On the Semicontinuity of the Mod 2 Spectrum of Hypersurface Singularities

J. ALGEBRAIC GEOMETRY 24 (2015) 379–398 S 1056-3911(2015)00640-3 Article electronically published on January 8, 2015

ON THE SEMICONTINUITY OF THE MOD 2 SPECTRUM OF HYPERSURFACE SINGULARITIES

MACIEJ BORODZIK, ANDRAS´ NEMETHI,´ AND ANDREW RANICKI

Abstract We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in Cn+1, using Seifert forms of high-dimensional non-spherical links, the Levine– Tristram signatures and the generalized Murasugi–Kawauchi inequality.

1. Introduction The present article extends the results of [BoNe12], valid for plane curves, to arbitrary dimensions. The main message is that the semicontinuity of the mod 2 Hodge spectrum (associated with local isolated singularities or with affine polynomials with some ‘tameness’ condition) is topological in nature, although its very definition and all known ‘traditional’ proofs sit deeply in the analytic/algebraic theory. Usually, the spectrum cannot be deduced from the topological data. In order to make clear these differences, let us briefly review the relevant invariants of a local isolated hypersurface singularity f :(Cn+1, 0) → (C, 0). The ‘homological package of the embedded topological type’ of an isolated hypersurface singularity contains information about the vanishing cohomology (cohomology of the Milnor fiber), the algebraic monodromy acting on it, and different polarizations of it including the intersection form and the (linking) Seifert form. In fact, the Seifert form determines all this homological data. See e.g. [AGV84]. On the other hand, the vanishing cohomology carries a mixed Hodge structure polarized by the intersection form and the Seifert form, and it is compat- ible with the monodromy action too. It has several definitions, but all of them are analytic [Arn81, Stee76, Stee85, Var83a, Var83b]. The equivariant Hodge

Received October 5, 2012.

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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). Arnold conjectured [Arn81] and Varchenko [Var83a, Var83b] and Steenbrink [Stee85] proved that the spectrum behaves semicontinuously under deformations. In this way it becomes a very strong tool e.g. in the treatment of the adjacency problem of singularities. More precisely, in the presence of a deformation ft,wheret is the deformation parameter t ∈ (C, 0), the semicontinuity guarantees that |Sp(f0) ∩ I| |Sp(ft=0 ) ∩ I| for certain semicontinuity domains I. Arnold conjectured that I =(−∞,α] is a semicontinuity domain for any α ∈ R; Steenbrink and Varchenko proved the statement for I =(α, α + 1], which implies Arnold’s conjecture. Additionally, for some cases, Varchenko verified the stronger version, namely, semicontinuity for I =(α, α + 1) [Var83a]. The relation between the Hodge invariants and the Seifert form was es- tablished in [Nem95], proving that the collection of mod 2 spectral pairs is equivalent to the real Seifert form. Therefore, the real Seifert form determines the mod 2 spectrum, that is, the collection of numbers α mod 2 in (0, 2], where α runs over Sp(f). Clearly, for plane curve singularities, i.e. when n =1,the mod 2 reduction does not lose information. Surprisingly, this correspondence can be continued: in [BoNe12] it is proved that if n = 1, then the semicontinuity property of the mod 2 spectrum is also topological: it can be proved independently of Hodge theoretical tools, following from classical link theory. Its precise statement is the following: length one ‘intervals’ intersected by the mod 2 spectrum, namely, sets of type Sp ∩ (α, α +1)and(Sp ∩ (0,α)) ∪ (Sp ∩ (α +1, 2]), for α ∈ [0, 1], satisfy semicontinuity properties whenever this question is well–posed (and under certain mild extra assumptions regarding the roots of the monodromy operator). The tools needed in this topological proof were the following: properties of the Levine–Tristram signature, its connection with the spectrum, and the Murasugi–Kawauchi inequality valid for it. It was very natural to ask for a possible generalization of this fact to arbitrary dimensions. This was seriously obstructed by two facts: the existence of a Seifert form up to S–equivalence and the analogue of the Murasugi– Kawauchi inequality valid for any 2–codimension embedded manifold of S2n+1. This article has the following plan. First, we collect and prove all the other steps which are needed in the verification of the semicontinuity, except these two main obstructions. Then, we handle the proofs of these two obstructions at two different levels of difficulty.

