link.

Definition 2.4.1. Let L be a link and S its Seifert matrix. The Tristram–Levine signature function is the mapping from S^{1} \ {1} = {ζ ∈ ℂ: |ζ| = 1, ζ ≠ 1} to ℤ given by

σ_{L}(ζ) = signature [(1 − ζ)S + (1 − \overline{ζ})S^{T} ] .

The nullity n_{L}(ζ) is the nullity of the same form (1 − ζ)S + (1 − \overline{ζ})S^{T}, while the normalized nullity, ñ_{L}(ζ), is defined as n_{L}(ζ) − Irr. For completeness we extend the definitions for ζ = 1, too. First, we set σ_{L}(1) = 0. Then notice that for any ζ ≠ 1, ñ_{L}(ζ) equals to the multiplicity of the root of Δ^{h} at ζ. We define ñ_{L}(1) by this characterization for ζ = 1.

We have the following relation between H–numbers, signatures and nullities of the link.

Proposition 2.4.2 ([BN, (4.4.6) and (4.4.9)]). Let Sp = Sp_{HVS} be the real part of the spectrum as in Definition 2.1.4. Let x ∈ (0, 1) and ζ = e^{2πix}. Then

σ_{L}(ζ) = −|Sp ∩ (x, x + 1)| + |Sp \ [x, x + 1]| + ∑_{n=1}^{∞} ∑_{u=±1} up_{ζ}^{2n}(u)
ñ_{L}(ζ) = ∑_{k,u} p_{ζ}^{k}(u).

In particular,

(2.4.3)  −σ(ζ) + ñ(ζ) ≥ |Sp ∩ (x, x + 1)| − |Sp \ [x, x + 1]|.

Remark 2.4.4. In the cases x ∈ {0, 1}, the inequality (2.4.3) still holds. Indeed, σ_{L}(1) = 0, and the right hand side is zero, as well, because Sp \ ℤ is symmetric. Moreover, if 1 is not a root of Δ^{h}, then (2.4.3) is an equality for x = 1.

Let us denote

D = |Sp ∩ {x, x + 1}| ≥ 0.

Assume that Δ^{h}, the characteristic polynomial of the link, has no roots outside the unit circle. Then deg Δ^{h} = |Sp| = |Sp ∩ (x, x + 1)| + |Sp \ [x, x + 1]| + D, hence one also has

(2.4.5) deg Δ^{h} − σ(ζ) + ñ(ζ) = 2|Sp ∩ (x, x + 1)| + ∑_{\begin{matrix}k odd \\ u=±1 \end{matrix}} p_{ζ}^{k}(u) + ∑_{k even} 2p_{ζ}^{k}(−1) + D.

For any x ∈ [0, 1], parallel to the set Sp ∩ (x, x + 1), we will also consider the set Sp \ [x, x + 1] = Sp ∩ (0, x) + Sp ∩ (1 + x, 2]. These two types cover all the ‘length one open intervals’ of the mod 2 spectrum.

The following corollary will be used extensively in the sequel.

Corollary 2.4.6. Let L be a link and Δ^{h} its characteristic polynomial. Assume that Δ^{h} has no roots outside the unit circle. If ζ = e^{2πix}, for x ∈ [0, 1), is not a root of Δ^{h} then

|Sp ∩ (x, x + 1)| = \frac{1}{2} (deg Δ^{h} − σ(ζ)) and |Sp \ [x, x + 1]| = \frac{1}{2} (deg Δ^{h} + σ(ζ)) .

Moreover, for arbitrary x ∈ [0, 1]:

\frac{1}{2} (deg Δ^{h} − σ(ζ) + ñ(ζ)) ≥ |Sp ∩ (x, x + 1)|

\frac{1}{2} (deg Δ^{h} + σ(ζ) + ñ(ζ)) ≥ |Sp \ [x, x + 1]|.

(2.4.7)

2.5. Morse theory of plane curves. For any ξ ∈ ℂ^{2} and r > 0 let B(ξ, r) be the ball centered at ξ and with radius r, also S^{3}(ξ, r) := ∂B(ξ, r). For an algebraic curve C sitting in ℂ^{2}, we write (C ∩ B(ξ, r))^{∧} for the normalization of C ∩B(ξ, r), and the genus of C ∩B(ξ, r) is the genus of its normalization.

For any link L, we denote by c_{L} its number of components, and we set

w_{L}(ζ) := −σ_{L}(ζ) + 1 − c_{L} + n_{L}(ζ)
−u_{L}(ζ) := σ_{L}(ζ) + 1 − c_{L} + n_{L}(ζ).

Remark 2.5.1. The convention used in [Bo] is that n_{L} is the dimension of the kernel of (1−ζ)S +(1−\overline{ζ})S^{T} increased by 1, this explains the formal differences compared with [Bo].

We will also fix ζ ∈ S^{1} \ {1}. Let us begin by citing a result from [Bo].

Proposition 2.5.2 ([Bo, Proposition 6.8]). Let ξ be a generic point of ℂ^{2} and r_{0} < r_{1} two values such that the intersections L_{i} := C ∩ S^{3}(ξ, r_{i}) are transverse (i = 0, 1). With the notations c_{i} = c_{L_{i}}, g_{i} = the genus of C_{i} := C ∩ B(ξ, r_{i}) and k_{i} = the number of connected components of C_{i}^{∧}, one has

w_{L_{1}}(ζ) − ∑ w_{L_{k}^{sing}}(ζ) − w_{L_{0}}(ζ) ≥ −2(g_{1} − g_{0} + c_{1} − c_{0} − k_{1} + k_{0}),
− (u_{L_{1}}(ζ) − ∑ u_{L_{k}^{sing}}(ζ) − u_{L_{0}}(ζ)) ≥ −2(g_{1} − g_{0} + c_{1} − c_{0} − k_{1} + k_{0}),

(2.5.3)

where L_{k}^{sing} are the links of singularities of C, which lie in B(ξ, r_{1}) \ B(ξ, r_{0}).

We use Proposition 2.5.2 in two special cases.

Corollary 2.5.4. Let C_{0} and C_{1} be as in 2.5.2. If C_{01} = C_{1} \ C_{0} is smooth then

−σ_{L_{1}}(ζ) + n_{L_{1}}(ζ) − (−σ_{L_{0}}(ζ) + n_{L_{0}}(ζ)) ≥ χ(C_{01}).
σ_{L_{1}}(ζ) + n_{L_{1}}(ζ) − (σ_{L_{0}}(ζ) + n_{L_{0}}(ζ)) ≥ χ(C_{01}).

Proof. Use the definition of w, (2.5.3) and C_{01}^{∧} = C_{01} for the first inequality. For the second one we use −u instead of w. □

The other important application is in the case when r_{0} is small, hence L_{0} is an unknot.

Proposition 2.5.5. Fix r such that the intersection C ∩ S(ξ, r) is transverse, and set L := C ∩ S(ξ, r). Let C_{smooth} be the smoothing of C ∩ B(ξ, r) (e.g. if C is given by F^{−1}(0) for some reduced polynomial, then C_{smooth} can be taken as F^{−1}(ε)∩B(ξ, r) for ε sufficiently small). Let z_{1}, …, z_{k} be the singular points of C ∩ B(ξ, r) with links L_{1}^{sing}, …, L_{k}^{sing}, Milnor numbers μ_{1}, …, μ_{k}, number of branches c_{1}, …, c_{k}, and signatures σ_{1}(ζ), …, σ_{k}(ζ). Then

−σ_{L}(ζ) + n_{L}(ζ) + (1 − χ(C_{smooth})) ≥ ∑_{j=1}^{k} (−σ_{L_{j}^{sing}}(ζ) + n_{j}(ζ) + μ_{j})

σ_{L}(ζ) + n_{L}(ζ) + (1 − χ(C_{smooth})) ≥ ∑_{j=1}^{k} (σ_{L_{j}^{sing}}(ζ) + n_{j}(ζ) + μ_{j}) .

(2.5.6)

Proof. We prove only the first part, in the second one we use −u_{L} instead of w_{L}.

Let r_{min} be minimal with C ∩ S(ξ, r) non-empty, and set r_{0} := r_{min} + ε for ε sufficiently small. Then L_{0} is an unknot with w_{L_{0}}(ζ) ≡ 0, c_{0} = k_{0} = 1, thus (2.5.3) gives

− σ_{L}(ζ) + n_{L}(ζ) + 1 − c_{L} ≥

≥ ∑_{j=1}^{k}(−σ_{j}(ζ) + n_{j}(ζ) + μ_{j}) − ∑_{j=1}^{k}(μ_{j} + c_{j} − 1) − 2g(C) − 2c_{L} + 2k_{1}.

The proof is completed by applying the genus formula 2(g(C_{smooth}) − g(C)) = ∑_{j=1}^{k}(μ_{j} + c_{j} −1), the fact that b_{1}(C_{smooth}) = 2g(C_{smooth})+c_{L}−1 and observing that b_{0}(C_{smooth}) ≤ 2k_{1} (it is even bounded by k_{1} alone). □

Remark 2.5.7. The cited result (i.e. Proposition 2.5.2) does not really require Morse theoretical arguments, although they are very convenient. We could deduce it — with approximately the same amount of work — from the Murasugi inequality [Kaw, Theorem 12.3.1], too. The argument is that C_{01} = C ∩ (B(ξ, r_{1}) \ B(ξ, r_{0})) induces a cobordism between the links L′_{0} := L_{0} ⨿ L_{1}^{sing} ⨿ … ⨿ L_{j}^{sing} and L_{1}. In this way we do not use anywhere that C is a complex curve, only that its genus is the difference of the genera of the minimal Seifert surfaces of L_{1} and L′_{0}.

3. The Seifert form and the MHS of a polynomial at infinity

In this section we compare the Hodge–spectrum associated with the limit mixed Hodge structure of a polynomial map at infinity with the HVS–spectrum provided by its regular link at infinity. In this way we recover the Hodge–spectrum from the ‘Seifert form at infinity’. For results concerning the limit mixed Hodge structure (MHS) and the Hodge spectrum at infinity the reader might consult [SSS, Di2, Br].

3.1. Basic definitions. Let F: ℂ^{2} → ℂ be a reduced polynomial with critical values x_{1}, …, x_{N}. Since ℂ^{2} is not compact, the topology of a fiber F^{−1}(y) can be different for different regular values y of F.

Definition 3.1.1. ([Neu3]) The fiber F^{−1}(c) is called regular at infinity if there exists a (small) disk D ∋ c in ℂ and a (large) ball B ⊂ ℂ^{2} such that F restricted to F^{−1}(D) \ B is a C^{∞} trivial fibration. The fiber is called irregular at infinity if it is not regular at infinity.

Consider all the values y_{1}, …, y_{M} such that F^{−1}(y_{k}) is not regular at infinity. Set ρ ∈ ℝ with ρ > max_{k,l}{|x_{k}|, |y_{l}|} and set γ = {z ∈ ℂ: |z| = ρ}. Then F restricted to F^{−1}(γ) is a C^{∞} locally trivial fibration, called the fibration of F at infinity. It will be denoted Fib_{∞}. The fiber of Fib_{∞} is the (generic) fiber Y_{∞} := F^{−1}(ρ) of F. The induced algebraic monodromy over γ, called the monodromy of F at infinity, will be denoted by

(3.1.2)  h_{∞}: H_{1}(Y_{∞}, ℤ) → H_{1}(Y_{∞}, ℤ).

Furthermore, we also consider on H_{1}(Y_{∞}, ℤ) the intersection form b_{∞}. Already this isometric structure (H_{1}(Y_{∞}); b_{∞}, h_{∞}) contains important information about the behavior of F at infinity, nevertheless, we will enhance it in two different ways. The first is topological: we investigate the possibility of extending the pair (b_{∞}, h_{∞}) to a variation structure (this, strictly speaking, in general, only ‘partially’ is possible). The candidate for the variation operator is the inverse transpose of the Seifert matrix of the link at infinity. The second is algebraic: one lifts the pair (b_{∞}, h_{∞}) to the level of a polarized mixed Hodge structure by considering the limit mixed Hodge structure of F at infinity.

First we start with the topological part.

Fix a fiber F^{−1}(c) which is regular at infinity. For sufficiently large R the intersection F^{−1}(c) with ∂B(R) is transverse. This link F^{−1}(c) ∩ ∂B(R) ⊂ ∂B(R), denoted by L_{reg}^{∞}, is independent (up to an isotopy) of R and c. It is called the regular link at infinity of F.

According to [Neu3, Theorem 5] we can associate with L_{reg}^{∞} the so–called fundamental multilink at infinity L_{fund}, which is fibred. This means the following: there exists a link L_{fund} with components {L_{fund,i}}_{i=1}^{ν} and positive multiplicities n = {n_{i}}_{i=1}^{ν} such that there is a fibration ϕ : S^{3} \ L_{fund} → S^{1} with the following property: for any closed loop τ ∈ S^{3} \ L_{fund}, ϕ_{∗}([τ]) ∈ H_{1}(S^{1}) = ℤ equals the linking number of [τ] with ∑_{i} n_{i}L_{fund,i}. Furthermore, the closure \overline{Y_{t}} of the fiber Y_{t} = ϕ^{−1}(e^{2πit}) (t ∈ [0, 1]) is not a manifold with boundary, but homologically \overline{Y_{t}} \ Y_{t} is the multilink ∑_{i} n_{i}L_{fund,i}.

Finally, the connection between the multilink L_{fund} and the link at infinity L_{reg}^{∞} is the following. Let T = T (L_{fund}) be a closed small tubular neighbourhood of L_{fund}. Then L_{reg}^{∞} is the intersection of a fiber Y_{0} with ∂ T (L_{fund}).

Lemma 3.1.3. For any i ∈ {1, …, ν}, let l_{i} be the linking number

l_{i} = lk(L_{fund,i}, ∑_{j≠i} n_{j}L_{fund,j})

and n′_{i} the (positive) greatest common divisor of n_{i} and l_{i}. Then the number of components of Y_{0} ∩ ∂ T (L_{fund,i}) is exactly n′_{i}. Hence, L_{reg}^{∞} has ∑_{i=1}^{ν} n′_{i} components. Moreover, the components of Y_{0} ∩ ∂ T (L_{fund,i}) are cyclically permuted by the geometric monodromy of ϕ.

Proof. See [EN, § 3 and 4]. □

Another important point about L_{fund} is that its fiber Y_{0} can be identified with the generic fiber Y_{∞} of the polynomial F [Neu3, Theorem 4]. In fact, by [AC, Theorem 1.1] one has

Lemma 3.1.4. The multilink fibration S^{3} \ L_{fund} → S^{1} associated with (L_{fund}, n) and the fibration F ib_{∞} of F are isomorphic.

By [EN, page 37], Y_{0} has d = gcd_{i}{n_{i}} connected components. In the sequel we will assume that d = 1, that is, the generic fiber of F is connected.

3.2. The multilink Seifert form of L_{fund}. The surface Y_{0}, the fiber of the multilink (L_{fund}, n), is a generalized Seifert surface of the multilink, cf. [EN, pages 28-29]. In the sequel we refer to it as the multilink Seifert surface. Using this surface, one can define the multilink Seifert form associated with Y_{0}, cf. [EN, §15]. It is a bilinear form on H_{1}(Y_{0}, ℤ) defined similarly as the classical Seifert form, namely S_{fund}(α, β) for α, β ∈ H_{1}(Y_{0}, ℤ) is the linking number lk(α, β^{+}), where β^{+} is the push-forward of β in the positive direction.

If all the multiplicities {n_{i}}_{i} equal 1, then L_{fund} is a fibred link, and S_{fund} is its classical Seifert form, hence it has determinant ±1. In the case of general multiplicity system n this is not the case anymore. In fact, S_{fund} can be even degenerate. Nevertheless, some parts of the classical theory survive.

Lemma 3.2.1. Let H^{∗} denote the dual of H, T^{o} the interior of T, and \overline{Y}_{[a,b]} := ⋃_{a≤t≤b} \overline{Y_{t}}.