The first level proof (the ‘strong version’) is completely topological and uses only the isotopy types of the corresponding higher dimensional links. It is rather long, and the required tools are different from those needed to verify the semicontinuity. Therefore, we decided to include them in another paper [BNR12]. We wish to emphasize that the results of [BNR12] combined with the present article show that the mod 2 spectrum is a purely topological invariant, its semicontinuity is a topological phenomenon and the proof of semicontinuity also can be done topologically. For more precise comments see the paragraphs after Assumption 2.1.14. The second proof is adapted to the present application: it assumes the existence of concrete Seifert surfaces and of concrete cobordisms between them, an assumption that is automatically satisfied in our present situation, since the links are cut out by holomorphic/polynomial equations. In this way the proof becomes considerably simpler and more transparent. This ‘weak version’ of Murasugi–Kawauchi type inequality is given here in Subsection 2.1. Some properties of the links occurring, e.g. the existence of the Milnor fiber, which are rather standard in the theory of local singularities of affine polynomial maps, are used several times. Therefore, from the point of view of the present application, the article can be considered complete and self–contained. The article is organized as follows. In Section 2 we review the theory of higher dimensional links and their Seifert forms. Then we discuss the generalization of the Murasugi–Kawauchi inequality: we state the general version of it from [BNR12], and we give here a simpler proof using some additional assumptions (which are satisfied in the present case). In the last subsection we show how they can be applied for complex affine hypersurfaces. Section 3 reviews facts about hermitian variation structures, the tool which connects Seifert forms with the spectral numbers via the Levine–Tristram signatures. Then we discuss how the machinery works in the case of a global affine hypersurface, provided that the corresponding polynomial map satisfies some regularity conditions, which guarantees similar properties at infinity that are valid for local isolated singularities (the ‘Milnor package’). In this section we review some needed facts about these ‘tameness’ conditions as well. Section 4 contains the three semicontinuity results: (a) the local case of deformation of isolated singularities, (b) semicontinuity of the mod 2 spectrum of the mixed Hodge structure at infinity of ‘nice’ polynomials, and (c) an inequality which compares the mod 2 spectrum at infinity of an affine fiber with the local spectrum of its singular points. We wish to emphasize that this type of semicontinuity (which produces bounds for the local spectra in terms of the spectrum at infinity) is unknown in Hodge theory.

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Certain proofs and parts of the note show similarities with [BoNe12]; therefore some arguments are shortened, although we have tried to provide a pre- sentation emphasizing all the basic steps of the proof. Notation. For a ﬁnite set A,wedenoteby|A| the cardinality of A;fora subset X of Cn+1,intX denotes its interior and X its closure.

2. High dimensional links and their signatures 2.1. A quick trip through high dimensional link theory. Here we present, in a condensed form, results about high dimensional links,thatis, codimension 2 embeddings M ⊂ S2n+1,whereM is a closed oriented (2n−1)– dimensional manifold, for any n 1. The theory resembles the ‘classical’ theory of embeddings of disjoint unions of copies of S1’s into S3, the special case n = 1, although the proofs of the corresponding statements are more involved. Unless specified otherwise, the results are from [BNR12]. The first one, however, is older. It dates back to Erle [Erle69]; see also [BNR12]. Proposition 2.1.1. Let M 2n−1 ⊂ S2n+1 be a link. Then there exists a compact, oriented, connected 2n–dimensional manifold Σ ⊂ S2n+1 such that ∂Σ=M. We shall call such Σ a Seifert surface for M; in general, there is not a canonical choice of Σ, although there is a canonical choice if M is the link of a singularity. Given a Seifert surface Σ, let FHn be the torsion free quotient of Hn(Σ, Z). Let α1,...,αm be the basis of FHn and let us represent it by { }m cycles denoted also by α1,...,αm. We can define a matrix S = sij i,j=1 + + with sij =lk(αi,αj ), where αj is a cycle αj pushed slightly off Σ in the positive direction. S is called the Seifert matrix of M relative to Σ. Remark 2.1.2. Calling Σ a Seifert surface might be an abuse of language since dim Σ = 2n. However, this terminology is standard in the literature devoted to higher dimensional knot theory; see e.g. [BlMi97, Ker71]. Proposition 2.1.3 (See [BNR12, Main Theorem 1]). Any two Seifert matrices of a given link M, corresponding to possibly different Seifert surfaces, are S-equivalent. We refer to [Kaw96, Chapter 5.2] for the definition of the S-equivalence of Seifert matrices. The above result is standard for knots S2n−1 ⊂ S2n+1 (when b = S +(−1)nST is invertible), but is less familiar for links M 2n−1 ⊂ S2n+1. Let us fix a link M and let Σ be one of its Seifert surfaces, with Seifert matrix S. Following an algebraic result of Keef [Keef83] (see also [BoNe13,

Section 3]), over the ﬁeld of rational numbers changing S by a matrix S- equivalent to it we can write

(2.1.4) S = Sndeg ⊕ S0,

where det Sndeg = 0 and all the entries of S0 are zero. Let us deﬁne

(2.1.5) n0 := size of S0. Moreover, the right hand side of (2.1.4) does not depend on a particular choice of the Seifert matrix in its S-equivalence class; more precisely, the integer n0 and the rational congruence class of the matrix Sndeg are well defined. Definition 2.1.6. Let M 2n−1 ⊂ S2n+1 be a link with a Seifert surface Σ2n ⊂ S2n+1, S the Seifert matrix and ST the transposed matrix. The Alexander polynomial of M is defined by · − n T ΔM (t):=det(Sndeg t +( 1) Sndeg). For any ξ ∈ S1 \{1} define the Levine–Tristram signature and nullity of M at ξ: n+1 T σM (ξ) := signature [ (1 − ξ)S +(−1) (1 − ξ)S ] n+1 T nM (ξ) := nullity [ (1 − ξ)S +(−1) (1 − ξ)S ] . Remark 2.1.7. By convention, used e.g. in [Nem95, Section 4], the signature of a skew-hermitian form C is the signature of the hermitian form i · C.