(a) The groups H_{1}(\overline{Y_{0}}, ℤ) and H_{1}(Y_{0}, ℤ)^{∗} are isomorphic. In fact one has the following sequence of isomorphisms, denoted by s:

H_{1}(\overline{Y_{0}}) ⟶^{∂^{−1}} H_{2}(S^{3}, \overline{Y_{0}}) ⟶^{(1)} H_{2}(S^{3}, \overline{Y}_{[0, \frac{1}{2} ]}) ⟶^{(2)} H_{2}(\overline{Y}_{[ \frac{1}{2} ,1]}, \overline{Y_{1/2}} ∪ \overline{Y_{1}}) ⟶^{(3)}

H_{2}(\overline{Y}_{[ \frac{1}{2} ,1]}, \overline{Y_{1/2}} ∪ \overline{Y_{1}} ∪ (T ∩ \overline{Y}_{[ \frac{1}{2} ,1]})) ⟶^{(4)} H_{2}(\overline{Y}_{[ \frac{1}{2} ,1]} \ T^{o}, ∂(\overline{Y}_{[ \frac{1}{2} ,1]} \ T^{o}))

⟶^{(5)} H_{1}(\overline{Y}_{[ \frac{1}{2} ,1]} \ T^{o})^{∗} ⟶^{(6)} H_{1}(\overline{Y_{1}} \ T^{o})^{∗} ⟶^{(7)} H_{1}(Y_{1})^{∗} = H_{1}(Y_{0})^{∗}.

(b) Let j : H_{1}(Y_{0}, ℤ) → H_{1}(\overline{Y_{0}}, ℤ) be induced by the inclusion. Then the composition

H_{1}(Y_{0}, ℤ) ⟶^{j} H_{1}(\overline{Y_{0}}, ℤ) ⟶^{s} H_{1}(Y_{0}, ℤ)^{∗}

can be identified with the multilink Seifert form S_{fund}.

(c) Identify the isometric structure (b_{∞}, h_{∞}) with the intersection form and monodromy of H_{1}(Y_{0}) (by 3.1.4). Then, in matrix notation,

b_{∞} = S_{fund} − S_{fund}^{T}, and S_{fund}^{T}h_{∞} = S_{fund}.

In particular, h_{∞} is an automorphism of S_{fund}, that is h_{∞}^{T}S_{fund}h_{∞} = S_{fund}.

Proof. In the sequence of isomorphisms ∂^{−1} comes from the exact sequence of the pair; (1), (3), (6) and (7) are induced by deformation retracts; (2) and (4) are excisions, while (5) is provided by duality of the manifold with boundary \overline{Y}_{[ \frac{1}{2} ,1]} \ T^{o}. Part (b) and (c) follow by similar argument as in the classical case, see e.g. the survey [Nem5, (3.15)]. □

In the above composition, although s is an isomorphism, j in general is not. Since, by our assumption H̃_{0}(Y_{0}, ℤ) = 0, the morphism j can be inserted in the following long exact sequence:

(3.2.2)  0 → H_{2}(\overline{Y_{0}}) → H_{2}(\overline{Y_{0}}, Y_{0}) → H_{1}(Y_{0}) ⟶^{j} H_{1}(\overline{Y_{0}}) → H_{1}(\overline{Y_{0}}, Y_{0}) → 0.

Lemma 3.2.3.

\begin{aligned}(a) & H_{2}(\overline{Y_{0}}, Y_{0}, ℤ) & = ⊕_{i=1}^{ν}ℤ^{n′_{i}−1}, \\ (b) & H_{1}(\overline{Y_{0}}, Y_{0}, ℤ) & = ⊕_{i=1}^{ν}( ℤ^{n′_{i}−1} ⊕ ℤ_{n_{i}/n′_{i}}). \end{aligned}

In particular, H_{2}(\overline{Y_{0}}, ℤ) = 0 and

(3.2.4)  dim ker j = ∑_{i=1}^{ν}(n′_{i} − 1).

Proof. By excision and deformation retract argument H_{q}(\overline{Y_{0}}, Y_{0}) = ⊕_{i}H_{q}(A_{i}, B_{i}), where

(A_{i}, B_{i}) := (\overline{Y_{0}} ∩ T (L_{fund,i}), Y_{0} ∩ ∂T (L_{fund,i})).

Note that the homotopy type of A_{i} is L_{fund,i}, while of B_{i} is n′_{i} copies of S^{1}. Each of these copies maps (via the inclusion B_{i} ↪ A_{i}) onto L_{fund,i} as the n_{i}/n′_{i}–covering. Therefore, the inclusion B_{i} ↪ A_{i} at H_{1}–level is ℤ^{n′_{i}} → ℤ, \{a_{1}, …, a_{n′_{i}}\} ↦ \frac{n_{i}}{n′_{i}} · ∑_{k} a_{k}. This gives (a) and (b). The rest follow by rank computation argument from (3.2.2). □

Next, we will consider another compactification Ỹ_{0} of Y_{0}. Denote Y_{0}^{o} := Y_{0} \ T (L_{fund})^{o}, the complement of the interior of the tube. ∂Y_{0}^{o} consists of ∑_{i} n′_{i} copies of S^{1}. Let Ỹ_{0} be obtained from Y_{0}^{o} by gluing to each boundary circle a 2–disc, in this way obtaining a compact smooth surface. In fact, the fibration at infinity F over γ can be compactified (even algebraically) to a fibration F̃ over γ with smooth compact fibers Ỹ_{0}, where in this language the compact fiber consists of Y_{0} with additionally ∑_{i} n′_{i} ‘points at infinity’. This point of view is used in Hodge theoretical computations, see e.g. [Di1] or [Di2, §3].

One has the following exact sequence:

(3.2.5)  0 → ℤ → ⊕_{i}ℤ^{n′_{i}} → H_{1}(Y_{0}, ℤ) → H_{1}(Ỹ_{0}, ℤ) → 0.

Above, ⊕_{i}ℤ^{n′_{i}} is generated by the discs, their images in H_{1}(Y_{0}) are the classes of the circles ∂Y_{0}^{o}. ℤ from the left is H_{2}(Ỹ_{0}); its image is generated by ∂Y_{0}^{o}. The monodromy extends to H_{1}(Ỹ_{0}) (or to F̃) (and will be denoted by h̃_{∞}), and also to the discs/points at infinity: it acts trivially on ℤ, on ℤ^{n′_{i}} acts by permutation of the base elements (denoted by h_{per}).

Let b̃_{∞} be the intersection form on H_{1}(Ỹ_{0}).

Lemma 3.2.6.

(a) h̃_{∞} has no eigenvalue 1, and all its Jordan blocks have size not larger than two.

(b) The exact sequence (3.2.5), together with the algebraic monodromy action on it, splits. That is, h_{∞} has no Jordan block of size three, and the blocks of size two of h_{∞} and h̃_{∞} agree. In other words, over ℚ, one has a direct sum decomposition:

(3.2.7)  (H_{1}(Y_{0}); b_{∞}, h_{∞}) = (H_{1}(Ỹ_{0}); b̃_{∞}, h̃_{∞}) ⊕ (⊕_{i}ℚ^{n′_{i}}/ℚ; 0, h_{per}).

Moreover, (b̃_{∞}, h̃_{∞}) is a non–degenerate isometric structure.

(c) All roots of h_{∞} are roots of unity.

Proof. The statements follow from the mixed Hodge theory of the degeneration at infinity of F and F̃. Part (a) is proved e.g. in [Di1, Di2]. Part (b) follows from the spectral pair computation of the mixed Hodge structure carried on H^{1}(Y_{0}, ℂ). More precisely, there is a cohomological analogue of the sequence (3.2.5) which carries mixed Hodge structure compatible with the action of the monodromy, see again [Di2, § 3]. The number of Jordan blocks of size two correspond to those spectral pairs (α, w) for which w = 0. These are computed for both H^{1}(Y_{0}, ℂ) and H^{1}(Ỹ_{0}, ℂ) in [Br], and their number agree. For (c) use the Monodromy Theorem for F̃ at infinity. □

Finally, we summarize the properties of L_{fund} in the following proposition. As usual, if h is an automorphism of the vector space V, then V_{λ=1} denotes the generalized eigenspace corresponding to eigenvalue 1, while V_{λ≠1} is the direct sum of the other generalized eigenspaces.

Proposition 3.2.8. Set U := H_{1}(Y_{0}, ℚ) and let b_{∞} and h_{∞} be the intersection form and the algebraic monodromy induced by the multilink fibration ϕ : S^{3} \ L_{fund} → S^{1}.

Then the following facts hold:

(a) Y_{0} is the minimal multilink Seifert surface of the multilink (L_{fund}, n), and all minimal multilink Seifert surfaces of (L_{fund}, n) are isotopic to Y_{0}.

(b) One has a direct sum decomposition (Keef decomposition, cf. (2.2.1)):

(3.2.9)  (U, S_{fund}) = (U_{0} ⊕ U_{ndeg}, S_{fund,0} ⊕ S_{fund,ndeg})

such that S_{fund,0} = 0 of size Irr = ∑_{i=1}^{ν} (n′_{i} − 1), and S_{fund,ndeg} is non–degenerate. A possible free generator set for U_{0} is the collection of the cycles L_{reg,i,k}^{∞} − L_{reg,i,k+1}^{∞}

(1 ≤ i ≤ ν; 1 ≤ k < n′_{i}), where {L_{reg,i,k}^{∞}}_{k=1}^{n′_{i}} are the components of Y_{0} ∩ ∂T (L_{fund,i}). (c) The compatibility of the decompositions (3.2.9) and (3.2.7) is the following:

(c.1) (U_{ndeg})_{λ≠1} = H_{1}(Ỹ_{0}). On this space (S_{fund,ndeg})_{λ≠1} completes the non–degenerate isometric structure (b̃_{∞}, h̃_{∞}) to a simple (−1)–variation structure.

(c.2) (⊕_{i}ℚ^{n′_{i}}/ℚ; 0, [h_{per}]) = ((U_{ndeg})_{λ=1}; 0, Id) ⊕ (U_{0}; 0, h_{∞}|_{U_{0}}) (and this is an eigenspace decomposition).

(c.3) (U_{ndeg})_{λ=1} has dimension ν −1, on it the restriction of b_{∞} is trivial, the restriction of h_{∞} is the identity, and this degenerate isometric structure is completed by (S_{fund,ndeg})_{λ=1} to a simple variation structure.

(c.4) On U_{0} the restrictions of b_{∞} and S_{fund} are trivial (hence all the equivariant signature type invariants including the Tristram–Levine signatures of restrictions of S_{fund} and (b_{∞}, h_{∞}) are the same). Nevertheless, the restriction of h_{∞} is non–trivial (in fact, it has no eigenvalue 1), hence the isometric structure cannot be completed to a variation structure. The characteristic polynomial of the restriction of h_{∞} is

det(h_{∞}|_{U_{0}} − t Id) = ∏_{i} \frac{t^{n′_{i}} − 1}{t − 1}.

Proof. (a) follows from [EN, (4.1)]. For (b) note that the generators listed are in the kernel of j. For this use e.g. the proof of (3.2.3), where {L_{reg,i,k}^{∞}}_{k=1}^{n′_{i}} are exactly the components of B_{i}. Another possibility is a direct verification of the fact that S_{fund}(L_{reg,i,k}^{∞}− L_{reg,i,k+1}^{∞}, β) = S_{fund}(β, L_{reg,i,k}^{∞} − L_{reg,i,k+1}^{∞}) = 0 for any β. Indeed, if T is a sufficiently small tubular neighborhood, then it does not intersect β, on the other hand inside of T the circles L_{reg,i,k}^{∞} and L_{reg,i,l}^{∞} are homologous. For part (c) use Lemmas 3.2.1(c) and 3.2.6; for the characteristic polynomial use the fact that the components {L_{reg,i,k}^{∞}}_{k} are cyclically permuted, cf. Lemma 3.1.3. □

The multilink structure (L_{fund}, S_{fund}) now will be used in two different aspects. First, it can be related with the link L_{reg}^{∞}; in fact, one can recover it from L_{reg}^{∞}, see (3.3). On the other hand, the multilink fibration of L_{fund} can be identified with the fibration at infinity F ib_{∞} of F, cf. Lemma 3.1.4. In this way L_{fund} creates the bridge between L_{reg}^{∞} and F ib_{∞}.

3.3. The Seifert form of L_{reg}^{∞}. Set Y_{0}^{o} := Y_{0} \ T (L_{fund})^{o} as above. Obviously, Y_{0}^{o} ↪ Y_{0} admits a deformation retract, hence H_{1}(Y_{0}^{o}, ℤ) = H_{1}(Y_{0}, ℤ) canonically.

Lemma 3.3.1. One has the following facts:

(a) Y_{0}^{o} is the minimal Seifert surface of L_{reg}^{∞}, and all minimal Seifert surfaces of L_{reg}^{∞} are isotopic to Y_{0}^{o}.

(b) The Seifert form S_{reg} of L_{reg}^{∞} associated with Y_{0}^{o} is identical with S_{fund} (under the identification H_{1}(Y_{0}^{o}, ℤ) = H_{1}(Y_{0}, ℤ)). In particular, all the result listed in Proposition 3.2.8 about S_{fund} are valid for S_{reg} as well.

(c) Let h_{reg} be the monodromy of the variation structure associated with S_{reg,ndeg} = S_{fund,ndeg} (as in 2.2). Then the higher Alexander polynomials Δ_{k} of L_{reg}^{∞} satisfies the following identities: Δ_{k} ≡ 0 for 0 ≤ k < Irr, Δ_{Irr}(t) = det(h_{reg} − t Id).

(d) All the roots of the (higher) Alexander polynomial Δ_{Irr} of L_{reg}^{∞} are roots of unity.

Proof. (a)–(b)–(c) follows from Proposition 3.2.8 and from the construction of Y_{0}^{o}. For (d) use either Lemma 3.2.6(c) or note that the multilink fibration of L_{fund} can be represented by a splice diagram [Neu3, Neu4], hence the characteristic polynomial of h_{fund} is a product of cyclotomic polynomials by [EN, Theorem 13.6]. □

3.4. The HVS–spectrum of the regular link at infinity, L_{reg}^{∞}.

For local isolated plane curve singularities we have the following classical result (consequences of the Monodromy Theorem and polarization properties), which in the language of H–numbers p_{λ}^{k}(±1) and q_{λ}^{k} of their local links can be formulated as follows (see e.g. [BN, Proposition 3.1.5, Lemma 3.1.6]).

Proposition 3.4.1. Let L be an algebraic link and p_{λ}^{k}(±1), q_{λ}^{k} its H–numbers. Then (a) q_{λ}^{k} = 0 for all k > 0 and |λ| < 1; moreover p_{λ}^{k}(±1) = 0 unless λ is a root of unity,

(b) p_{λ}^{k}(±1) = 0 for all k > 2. Moreover p_{1}^{2}(±1) = 0,

(c) p_{λ}^{2}(−1) = 0 and p_{1}^{1}(−1) = 0.

The fundamental mulitlink at infinity L_{fund}, or the regular link at infinity L_{reg}^{∞}, in general, cannot be realized by a local algebraic link. However, their H–numbers share similar properties as the H–numbers of local links.

Proposition 3.4.2. For the H–numbers of L_{reg}^{∞} the following facts hold.

(a) q_{λ}^{k} = 0 for all k > 0 and |λ| < 1, moreover p_{λ}^{k}(±1) = 0 unless λ is a root of unity,

(b) p_{λ}^{k}(±1) = 0 for all k > 2, and p_{1}^{2}(±1) = 0,

(c) p_{1}^{1}(−1) = 0,

(d) p_{λ}^{2}(1) = 0 for λ ≠ 1.

Proof. (a) and (b) follow from Lemma 3.2.6. Next we prove (c). First we recall that W_{1}^{1}(±1) = (ℂ; 0, Id, ∓1), hence we have to show that the restriction of S_{fund} on U_{λ=1} is negative definite. This follows from the more general Proposition 3.7.2 of section 3.7.

(d) By [Neu3, NeRu] L_{fund} can be represented by a splice diagram where all edges determinants are negative. Thus, L_{fund} has uniform twists (all positive) (see [EN, Chapter 14]). Therefore, by the discussion in [Neu1, Section 2] we have for each x ∈ U_{λ}

S_{fund}(λx, (h_{∞} − λ)x) ≥ 0.

This shows that p_{λ}^{2}(+1) cannot occur. □

Let Sp_{HVS}(L_{reg}^{∞}) be the HVS–spectrum associated with the link L_{reg}^{∞}.

Corollary 3.4.3. (a) The HVS–spectrum of U_{λ=1} consists of (ν − 1) copies of (1).

(b) All elements of Sp_{HVS}(L_{reg}^{∞}) are situated in (0, 2), and Sp_{HVS}(L_{reg}^{∞}) is symmetric with respect to 1.

Remark 3.4.4. The proofs of Propositions 3.4.1 and 3.4.2 rely on some key properties of the splice diagrams of the corresponding links. The common properties (which imply the common p_{1}^{1}(−1) = 0) is that in both cases the ‘multiplicities of the nodes’ and the ‘(near) weights’ are positive. The crucial difference between the diagrams is that in the local case the edge determinants are positive, while for the diagram at infinity they are negative. This implies the sign difference in the p_{λ}^{2}(±1)–vanishing. For more detail, see subsection 3.7.