Remark 2.1.8. Strictly speaking, ΔM (t) is the classical Alexander polynomial only if n0 = 0, i.e. if S = Sndeg.Otherwise,ifn0 > 0, then ΔM (t) is the (n0 + 1)–st classical Alexander polynomial. In this note we shall not insist on this distinction. The following result is completely analogous to its one-dimensional version. Lemma 2.1.9. If ξ ∈ S1 \{1} is not a root of the Alexander polynomial, 1 then nM (ξ)=n0.OtherwisenM (ξ) >n0.Forξ,η ∈ S \{1} if there exists an arc in S1 \{1} connecting them and not containing any root of the Alexander polynomial, then σM (ξ)=σM (η). For completeness of the argument we present a straightforward proof.

Proof. Let S = Sndeg ⊕ S0.ThenσM (ξ) = signature[(1 − ξ)Sndeg + − n+1 − T − − n+1 − T ( 1) (1 ξ)Sndeg]andnM (ξ) = null[(1 ξ)Sndeg+( 1) (1 ξ)Sndeg]+n0. Hence we can assume S = Sndeg.Butthen, (1 − ξ)S +(−1)n+1(1 − ξ)ST =(ξ − 1)(ξS +(−1)nST ). n+1 T Hence ΔM (ξ) = 0 if and only if (1−ξ)S +(−1) (1−ξ)S is non-degenerate. Furthermore, if the arc γ joins ξ and η, then the matrix n+1 T (1 − α)S +(−1) (1 − α)S ]α∈γ

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is a path in the space of non-degenerate sesquilinear forms along which the signature is constant. Our main tool is the following generalization of the classical Murasugi– Kawauchi inequality.

Theorem 2.1.10 (see [BNR12, Main Theorem 2]). Let (Y ; M0,M1) ⊂ 2n+1 2n+1 S × ([0, 1]; {0}, {1}) be a cobordism of links M0,M1 ⊂ S . For any 2n+1 1 Seifert surfaces Σ0, Σ1 ⊂ S for M0,M1 and ξ ∈ S \{1} we have (2.1.11) | − | ∪ ∪ − − σM0 (ξ) σM1 (ξ) bn(Σ0 M0 Y M1 Σ1) bn(Σ0) bn(Σ1)+nM0 (ξ)+nM1 (ξ),

where bn(·) denotes the n-th Betti number. Remark 2.1.12. In the classical case n = 1, the inequality looks slightly different (compare [Kaw96, Theorem 12.3.1]); namely, one has | − | | − | (2.1.13) σM0 (ξ) σM1 (ξ) + nM0 (ξ) nM1 (ξ) b1(Y,M0)=b1(Y,M1) . In this case one has a better interplay between the topology and nullities (see for example the quantities wL and uL in [Boro12, Section 5]). We do not know whether this stronger inequality holds in higher dimensions; the approach of [BNR12] seems to be insufficient to prove that. For the convenience of the reader, and also for completeness of the proof of the semicontinuity property, we present below a sketch of the proof of Theorem 2.1.10 under the following additional assumption. Assumption 2.1.14. There exists a smooth function f : S2n+1×[0, 1] → C such that: −1 2n+1 −1 • Y = f (0),inparticularM0 = S ×{0}∩f (0) and M1 = S2n+1 ×{1}∩f −1(0); • 2n+1× \ → 1 → f(x) the map arg f : S [0, 1] Y S given by x |f(x)| is surjective; • there is a non-critical value δ ∈ S1 of arg f and of the restriction 2n+1 of arg f to S ×{0, 1}\(M0 M1) such that if we define Ω= −1 2n+1 (arg f) (δ),thenΣi =Ω∩ S ×{i} for i =0, 1. If we study deformations of isolated singularities of hypersurfaces, the existence of function f will always be clear. The function will then be even complex analytic, although we only need smoothness to ensure that Sard’s lemma works. In fact, many arguments of [BNR12] rely on constructing a function like the one above using homological methods. The last condition of the assumptions is slightly weaker than the condition that arg f gives a 2n+1 1 fibration of S ×{i}\Mi → S with fiber isotopic to Σi for i =0, 1. Assumption 2.1.14 is satisfied in all the cases that we consider in the paper (see Remark 2.2.2 or Proposition 3.2.1 below). This enables us to prove all the results from Section 4 without referring to surgery theory from [BNR12].

However, in this approach, the signatures and their properties depend a priori on the function f and on the specific choice of the Seifert surface. It is not clear whether these properties are of a topological nature, in particular, whether the results from Section 4 are purely topological or not. The theorems from [BNR12] clarify this. The Levine–Tristram signatures are defined even if the link in question is not fibered and depend only on the isotopy of the link. Moreover, the behaviour of the Levine–Tristram signatures under cobordism depends only on homological properties of the manifold which realizes the cobordism between links. 2n+1 2n+1 Proof of Theorem 2.1.10. We denote C0 := S ×{0} and C1 := S × {1}.On(−C0) C1 (the minus sign denotes that we reverse the orientation) we can consider the linking form: if α, β are two disjoint n-dimensional cycles on C0 C1 we define

lk(α, β)=− lk(α0,β0) + lk(α1,β1),

where αi = α∩Ci, βi = β ∩Ci, i =0, 1. This definition allows us to define the Seifert pairing for M0 M1 with respect to the Seifert surface −Σ0 Σ1 by + + S(α, β) = lk(α, β ), where β is the cycle β pushed off slightly from Σ0 Σ1 in a positive normal direction. If S0 and S1 denote the Seifert pairings for M0 and M1, then clearly S =(−S0) ⊕ S1. Observe now that ∂Ω=Σ0 ∪Y ∪Σ1.Letk :Σ0 Σ1 → ∂Ω be the inclusion. We denote also j : ∂Ω → Ωandi = j ◦ k.