The next identity will be often used in the sequel.

Corollary 3.4.5. If Y_{∞} is a regular fiber of F, then 1 − χ(Y_{∞}) = deg Δ_{Irr}(L_{reg}^{∞}) + Irr .

Proof. By Lemma 3.3.1(a) the size of S_{reg} is equal to 1 − χ(Y_{∞}). On the other hand, deg Δ_{Irr} is equal to the size of S_{ndeg} by Lemma 3.3.1(c). The difference of the sizes of the two matrices is equal to Irr by Proposition 3.2.8(b) (cf. also Lemma 3.3.1(b)). □

3.5. The Hodge–spectrum of the fibration of F at infinity. Let Sp_{MHS,∞} be the spectrum associated with the limit mixed Hodge structure of F at infinity defined in a similar way as in subsection 2.3. The main result of this subsection shows that Sp_{MHS,∞} can be recovered from the rational Seifert form of L_{reg}^{∞} and from the integers {n′_{i}}_{i}. Conversely, Sp_{HVS}(L_{reg}^{∞}) is the maximal subset of Sp_{MHS,∞}, which is symmetric with respect to 1.

More precisely, in ℤ[ℚ] one has Theorem 3.5.1.

Sp_{MHS,∞} = Sp_{HVS}(L_{reg}^{∞}) + ∑_{i=1}^{ν} (\frac{1}{n′_{i}} ) + ⋯ + (\frac{n′_{i} − 1}{n′_{i}}).

Proof. Let us consider the decomposition given in Proposition 3.2.8:

(3.5.2)  (U_{0}; 0, h_{∞}|_{U_{0}}) ⊕ ((U_{ndeg})_{λ=1}; 0, Id) ⊕ ((U_{ndeg})_{λ≠1}; b̃_{∞}, h̃_{∞})

The last component carries a limit mixed Hodge structure which is polarized by b̃_{∞}, and it extends to a simple hermitian variation structure with (S_{fund,ndeg})_{λ≠1}. In such a situation, the HVS–spectrum agrees with the Hodge spectrum. The proof is absolutely the same as in the local case, see Proposition 2.3.4, or the original source [Nem2, (6.5)] (or the affine polynomial case in [GN2]).

For the middle component both HVS and Hodge spectrum consists of (ν − 1) copies of (1): in the HVS case see Corollary 3.4.3 as a consequence of Proposition 3.4.2(c), while for the Hodge case see [Br] or [Di2].

These two components provide the contribution from Sp_{HVS}(L_{reg}^{∞}). The remaining part, provided by the first summand, is computed in [Br], and it is the sum in the right hand side of the identity of Theorem 3.5.1. □

Example 3.5.3. Recall that F is ‘good at infinity’ if and only if L_{reg}^{∞} is a fibred link, that is n_{i} = 1 for all i, cf. [NeRu, Theorem 6.1]. By our result, in such a case one has Sp_{MHS,∞} = Sp_{HVS}(L_{reg}^{∞}).

Corollary 3.5.4. For any x ∈ [0, 1] one has

|Sp_{HVS}(L_{reg}^{∞}) ∩ (x, x + 1)| ≤ |Sp_{MHS,∞} ∩ (x, x + 1)| ≤ |Sp_{HVS}(L_{reg}^{∞}) ∩ (x, x + 1)| + Irr

and analogous inequality holds for Sp \ [x, x + 1].

3.6. An example. The above discussion might have been technically quite involved. We want to illustrate the occuring phenomena by investigating one example, the Briançon polynomial, which appeared in [ACL, Br, Di2, DN] (we remark that in [AC, Exemple 4.14] there is a different polynomial called Briançon polynomial, it has different link at infinity and different irregular fibers).

The splice diagram of the fundamental link at infinity is as follows

3 2  −7 −1  (3) (2)  (1) (1) (4)  (1) root

Here the numbers in parentheses are the multiplicities of the vertices and arrowheads (link components). The numbers not in parenthesis denote the weights of corresponding edges (those omitted are equal to 1). We have n_{1} = 4, n′_{1} = gcd(4, 6) = 2, and n_{2} = n′_{2} = 1.

Computing the Euler characteristics of a minimal Seifert surface of L_{fund} (as in [EN]) we get that this surface is a three times punctured torus (3 is the number of components of L_{fund}). The rank of H_{1}(Y_{∞}) is 4, while the ranks of U_{0} and (U_{ndeg})_{λ=1} are 1. The monodromy at infinity permutes L_{1,1}, L_{1,2} and fixes L_{2}. The characteristic polynomial of the monodromy on boundary components is therefore t^{2} − 1. The Alexander polynomial of L_{fund}, hence the characteristic polynomial of the monodromy at infinity, is (t^{2}−1)(t^{2}+t+1).

The equivariant signatures (which correspond to jumps of the Tristram–Levine signatures) of L_{fund} can be computed using [Neu2, Theorem 5.3 and Section 6]. Using [Neu2, Theorem 5.3] for the left-most splice component we compute that σ_{e^{2πi/3}}^{−} = −1 and σ_{e^{−2πi/3}}^{−} = 1 so the jumps of the Tristram–Levine signature function are respectively −2 and 2, in other words p_{e^{2πi/3}}^{1}(+1) = 0, p_{e^{2πi/3}}^{1}(−1) = 1, p_{e^{−2πi/3}}^{1}(+1) = 1, p_{e^{−2πi/3}}^{1}(−1) = 0 (compare [Br, Sections 3.5, 3.6]). On the other hand, a straightforward computation shows that the right splice component does not contribute to the equivariant signature at all. Hence, the non–trivial H–numbers are p_{e^{2πi/3}}^{1}(−1) = p_{e^{−2πi/3}}^{1}(+1) = p_{1}^{1}(1) = 1.

Concluding, the spectrum at infinity is equal to \{\frac{2}{3}, \frac{4}{3}, \frac{1}{2}, 1\} (cf. [Di2, Example 3.6(ii)]), where \{\frac{2}{3}, 1, \frac{4}{3}\} is the contribution from L_{reg}^{∞}.

3.7. The definiteness of ‘linking matrix’. The proof of (3.4.2)(c). We wish to prove that the restriction of S_{fund} on U_{λ=1} is negative definite. This follows from a more general combinatorial result, which we now state.

Let Γ be a rooted Eisenbud–Neumann diagram, cf. [Neu3]. For an edge we call the weight which is closer to the root vertex the near weight and the other one the far weight. For any two nodes v and w, if the geodesics connecting w and the root vertex contains v then we say that w is beyond v. We allow more than one near weight at each node to have weight different than 1. The linking numbers and multiplicities are determined from the diagram as in [EN, §10,11]. The arrowhead vertices will be denoted by L_{1}, …, L_{ν}, their multiplicities are n_{1}, …, n_{ν}.

Let ℚ^{ν} be the ℚ–vector space generated by {L_{i}}_{i}. The linking matrix {lk(L_{i}, L_{j})}_{ij} is defined as follows: for i ≠ j it is the standard linking pairing, while the self–linking lk(L_{i}, L_{i}) is defined via the identity lk(L_{i}, ∑_{j} n_{j}L_{j}) = 0. Equivalently,

(3.7.1)  lk(n_{i}L_{i}, n_{i}L_{i}) = − ∑_{j≠i} lk(n_{i}L_{i}, n_{j}L_{j}).

In particular, the null–space of the linking matrix is at least 1–dimensional.

Proposition 3.7.2. Let Γ be a rooted connected graph with the following properties (a) all near weights are positive and no far weight is allowed to be zero.

(b) if the far weight at a node v is negative, then all far weights of nodes beyond v are also negative (this property is weaker than negativity of edge determinants);

(c) the multiplicities of all arrowhead and non–arrowhead vertices are positive. Then, the linking matrix lk(L_{i}, L_{j}) is negative semi–definite with 1–dimensional null–space.

Proof. We begin with a following special case.

Lemma 3.7.3. The statement of Proposition 3.7.2 holds if lk(L_{i}, L_{j}) > 0 for all i ≠ j.