Claim 1. If α, β are disjoint and are elements of ker i∗ : Hn(Σ0 Σ1; Q) → Hn(Ω; Q), then S(α, β)=0. To prove the claim, we assume that α, β ∈ ker i∗. Then, there exist (n+1)- dimensional cycles A, B ⊂ Ω such that ∂A = α, ∂B = β.LetB+ be the cycle in S2n+1 × [0, 1] obtained by pushing B oﬀ Ω in a positive normal direction. Clearly ∂B+ = β+.Butthen lk(α, β+)=A · B+ =0, because A and B+ are disjoint. This proves the claim.

Now let k∗ and j∗ denote the induced maps on n-th homology with rational coeﬃcients. 1 Claim 2. We have dim ker j∗ = 2 bn(∂Ω). The proof of Claim 2 is a standard homological algebra (compare e.g. [BlMi97, Lemma 3.8]). The homology long exact sequence (with rational coeﬃcients) of the pair (Ω,∂Ω) can be split into two parts:

→ Hn+2(Ω,∂Ω) → Hn+1(∂Ω) → Hn+1(Ω) → Hn+1(Ω,∂Ω) → ker j∗ → 0

0 → im j∗ → Hn(Ω) → Hn(Ω,∂Ω) → Hn−1(∂Ω) → ... .

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Since in an exact sequence the alternating sum of dimensions is 0, we infer that

dim ker j∗ = − dim Hn+1(Ω,∂Ω) + dim Hn+1(Ω) − dim Hn+1(∂Ω) + ...

dim im j∗ = − dim Hn(Ω) + dim Hn(Ω,∂Ω) − dim Hn−1(∂Ω) + ... .

Poincar´e duality implies that dim ker j∗ = dim im j∗, which concludes the proofofClaim2.Wenowwrite

dim ker i∗ dim ker k∗ +(dimkerj∗ − dim coker k∗) 1 = b (∂Ω) − (b (∂Ω) − b (Σ Σ )). 2 n n n 0 1 ∈ 1 \{ } Consider now ξ S 1 and assume for simplicity that nM0 (ξ)=nM1 (ξ)= 0 (the proof in the general case is only slightly more complicated). This means that the form (1 − ξ)S +(1− ξ)ST is non-degenerate. By the claim, it vanishes on a subspace of dimension dim ker i∗; therefore the absolute value of its signature is bounded by

bn(Σ0 Σ1) − 2 dim ker i∗ = bn(∂Ω) − bn(Σ0 Σ1). 2.2. Application to affine or local complex hypersurfaces. Let X ⊂ Cn+1 be a complex hypersurface with at most isolated singularities. If x ∈ X 2n+1 is a singular point of X, consider a sufficiently small sphere Sx S centered at x. The intersection X ∩ Sx embedded in Sx is called the link of the hypersurface singularity (X, x) ⊂ (Cn+1,x). We shall denote it by Mx ⊂ Sx. Now let B be any ball in Cn+1 such that S = ∂B is transverse to X (in particular S omits the singular set of X). Let M := S ∩ X ⊂ S. It is a closed codimension 2 submanifold of S, i.e. a high dimensional link. Let x1,...,xk be those singular points of X that lie inside B. We have the following result. Proposition 2.2.1. Let Xs be the smoothing of X inside B (whose boundary will be identified via an isotopy with M too) and Σ a Seifert surface of M ⊂ S. Moreover, for j =1,...,k,letΣj be the Milnor fiber at xj.Then for all ξ ∈ S1 \{1}, σ (ξ)− σ (ξ)b (Xs∪ Σ)−b (Σ)− b (Σ )+n (ξ)+ n (ξ). M Mxj n M n n j M Mxj j j j Proof. For any j =1,...,k let us pick pairwise disjoint small Milnor balls ∩ ⊂ ⊂ Bj around xj ; hence ∂Bj X, ∂Bj X = Mxj and Σj ∂Bj .LetB0 B be a ball disjoint from X ∪ B1 ∪···∪Bk.Foreachj =1,...,k,letγj be a smooth, closed curve joining ∂B0 with ∂Bk such that γj is disjoint from Σj and from all other balls Bl and other curves γl for l = j. We also assume that

B1 γ ′ 1 S S B 0 γ B2 2 γ 3 B3

X X

Figure 1. The cobordism between local and global links

the relative interior of γj is disjoint from B0 and Bj (see Figure 1). Let Uj be a small tubular neighbourhood of γj and consider U = B0 ∪ (Bj ∪ Uj )andY = X ∩ B \ int Bj . j j

The assumptions on γj ’s guarantee that • U is diffeomorphic to a standard ball; • S := ∂U (after possibly smoothing corners) is a sphere transverse to X; • ∩ ··· M := S X is a disjoint union Mx1 Mxk of the local links; • Σ := Σ1 ··· Σk is a Seifert surface for M . In particular σ (ξ)= σ (ξ)andn (ξ)= n (ξ). M Mxj M Mxj j j We say that the cobordism of links (Y ; M,M) ⊂ (B \ int U; S, S) ≈ S2n+1 × ([0, 1]; {0}, {1}) is constructed by the ‘boleadoras’ trick. The generalized Murasugi–Kawauchi inequality of Theorem 2.1.10 gives σ (ξ)− σ (ξ) b (Y ∪Σ ∪Σ)−b (Σ)−b (Σ)+n (ξ)+ n (ξ). M Mxj n n n M Mxj j j s s Now Σj ≈ X ∩ Bj by [Miln68]; hence Y ∪ Σ ≈ X , and the statement follows. Remark 2.2.2. (a) If X is given by {f =0} for a complex polynomial f and there exists δ ∈ S1 such that Σ = (arg f)−1(δ) (for instance if the link M is fibred), then we can find B0 and the paths γ1,...,γk (usedinthe

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proof of Proposition 2.2.1) disjoint from (arg f)−1(δ). Then (arg f)−1(δ) ∩ S

is a disjoint union of the Seifert surfaces for Mx1 ,...,Mxk . This proves that Assumption 2.1.14 in this case is satisﬁed. (b) Proposition 2.2.1 is the main tool in the proof of the semicontinuity of the mod 2 spectrum; cf. Section 4. We remark that if n =1,thisproposition allows us to prove semicontinuity of the spectrum without referring to [Boro12] and [BNR12]. (The article [BoNe12] uses [Boro12].) (c) The space U, as drawn in Figure 1, resembles the South American throwing weapon boleadoras [Bol], hence the name of the cobordism construction in 2.2.1.

3. Hermitian variation structures and mixed Hodge structures 3.1. Generalities about hermitian variation structures. Variation structures were introduced in [Nem95]. As was shown in [BoNe13] and [BoNe12] they form a bridge between knot theory and Hodge theory. Let us recall briefly the definition, referring to [Nem95] or [BoNe13, Section 2] for all details and further references. Definition 3.1.1. Fix a sign ε = ±1. An ε–hermitian variation structure (in short: HVS) consists of the quadruple (U; b, h, V ), where U is a complex linear space, b: U → U ∗ is an ε–hermitian linear map (it can be regarded as an ε–symmetric pairing on U × U), h: U → U is an automorphism preserving b,andV : U ∗ → U is a linear map such that

∗ ∗ V ◦ b = h − I and V = −εV ◦ h .

Here · denotes the complex conjugate and ∗ the duality. We shall call an HVS simple if V is an isomorphism. In this case V ∗ determines b and h completely by the formulae h = −εV (V )−1 and b = ∗−1 −V −1 − εV . It was proved in [Nem95] that each simple variation structure is a direct sum of indecomposable ones; moreover the decomposable ones can be completely classiﬁed. More precisely, for any k 1andanyλ ∈ C | | Wk ± with λ = 1, there are two structures λ ( 1)(uptoanisomorphism).In their case h is a single Jordan block of size k with eigenvalue λ.Furthermore, for any k 1andanyλ ∈ C such that 0 < |λ| < 1 there exists a single V2k structure λ .Inthiscaseh is a direct sum of two Jordan blocks of size k: one block has eigenvalue λ, the other 1/λ¯. In particular for each HVS V there

exists a unique decomposition V k ·V2k ⊕ k ·Wk (3.1.2) = qλ λ pλ(u) λ (u), 0<|λ|<1 |λ|=1 k1 k1,u=±1 k k ± ·V where qλ and pλ( 1) are certain non–negative integers. Here we write m for a direct sum of m copies of V. Next we recall the definition of the spectrum associated with an HVS. V k ± k Definition 3.1.3. Let be an HVS. Let pλ( 1) and qλ be the integers defined by (3.1.2). The extended spectrum ESp is the union ESp = Sp∪ ISp, where (a) Sp,thespectrum, is a finite set of real numbers from the interval (0, 2] with integral multiplicities such that any real number α occurs in Sp precisely s(α)times,where ∞ 2n − 1 − u(−1)α s(α)= p2n−1(u)+np2n(u) , (e2πiα = λ). 2 λ λ n=1 u=±1 (b) ISp is the set of complex numbers from (0, 2]×iR, ISp∩R = ∅,where z = α + iβ occurs in ISp precisely s(z)times,where ⎧ ⎪ k · qk if α 1, β>0ande2πiz = λ ⎨ λ s(z)= · k 2πiz ¯ ⎪ k qλ if α>1, β<0ande =1/λ ⎩⎪ 0ifα 1andβ<0, or α>1andβ>0. We have the following relation: |ESp| =dimU =degdet(h − t Id). The main motivation for introducing HVS comes from singularity theory; cf. [AGV84, Nem95]. Lemma 3.1.4. Let X ⊂ Cn+1 be a complex hypersurface with an isolated singularity x ∈ X.LetSx be a small sphere centered at x and let 1 π : Sx \ X → S be the Milnor fibration with fiber Σ.LetU = Hn(Σ; C), b be the intersection form on U and h: U → U be the homological monodromy. Finally, let V : Hn(Σ,∂Σ; C) → Hn(Σ; C) be the Picard–Lefschetz variation operator. Then the quadruple (U; b, h, V ) constitutes a simple HVS with ε =(−1)n. Definition 3.1.5. The HVS defined in Lemma 3.1.4 is called the HVS of (X, x) ⊂ (Cn+1,x). Remark 3.1.6. The Seifert matrix (associated with Σ) is the inverse trans- pose of V . Conversely, any non-degenerate Seifert matrix S associated with any (topologically defined) link M ⊂ S2n+1 determines a simple HVS via V =(S−1)T .