Proof. The reasoning is exactly as in [Neu1, §3]: for L = ∑ ℓ_{j}n_{j}L_{j} one has

lk(L, L) = ∑_{i<j} 2ℓ_{i}ℓ_{j} lk(n_{i}L_{i}, n_{j}L_{j}) + ∑_{i} ℓ_{i}^{2} lk(n_{i}L_{i}, n_{i}L_{i}).

Substituting (3.7.1), we get

(3.7.4)  lk(L, L) = − ∑_{i<j}(ℓ_{i} − ℓ_{j})^{2} lk(n_{i}L_{i}, n_{j}L_{j}).

Hence lk(L, L) is zero if ℓ_{1} = ⋯ = ℓ_{n}, and negative otherwise. □

In general, if some far weights are negative, some of the linking numbers lk(L_{i}, L_{j}) might be negative as well; in these cases the proof is more involved.

Lemma 3.7.5. If ν ≥ 2 then the self–linking number lk(L_{i}, L_{i}) is negative for any i.

Proof. For each i, let v_{i} be a node supporting L_{i}, α_{i} denotes the far weight at v_{i} and β_{i1}, …, β_{ik_{i}} the near weights at v_{i}, with β_{i1} the near weight on the edge supporting L_{i}.

If lk(L_{i}, L_{j}) > 0 for all j ≠ i, then the statement follows from (3.7.1). Hence, assume that lk(L_{i}, L_{j}) < 0 for some j. Assume that L_{i} and L_{j} are supported by nodes v_{i} and v_{j} respectively (the case v_{i} = v_{j} is also possible). Let γ be a path in Γ joining L_{i} to L_{j}. Since lk(L_{i}, L_{j}) < 0, one of the vertices lying on γ, call it v_{γ}, must have a negative weight. This, by assumption (a), must be a far weight, hence there is a unique v_{γ} along the path with this property. Now, if v_{i} is beyond v_{γ}, then by (b) we have α_{i} < 0. Otherwise, v_{i} = v_{γ} and α_{i} < 0 by the definition of v_{γ}. Next, let M_{i} be the multiplicity of v_{i}, namely

M_{i} = ∑_{j} lk(v_{i}, n_{j}L_{j}) = α_{i}β_{i2} … β_{ik_{i}}n_{i} + ∑_{j≠i} lk(v_{i}, n_{j}L_{j}).

But for j ≠ i one has lk(v_{i}, n_{j}L_{j}) = β_{i1} lk(L_{i}, n_{j}L_{j}), hence

(3.7.6)  − lk(L_{i}, L_{i}) = \frac{1}{β_{i1}n_{i}} ∑_{j≠i} lk(v_{i}, n_{j}L_{j}) = \frac{M_{i}}{n_{i}β_{i1}} − \frac{α_{i}β_{i2} … β_{ik_{i}}}{β_{i1}} > 0

as M_{i} > 0. □

Corollary 3.7.7. If the diagram has one or two arrowheads, then the statement of Proposition 3.7.2 holds.

Proof. Use Lemma 3.7.5 and the fact that the null–space is not trivial. □

The proof is based on induction via reduction of the diagram (via two operations).

Definition 3.7.8. Let Γ be a rooted graph. Assume that the supporting node v_{i} of the arrowhead vertex L_{i} has the following properties: it is not the root vertex, there is no node beyond it, L_{i} is the unique arrowhead supported by v_{i}. Hence, all its adjacent vertices except L_{i} and another one (in the direction of the root) are leaves. As above, denote the valency of v_{i} by k_{i} + 1 (see the picture below).

A collapse of v_{i} is a graph Γ′ with v_{i} replaced by an arrowhead vertex L′_{i} with multiplicity n_{i}β, where β = β_{i2} · · · · · · · β_{ik_{i}} and all other weights and multiplicities are unchanged.

α^{i} β^{i1}β^{i2} β^{i3} β_{ik_{i}} L^{i} (n_{i})  L′i (βn_{i}) Collapse

Lemma 3.7.9. The linking matrices of Γ and Γ′ are congruent. Moreover, if Γ satisfies the assumptions (a), (b) and (c) of the proposition, then so does Γ′.

Proof. We shall use the notation lk_{Γ} and lk_{Γ′} for the linking forms on Γ and Γ′.

For any vertex v (node or arrowhead), different from the deleted ones, we have lk_{Γ}(v, L_{i}) = lk_{Γ′}(v, βL′_{i}). We claim that lk_{Γ′}(βL′_{i}, βL′_{i}) = lk_{Γ}(L_{i}, L_{i}). This follows from (3.7.1) applied for Γ and Γ′, and from the fact that in Γ′ the relation βn_{i}L′_{i} + ∑_{j≠i} n_{j}L′_{j} = 0 holds. Thus, the linking matrix of Γ written in the basis L_{1}, …, L_{i}, …, L_{ν} is the same as the linking matrix of Γ′ in the basis L_{1}, …, βL′_{i}, …, L_{ν}. This proves the first part. As for the other part, the multiplicities of all vertices (besides the deleted ones) are preserved. This shows that if Γ satisfies (c), then so does Γ′, while (a) and (b) are obvious. □

Definition 3.7.10. Let v_{0} be a node with no other node beyond it. Let L_{1}, …, L_{k} be the arrowheads adjacent to v_{0} (k ≥ 2), denote their multiplicities by n_{1}, …, n_{k}. v_{0} might have several adjacent leaves as well. We denote by β the product of their near weights. Assume that the overall number of vertices of Γ is at least three.

A squeeze of Γ is a graph arising from Γ by replacing two arrowheads supported by v_{0} (say, L_{1} and L_{2}) by a single one, denoted by L_{s}, with multiplicity defined as

n_{s} := n_{2}β_{1} + n_{1}β_{2}

and the near weight β_{s} := β_{1}β_{2}.

α β^{1}  (n_{1}) β^{2}  (n_{2}) β^{k}  (n_{k})  α β^{s}  (n_{s}) β^{3}  (n_{3}) β^{k}  (n_{k}) Squeeze

Lemma 3.7.11. Let Γ′ be a squeeze of arrowhead L_{1} and L_{2} from Γ. If Γ satisfies the assumptions (a), (b) and (c) of the proposition, then so does Γ′. Moreover, the rational linking matrix of Γ is a direct sum of the linking matrix of Γ′ and a negative definite 1– dimensional matrix.

Proof. As for the first part we observe that β_{s} and n_{s} were chosen in such a way that all multiplicities of vertices are preserved. Moreover, by construction we have

(3.7.12) lk_{Γ′}(L_{i}, L_{j}) = lk_{Γ}(L_{i}, L_{j})  if {i, j} ∩ {1, 2} = ∅ and i ≠ j, lk_{Γ′}(n_{s}L_{s}, L_{j}) = lk_{Γ}(n_{1}L_{1} + n_{2}L_{2}, L_{j}) if j ≥ 3.

We claim that lk_{Γ′}(L_{j}, L_{j}) = lk_{Γ}(L_{j}, L_{j}) for j ≥ 3. Indeed, this follows from (3.7.12) and (3.7.1) applied for both graphs. Now, let us define

Λ_{1} = β_{1}L_{1} − β_{2}L_{2} and Λ_{2} = xL_{1} + yL_{2},

where the rational numbers x and y will be determined later. By definition,

lk_{Γ}(Λ_{1}, L_{j}) = 0 for any j ≥ 3.

The self-linking of Λ_{1} is equal to

lk_{Γ}(Λ_{1}, Λ_{1}) = β_{1}^{2} lk_{Γ}(L_{1}, L_{1}) + β_{2}^{2} lk_{Γ}(L_{2}, L_{2}) − 2αββ_{1}β_{2} … β_{k}.

If α > 0, then the above expression is negative, because lk_{Γ}(L_{1}, L_{1}) and lk_{Γ}(L_{2}, L_{2}) are negative by Lemma 3.7.5. If α < 0, we use (3.7.6) to show that lk_{Γ}(Λ_{1}, Λ_{1}) = −M_{v_{0}}(\frac{β_{1}}{n_{1}} + \frac{β_{2}}{n_{2}} ) < 0, because the multiplicity of M_{v_{0}} is positive. Hence, in all cases, lk_{Γ}(Λ_{1}, Λ_{1}) < 0.