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In the above algebraic/analytic case, by the Monodromy Theorem, h has all eigenvalues on the unit circle. In particular ESp = Sp. The spectrum associated with the hermitian variation structure of an isolated singularity will be denoted by SpHVS. On the other hand, for an isolated singular point x one can define a mixed Hodge structure on the n-th cohomology of its Milnor fiber, which determines a spectrum denoted by SpMHS; cf. work of Steenbrink and Varchenko [Stee76, Stee85, Var83a]. (We will use the same normalization as in [BoNe12, 2.3]. In particular, SpMHS is a subset of (0,n+1).) The relation between the two sets of spectral numbers is given in Proposi- tion 3.1.8. Before we state it, we need to fix some terminology. Definition 3.1.7. We denote the mod 2 reduction of the spectrum SpMHS by SpMHS/2. This means that SpMHS/2 is a finite set of rational numbers from the interval (0, 2] with integral multiplicities, and for any α ∈ (0, 2] the multiplicity of the spectral number in SpMHS/2 is the sum of the multiplicities MHS of the spectral numbers {α +2j}j∈Z in Sp . Proposition 3.1.8 ([Nem95, Theorem 6.5]). SpHVS is the mod 2 reduction of SpMHS for any isolated hypersurface singularity. Note that the spectrum SpHVS, by its very definition, can be determined from the Seifert form of the link. In particular, the spectrum is a purely topological invariant of the link. Moreover SpHVS can be recovered from the Levine–Tristram signatures. Proposition 3.1.9 ([BoNe12, Corollary 2.9]). Let x be an isolated singular point of the hypersurface X ⊂ Cn+1,andV =(U; b, h, V ) the corresponding HVS. Let α ∈ [0, 1) be such that ξ = e2πiα is not an eigenvalue of the monodromy operator h. Then we have 1 |SpHVS ∩ (α, α +1)| = (dim U − σ (ξ)) 2 Mx 1 |SpHVS \ [α, α +1]| = (dim U + σ (ξ)). 2 Mx

Remark 3.1.10. The dimension dim U = bn(Σ) is the Milnor number of the singularity. Moreover, the condition that ξ is not eigenvalue of the

monodromy is equivalent to nMx (ξ) = 0. This follows from Lemma 2.1.9 and from the fact that the Seifert matrix of the link corresponding to the Milnor ﬁber is non-degenerate. In Proposition 3.1.9 we assume that α ∈ [0, 1). If α = 1, the statement of the proposition still holds. In fact SpHVS ∩ (1, 2) = SpHVS \ [0, 1] and SpHVS \ [1, 2] = SpHVS ∩ (0, 1). Hence the case α = 1 is equivalent to the case α =0.

3.2. Spectrally tame polynomials. We introduce now a new class of tame polynomials, namely spectrally tame. We add this new terminology (to the rather big variety of different versions of ‘tame’ polynomials) in order to make precise what assumptions are needed to obtain a topological proof for the semicontinuity of the mod 2 reduction of the MHS-spectrum at infinity associated with polynomial maps. Below, we shall give some examples of spectrally tame polynomials as well. Let P : Cn+1 → C be a polynomial map. Let B be the bifurcation set of P , a finite subset B⊂C such that the restriction of P is a C∞ locally trivial fibration over C \B.LetD ⊂ C be a sufficiently large closed disc centered at the origin so that B⊂D. Finally, take R sufficiently large such that for any n+1 R R the boundary of any closed ball BR ⊂ C centered at the origin intersects any P −1(c), c ∈ ∂D, transversally. −1 For a fixed value c ∈ ∂D set Xc := P (c), the generic fiber of P , and let Mc be the link at infinity Xc ∩ ∂BR. Notice that by the above choices, the −1 −1 restriction of P on P (∂D)oronP (∂D) ∩ BR is the fibration of P at infinity. Since Xc is Stein, Hi(Xc, Z)=0fori>nand Hn(Xc, Z) is free. Moreover, n H (Xc, Q) carries an MHS, the mixed Hodge structure of P at infinity;see e.g. [Dim00, NeSa99, Sab99]. We add to these facts the following. −1 Proposition 3.2.1. Consider φ(z):=P (z)/|P (z)| : ∂BR \ P (int D) → S1, the Milnor map restricted on the complement of P −1(int D).Then

−1 −1 1 φ :(∂BR \ P (int D),∂BR ∩ P (∂D)) → S

is a C∞ locally trivial fibration of pairs of spaces over S1. This fibration is C∞ equivalent with the fibration at infinity associated with P :

−1 −1 P :(P (∂D) ∩ BR,P (∂D) ∩ ∂BR) → ∂D.

−1 In particular, Mc ⊂ ∂BR admits a Seifert surface, namely Σ:=φ (c/|c|), which is diffeomorphic to Xc ∩ BR. Proof. The proof is similar to the proofs of Theorems 10 and 11 from [NeZa92], with the only modification being that the arbitrary disc D used in [NeZa92] for semitame polynomials should be replaced by a sufficiently large disc D containing all the bifurcation values. Next, we analyze the Seifert form S associated with the Seifert surface Σ defined above. D −1 ∩ −1 2πit Proposition 3.2.2. (a) Set :=P (D) ∂BR and ΦI := t∈I φ (e ) for any subset I ⊂ [0, 1].WewriteΦ1 for Φ{1} (in particular Φ1 is a Seifert ∗ surface). Then the groups Hn(Φ1 ∪D, Z) and Hn(Φ1, Z) are isomorphic. In

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fact one has the following sequence of isomorphisms, denoted by s:

−1 ∂ 2n+1 (1) 2n+1 (2) Hn(Φ1 ∪D) −→ Hn+1(S , Φ1 ∪D) −→ Hn+1(S , Φ 1 ∪D) −→ [0, 2 ]

(3) ∗ (4) ∗ Hn+1(Φ 1 ,∂Φ 1 ) −→ Hn(int Φ 1 ) −→ Hn(Φ1) , [ 2 ,1] [ 2 ,1] [ 2 ,1] where ∂−1 comes from the exact sequence of the pair, (1) and (4) are induced by deformation retracts, (2) is an excision, while (3) is provided by Lefschetz duality. (b) Let j : Hn(Φ1, Z) → Hn(Φ1 ∪D, Z) be induced by the inclusion. Then the composition j s ∗ Hn(Φ1, Z) −→ Hn(Φ1 ∪D, Z) −→ Hn(Φ1, Z)

can be identified with the Seifert form S associated with Φ1 =Σ⊂ ∂BR. (c) Let b∞ and h∞ be the intersection form and monodromy of Hn(Φ1, Z)= Hn(Xc, Z);cf.3.2.1. Then, in matrix notation, T T b∞ = −εS − S and − εS h∞ = S. T In particular, h∞ is an automorphism of S, that is, h∞Sh∞ = S. Proof. Part (a) is clear, while parts (b) and (c) follow by a similar argument as in the classical case; see e.g. the survey [Nem96, (3.15)]. Definition 3.2.3. P is called spectrally tame if the following conditions are satisfied: (S1) Hn−1(Xc, Q)=0; (S2) the natural inclusion induces an isomorphism −1 Hn(Xc ∩ BR) → Hn((Xc ∩ BR) ∪ (P (D) ∩ ∂BR));

(S3) the spectrum of the HVS associated with the link Mc (and the Seifert form S) is the mod 2 reduction of the MHS spectrum of P at infinity. Note that the conditions are ‘only’ (co)homological, a fact which allows more possibilities for their verification and for applications. By excision and the long homological exact sequence of the pair, (S2) is the consequence of the vanishings (for c ∈ ∂D) −1 −1 (S2 ) Hq(P (D) ∩ ∂BR,P (c)∩∂BR)=0 forq = n, n +1. By Proposition 3.2.2(b), (S2) is equivalent with the non–degeneracy of the Seifert form S. Remark 3.2.4. If all the fibers of P have only isolated singularities and P is regular at infinity, then conditions (S1) and (S2) are satisfied; see e.g. [Nem99]. For (S3) one needs additionally the fact that the MHS at infinity is polarized by the intersection form (for monodromy eigenvalues λ =1)andby the Seifert form (for eigenvalue λ =1).Forλ = 1 this usually follows from

the standard properties of a projectivization/compactification of the fibers of P and the Hodge–Riemann polarization properties, while for λ = 1 one needs an additional Thom–Sebastiani type argument (that is, adding e.g. zN ,where z is a new variable) in order to reduce the situation to the λ =1case.This means that P + zN also should satisfy certain/similar regularity conditions at infinity. Proposition 3.2.5. The conditions (S1)–(S3) are guaranteed for several ‘tame’ polynomials present in the literature: (a) ∗–polynomials [GaNe96], (b) M–tame polynomials [NeZa90], (c) cohomologically tame polynomials [Sab99]. Proof. (a) The topological part follows from [GaNe96], (S3) from [GaNe99, Section 5]. (b) The topological part follows from [NeZa90,NeZa92], the Hodge theoretical part from [NeSa99]. (c) follows from [Sab99, NeSa99]. An immediate application of (S1)–(S2) is the following:

Lemma 3.2.6. With the notation of Deﬁnition 3.2.3,letΣ ∪ Xc be the smooth closed manifold obtained from Σ and Xc ∩ BR by gluing them together along their boundary Mc.Thenbn(Σ ∪ Xc)=bn(Σ) + bn(Xc).

Proof. In the long homological exact sequence of the pair (Σ ∪ Xc, Σ) one n−1 has Hn+1(Σ ∪ Xc, Σ) = Hn+1(Xc,Mc)=H (Xc)=0andHn−1(Σ) = Hn−1(Xc)=0.

4. Semicontinuity results

n+1 4.1. Local case. Let Pt :(C , 0) → (C, 0) be a family of germs of analytic maps depending smoothly on a parameter t ∈ (C, 0). Assume that −1 ∈ Cn+1 P0 (0) has an isolated singularity at 0 . Let us ﬁx a small Milnor ball n+1 B ⊂ C centered at 0, and let h0 be the homological monodromy operator of the Milnor ﬁbration of P0.