Since lk_{Γ}(Λ_{1}, Λ_{1}) < 0 and lk_{Γ}(L_{1}, L_{2}) ≠ 0, there exist x and y such that lk_{Γ}(Λ_{2}, Λ_{1}) = 0 and Λ_{1}, Λ_{2} are linearly independent. Such x and y are determined up to a multiplicative constant. To choose it observe that

lk_{Γ}(Λ_{2}, L_{j}) = x lk_{Γ}(L_{1}, L_{j}) + y lk_{Γ}(L_{2}, L_{j}) = (x + y \frac{β_{1}}{β_{2}} ) lk_{Γ}(L_{1}, L_{j})

and lk_{Γ′}(L_{s}, L_{j}) = \frac{1}{β_{2}} lk_{Γ}(L_{1}, L_{j}). We normalize the rational numbers x and y so that x + y \frac{β_{1}}{β_{2}} = \frac{1}{β_{2}} . Then, we have for all j ≥ 3

(3.7.13)  lk_{Γ}(Λ_{2}, L_{j}) = lk_{Γ′}(L_{s}, L_{j}).

Finally, we show that lk_{Γ}(Λ_{2}, Λ_{2}) = lk_{Γ′}(L_{s}, L_{s}). This is done as follows. First, on Γ we have the relation n_{1}L_{1} + n_{2}L_{2} + ∑ n_{j}L_{j} = 0, which can be rewritten as

λ_{1}Λ_{1} + λ_{2}Λ_{2} + ∑_{j≥3} n_{j}L_{j} = 0

for some λ_{1} and λ_{2}. On the other hand, on Γ′ we have n_{s}L_{s} + ∑_{j≥3} n_{j}L_{j} = 0. Now taking the linking numbers with L_{r} for some r ≥ 3 we obtain

0 = ∑_{j≥3} n_{r} lk_{Γ}(L_{r}, L_{j}) + λ_{2} lk_{Γ}(L_{r}, Λ_{2}) = ∑_{j≥3} n_{r} lk_{Γ′}(L_{r}, L_{j}) + n_{s} lk_{Γ′}(L_{r}, L_{s}).

Now by (3.7.12), since r ≥ 3 the above equation simplifies to

λ_{2} lk_{Γ}(L_{r}, Λ_{2}) = n_{s} lk_{Γ′}(L_{r}, L_{s}).

¿From (3.7.13) and lk_{Γ′}(L_{r}, L_{s}) ≠ 0 it follows that n_{s} = λ_{2}. But then we have

lk_{Γ}(Λ_{2}, λ_{2}Λ_{2}) = − ∑_{j≥3} lk_{Γ}(Λ_{2}, n_{j}L_{j}) = − ∑_{j≥3} lk_{Γ′}(L_{s}, n_{j}L_{j}) = lk_{Γ′}(L_{s}, n_{s}L_{s}).

As n_{s} = λ_{2} we conclude that lk(Λ_{2}, Λ_{2}) = lk(L_{s}, L_{s}). Hence the linking form on Γ restricted to Λ_{2}, L_{3}, …, L_{n} is the same as the linking form on Γ′ written in basis L_{s}, L_{3}, …, L_{n}, while the element Λ_{1} splits out completely as an orthogonal summand. □

Finishing the proof of Proposition 3.7.2. By applying collapses and squeezes to Γ we end up with a diagram, for which no further collapse or squeeze is possible. This diagram has one or two arrowheads and we conclude the proof by Corollary 3.7.7. □

Remark 3.7.14. If we assume that the multiplicities of nodes of Γ are only non-negative (not just positive), we can still prove semidefiniteness of the linking matrix, possibly with higher dimensional null–space. We omit the details.

4. Semicontinuity results

Now we are ready to prove various semicontinuity results. In subsection 4.1 we recover (in a slightly weaker form) the classical semicontinuity results valid in the local case of algebraic plane curve singularities (proved by Varchenko in [Var], see also [St2]). Next, in 4.2, we analyse the behaviour of the spectrum under a degeneration of affine plane curves in the spirit of [NS]. Finally, we consider an affine plane curve, and we relate its spectrum at infinity with the spectrum of its local singularities, see 4.3. This type of comparison is unknown in Hodge theory.

4.1. Semicontinuity of the local singularity spectrum. Recall that in the local case Sp_{MHS} = Sp_{HVS} (cf. 2.3.4), which will be denoted just by Sp.

Let us consider now the following situation. Let f_{t}(x, y) be a smooth family of holomorphic functions in two local coordinates depending on a local parameter t. Assume that f_{0}(x, y) = 0 has an isolated singularity at the origin. Let us introduce the following notation.

• B is a small ball centered at the origin such that f_{0}^{−1}(0) is transverse to ∂B and f_{0}(z)/|f_{0}(z)|: ∂B \ f_{0}^{−1}(0) → S^{1} is a Milnor fibration;

• L_{0} = f_{0}^{−1}(0) ∩ ∂B is the link of f_{0} at 0, and Sp_{0} the spectrum of the link;

• t ≠ 0 and |t| is sufficiently small so that f_{t}^{−1}(0) ∩ ∂B is a transversal intersection, and this link is isotopic in ∂B to L_{0};

• C = f_{t}^{−1}(0) ∩ B;

• z_{1}, …, z_{k} are singular points of C, L_{1}^{sing}, …, L_{k}^{sing} the corresponding local links of these singularities, and Sp_{1}, …, Sp_{k} denote the spectrum of L_{1}^{sing}, …, L_{k}^{sing} respectively.

Proposition 4.1.1. Fix x ∈ [0, 1] such that e^{2πix} is not a root of the Alexander polynomial of L_{0}. Then

\begin{aligned}|Sp_{0} ∩ (x, x + 1)| & ≥ ∑_{j} |Sp_{j} ∩ (x, x + 1)| \\ |Sp_{0} \ [x, x + 1]| & ≥ ∑_{j} |Sp_{j} \ [x, x + 1]|. \end{aligned} ; (4.1.2)

Proof. Assume x ≠ 0, 1. We shall prove only the first inequality, the second one is completely analogous (in Section 2.5 all inequalities are given in pairs, we use the first one to prove results about Sp ∩ (x, x + 1), the other one to prove results about Sp \ [x, x + 1]). As L_{j}^{sing} is an algebraic links, μ_{j} is the degree of the Alexander polynomial of L_{j}^{sing}. Hence μ_{j} − σ_{L_{j}^{sing}}(ζ) + n_{L_{j}^{sing}}(ζ) ≥ 2|Sp_{j} ∩ (x, x + 1)| by Corollary 2.4.6.

By assumption n_{L_{0}}(ζ) = 0. Since L_{0} is also an algebraic link, 1−χ(C_{smooth}) is the degree of the Alexander polynomial of L_{0}. Thus, again by Corollary 2.4.6, one gets −σ_{L_{0}}(ζ) + (1 − χ(C_{smooth})) = 2|Sp_{0} ∩ (x, x + 1)|. Then we conclude by the inequality (2.5.6).

Next, assume that x ∈ {0, 1}. The assumption that e^{2πix} is not a root of the Alexander polynomial means that L_{0} is a knot and |Sp_{0} ∩ (0, 1)| = |Sp_{0} ∩ (1, 2)| is the delta invariant δ_{0}. For any singularity link, hence for L_{j}^{sing}, δ_{j} = |Sp ∩ (0, 1]| ≥ |Sp_{j} ∩ (0, 1)|. Hence the statement follows from δ_{0} ≥ ∑ δ_{j}. □

4.2. Semicontinuity of spectrum at infinity of families of affine curves. The methods described in this paper allow us also to prove the results on semicontinuity of the spectrum at infinity in the sense of [NS].

Let F_{t}: ℂ^{2} → ℂ be a smooth family of polynomials with a local deformation parameter t. Let Sp_{t} be corresponding Hodge spectrum at infinity and Irr_{t} be the irregularity of the link at infinity L_{t,reg}^{∞} associated with F_{t}.

Note that over a small punctured disc D^{∗} ∋ t the spectrum Sp_{t} is constant.

Theorem 4.2.1. Fix x ∈ [0, 1] such that {x, x + 1} ∩ Sp_{t} = ∅ for t ∈ D^{∗}. Then

|Sp_{t} ∩ (x, x + 1)| + Irr_{t} ≥ |Sp_{0} ∩ (x, x + 1)|

and the same statement holds for Sp \ [x, x + 1] instead of Sp ∩ (x, x + 1).

Proof. Let us assume first that x is not an integer. We write ζ := e^{2πix} ∈ S^{1} \ {1}.