Theorem 4.1.1. For any t with 0 < |t|1,letx1,...,xk be all the −1 ∩ ∈ 2πiα singular points of Pt (0) B,andletα [0, 1] be chosen so that ξ = e is not an eigenvalue of h0.Then | MHS/2 ∩ | | MHS/2 ∩ | Sp0 (α, α +1) Spj (α, α +1), j (4.1.2) | MHS/2 \ | | MHS/2 \ | Sp0 [α, α +1] Spj [α, α +1], j

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MHS/2 MHS/2 where Sp0 (respectively Spj )isthemod 2 reduction of the MHS– spectrum associated with the singularity at 0 of P0 (respectively with the singularity xj of Pt). −1 Proof. Let Mxj be the link of (Pt (0),xj) with Seifert surface (i.e. Milnor −1 ﬁber) Σj , and let M be the link of (P0 (0), 0) with Seifert surface Σ in ∂B. First assume that α is chosen so that ξ = e2πiα is not an eigenvalue of the −1 homological monodromy operators hj of (Pt (0),xj) for any j =1,...,k. Then, the nullities n (ξ)andn (ξ) are all zero (see Remark 3.1.10). Then Mxj M we apply Proposition 2.2.1 to obtain σ (ξ) − σ (ξ) b (Xs ∪ Σ) − b (Σ) − b (Σ ), M Mxj n n n j j j

s −1 ∩ where X is a smoothing (nearby smooth ﬁber) of f0 (0) B. By removing s s the absolute value sign and using bn(X )=bn(Σ) (by [Miln68]) and bn(X )+ s bn(Σ) = bn(X ∪Σ) (by the local analogue of Lemma 3.2.6 proved in the same way) we obtain (4.1.3) −σ (ξ)+b (Σ) (−σ (ξ)+b (Σ )). M n Mxj n j j

In this way, via Proposition 3.1.9, we prove the first inequality of (4.1.2). The secondoneisprovedifweremovetheabsolutevaluesignintheoppositeway. 2πiα If ξ = e happens to be an eigenvalue of hj for some j but is not 2πiα an eigenvalue of h0, then for all α sufficiently close to α, e is not an eigenvalue either of h0 or of any hj . We can deduce the inequality (4.1.2) for →| MHS/2 ∩ α from the fact that it holds for α and that the function α Spj (α, α +1)| is lower semicontinuous. 4.2. Semicontinuity at infinity. We prove topologically that the mod 2 reduction of the result of [NeSa99] is valid for polynomial maps which are ‘tame’ at infinity.

Theorem 4.2.1. Let Pt be a smooth family of spectrally tame polynomial maps, where t ∈ (C, 0). Then, for any t with 0 < |t|1, and for any α ∈ [0, 1] such that ξ = e2πiα is not a root of the homological monodromy operator at inﬁnity of Pt, we have | MHS/2 ∩ | | MHS/2 ∩ | Spt (α, α +1) Sp0 (α, α +1) | MHS/2 \ | | MHS/2 \ | Spt [α, α +1] Sp0 [α, α +1],

MHS/2 where Spt denotes the mod 2 spectrum of the MHS at inﬁnity associated with Pt.

−1 Proof. We fix a regular fiber X0 = P0 (c)ofP0.LetB be a large ball so ∩ | | that X0 ∂B = MΣ0 is the link at infinity of P0. For any t with 0 < t 1, −1 the intersection of Xt = Pt (c) with ∂B is isotopic to MΣ0 . After perturbing c, if necessary, we can assume that Xt is smooth. Let Bt ⊃ B be a ball such that Xt ∩ ∂Bt is the link at infinity of Xt.Then

Y = Xt ∩ (Bt \ int B)

realizes a cobordism between MΣ0 and MΣt . The proof is similar to the proof of Theorem 4.1.1 based on assumptions (S1)–(S3), (S2’) and Proposition 2.2.1. 4.3. Local to global case. We can also compare the mod 2 spectrum of all the local singularities of a fixed fiber with the spectrum at infinity of a spectrally tame polynomial. Theorem 4.3.1. Let P : Cn+1 → C be a spectrally tame polynomial. Let X be one of its fibers with (isolated) singular points x1,...,xk.Wedenoteby MHS/2 Mx1 ,...,Mxk the corresponding links of local singularities and Sp1 ,..., MHS/2 MHS/2 Spk the mod 2 reduction of their MHS–spectrum. Let Sp∞ be the mod 2 reduction of the MHS–spectrum at infinity of P .Then,ifα ∈ [0, 1] is chosen so that ξ = e2πiα is not an eigenvalue of the monodromy of P at infinity, then | MHS/2 ∩ | | MHS/2 ∩ | Sp∞ (α, α +1) Spj (α, α +1) j | MHS/2 \ | | MHS/2 \ | Sp∞ [α, α +1] Spj [α, α +1]. j

Proof. We follow closely the proof of Theorem 4.1.1, with the modiﬁcation that now B is a large ball such that X ∩ ∂B is the link at inﬁnity of P .

Acknowledgements The ﬁrst author was supported by the Foundation for Polish Science FNP and also by Polish OPUS grant No. 2012/05/B/ST1/03195. The second author was partially supported by OTKA Grant K67928. The authors express their thanks to the Research Fund of the Edinburgh Mathematical Society for supporting the visit of the ﬁrst and second authors to Edinburgh in March 2012.

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Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 War- saw, Poland E-mail address: [email protected] A. Renyi´ Institute of Mathematics, Realtanoda´ u. 13-15, 1053 Budapest, Hun- gary E-mail address: [email protected] School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scot- land, United Kingdom E-mail address: [email protected]