Let us choose c such that C_{0} = F_{0}^{−1}(c) is smooth and regular at infinity. Furthermore, choose ξ and r_{0} such that S^{3}(ξ, r_{0}) ∩ C_{0} is the regular link of F_{0} at infinity, denoted by L_{0}.

By openness of transversality condition, there exist D, an open neighbourhood of 0, and W_{c}, an open neighbourhood of c, such that for any w ∈ W_{c} and t ∈ D, the intersection F_{t}^{−1}(w) ∩ S^{3}(ξ, r_{0}) is transverse and isotopic to L_{0}. Let us take any t ∈ D^{∗} and choose w ∈ W_{c} such that C_{t} = F_{t}^{−1}(w) is smooth and regular at infinity. Finally, choose r_{t} such that L_{t} := S^{3}(ξ, r_{t}) ∩ C_{t} is the regular link at infinity of C_{t}.

Since C_{t} ∩ B(ξ, r_{0}) is isotopic to C_{0} ∩ B(ξ, r_{0}), by Corollary 2.5.4 we get

−σ_{L_{t}}(ζ) + n_{L_{t}}(ζ) + 1 − χ(C_{t} ∩ B(ξ, r_{t})) ≥ −σ_{L_{0}}(ζ) + n_{L_{0}}(ζ) + 1 − χ(C_{0} ∩ B(ξ, r_{0})).

By assumption, ζ is not a root of Δ_{Irr_{t}}(L_{t}). Hence, applying 3.4.5 for F_{t} and F_{0} we obtain

−σ_{L_{t}} + deg Δ_{Irr_{t}}(L_{t}) + 2 Irr_{t} ≥ −σ_{L_{0}}(ζ) + ñ_{L_{0}}(ζ) + deg Δ_{Irr_{0}} + 2 Irr_{0} .

Then Corollary 2.4.6 implies

|Sp_{HVS}(L_{t}) ∩ (x, x + 1)| + Irr_{t} ≥ |Sp_{HVS}(L_{0}) ∩ (x, x + 1)| + Irr_{0} .

Finally, Corollary 3.5.4 provides the result. To show the statement for Sp \ [x, x + 1] we use the same argument.

If x = 0 (or x = 1) and x satisfies the assumption of the theorem, then for any t ≠ 0, and for any θ > 0 sufficiently small, we have

|Sp_{t} ∩ (θ, 1 + θ)| = Sp_{t} ∩ (0, 1)|.

On the other hand, for θ > 0 sufficiently small, we have |Sp_{0} ∩ (θ, 1 + θ)| ≥ |Sp_{0} ∩ (0, 1)| (we have an equality iff Sp_{0} ∩ {1} = ∅). Hence the statement of the theorem for x = 0, 1 follows from the statement for x ∈ (0, 1). □

4.3. Spectrum at infinity of a singular curve. Let C ⊂ ℂ^{2} be an irreducible plane algebraic curve given by zero set of a reduced polynomial F. Let z_{1}, …, z_{k} be its singular points and Sp_{1},…,Sp_{k} their (Hodge or HVS) spectrum. Let Sp_{∞} be the Hodge spectrum of F at infinity. Similarly, let L_{reg}^{∞} be the regular link of F at infinity, Sp_{HVS}(L_{reg}^{∞}) its HVS–spectrum and Irr be as defined in (3.2.4).

Theorem 4.3.1. With the above notations, for all x ∈ [0, 1] such that e^{2πix} is not a root of the Alexander polynomial of L_{reg}^{∞} we have

|Sp_{HVS}(L_{reg}^{∞}) ∩ (x, x + 1)| + Irr ≥ ∑_{j} |Sp_{j} ∩ (x, x + 1)|

|Sp_{∞} ∩ (x, x + 1)| + Irr ≥ ∑_{j} |Sp_{j} ∩ (x, x + 1)| .

(4.3.2)

Moreover, the analogous statement holds if we replace Sp ∩ (x, x + 1) by Sp \ [x, x + 1].

In the good case, if the regular link at infinity is fibred (e.g. if it is a knot), then Irr = 0 and the second inequality of (4.3.2) takes the form |Sp_{∞} ∩ (x, x + 1)| ≥ ∑_{k=1}^{n} |Sp_{k}(x, x + 1)|.

Proof. First we assume that x ∈ (0, 1). We focus on the case Sp ∩ (x, x + 1); the case Sp \ [x, x + 1] is analogous.

If C is regular at infinity, the inequality (2.5.6) reads as

(4.3.3)  −σ_{L_{reg}^{∞}}(ζ) + n_{L_{reg}^{∞}}(ζ) + (1 − χ(C_{smooth})) ≥ ∑_{j}(−σ_{j}(ζ) + n_{j}(ζ) + μ_{j}),

where C_{smooth} is the smoothing of C. Since each link L_{j}^{sing} is algebraic, −σ_{j}(ζ)+n_{j}(ζ)+μ_{j} ≥ 2|Sp_{j} ∩ (x, x + 1)|. On the other hand, by Proposition 3.4.5 we get

(4.3.4)  1 − χ(C_{smooth}) = Irr + deg Δ_{Irr}.

By Lemma 3.3.1(d) Δ_{L_{reg}^{∞}}^{h} has no roots outside the unit circle, hence Corollary 2.4.6 applies. Since ζ is not a root of Δ_{L_{reg}^{∞}}^{h}, ñ(ζ) = 0, hence

(4.3.5)  −σ_{L_{reg}^{∞}}(ζ) + deg Δ_{L_{reg}^{∞}}^{h} = 2|Sp_{L_{reg}^{∞}} ∩ (x, x + 1)|,

and n(ζ) = Irr (see 3.2). Then (4.3.3), (4.3.4) and (4.3.5) prove the statement in this case.

If C is not regular at infinity, we argue as follows. We take an r_{0} such that C ∩ S^{3}(ξ, r_{0}) is the link of C at infinity, denoted by L_{C}. Then (2.5.6) yields

(4.3.6)  −σ_{L_{C}}(ζ) + n_{L_{C}}(ζ) + (1 − χ(C_{smooth}^{r_{0}})) ≥ ∑_{j}(−σ_{j}(ζ) + n_{j}(ζ) + μ_{j}),

where C_{smooth}^{r_{0}} := C_{ε} ∩ B(ξ, r_{0}) is the smoothing of C in B(ξ, r_{0}). Here C_{ε} := F^{−1}(ε) is smooth and regular at infinity (for ε non-zero and sufficiently small). Moreover, we can assume that the links C ∩ S^{3}(ξ, r_{0}) and C_{ε} ∩ S^{3}(ξ, r_{0}) are isotopic. Let r_{1} be such that C_{ε} ∩ S^{3}(ξ, r_{1}) is the regular link of F at infinity. Corollary 2.5.4 applied to C_{ε} yields

(4.3.7)  −σ_{L_{reg}^{∞}}(ζ) + n_{L_{reg}^{∞}}(ζ) − (−σ_{L_{C}}(ζ) + n_{L_{C}}(ζ)) ≥ χ(C_{01}),

where C_{01} = C_{ε} ∩ (B(ξ, r_{1}) \ B(ξ, r_{0})). (4.3.6) and (4.3.7) combined yields

−σ_{L_{reg}^{∞}}(ζ) + n_{L_{reg}^{∞}}(ζ) + (1 − χ(C_{ε})) ≥ ∑_{j}(−σ_{j}(ζ) + n_{j}(ζ) + μ_{j}).

This inequality is identical to (4.3.3) and we proceed further as in the previous case.

Assume that x = 0. Then, by the assumption, 1 is not a root of the Alexander polynomial of L_{reg}^{∞}, hence L_{reg}^{∞} is a knot (because U_{λ=1} is trivial, but its dimension is ν − 1 by Proposition 3.2.8). Therefore the link at infinity is good, Irr = 0 and Sp_{HVS}(L_{reg}^{∞}) = Sp_{∞}.

For θ > 0 sufficiently small |Sp_{∞} ∩(0, 1)| = |Sp_{∞} ∩(θ, 1+θ)| (because 1 ∉ Sp_{HVS}(L_{reg}^{∞}) = Sp_{∞}). On the other hand, in the local case, |Sp_{j} ∩ (0, 1)| ≤ |Sp_{j} ∩ (θ, 1 + θ)|, hence the statement follows from the case x ∈ (0, 1).

The case x = 1 follows by the same argument with θ < 0. □

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Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

E-mail address: mcboro@mimuw.edu.pl

A. Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13-15, Hungary.

E-mail address: nemethi@renyi.hu