New York Journal of Mathematics

New York J. Math. 1 (1995) 149–177.

Spectral Pairs, Mixed Hodge Modules, and Series of Plane Curve Singularities

A. Némethi and J.H.M. Steenbrink

Abstract. We consider a mixed Hodge module M on a normal surface singularity (X, x) and a holomorphic function germ f : (X, x) → (C, 0). For the case that M has an abelian local monodromy group, we give a formula for the spectral pairs of f with values in M. This result is applied to generalize the Sebastiani-Thom formula and to describe the behaviour of spectral pairs in series of singularities.

Contents

1. Introduction 150 2. Mixed Hodge Modules and Spectral Pairs 151 3. The General Setup 154 4. The Definition of Spp_{Γ} 154 5. The Main Result 157

5.1. The proof of Theorem 5.1 158 6. Examples 162

6.1. Abelian coverings 162 6.2. The case of the trivial mixed Hodge module 164

7. Topological Series of Curve Singularities 165 7.1. Geometric meaning 166 7.2. Topologically trivial series 166 7.3. Intrinsic invariants 167

8. Topological Series of Plane Singularities with Coefficients in a Mixed Hodge Module 167 8.1. Limit mixed Hodge structures 168 8.2. Intrinsic meaning 168 8.3. Further computations 168

9. The Spectral Pairs of Series of Plane Singularities 169

Received December 15, 1994. Mathematics Subject Classification. 14B, 14C. Key words and phrases. singularity spectrum, series of singularities. A. Némethi was supported by the Netherlands Organisation for the Advancement of Scientific

Research N.W.O.

\begin{matrix}c \\ ◯ \end{matrix}1995 State University of New York

ISSN 1076-9803/95

9.1. General formula 169 9.2. The case of topologically trivial series 170

10. Spectral Pairs of Series of Composed Singularities 172 10.1. Some invariants of an ICIS with 2-dimensional base space 172 10.2. Topological series of composed singularities 173

11. A Generalized Sebastiani-Thom Type Result 174 References 176

1. Introduction

Spectral pairs were introduced first in [17] as discrete invariants of the mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singularity. The spectral pairs which are considered in this article are defined following a slightly different convention, as in [11]. This invariant encodes the dimensions of the eigenspaces of the semisimple part T_{s} of the monodromy acting on each subquotient Gr_{p+q}^{W}Gr_{F}^{p} of the vanishing cohomology, and takes its values in the group ring Z[Q × Z].

Instead of vanishing cycles with constant coefficients one may consider vanishing cycles with coefficients in a mixed Hodge module [10]. We are led to consider these in the study of composed functions f = p ∘ ϕ where ϕ : (X, x) → (C^{2}, 0) is a 2-parameter smoothing of an isolated complete intersection singularity and p : (C^{2}, 0) → (C, 0) is a holomorphic function germ. The main result of this article gives a formula for the spectral pairs for such p at 0 with values in a mixed Hodge module on (C^{2}, 0) in terms of a decorated graph associated with p^{−1}(0) ∪ Δ, where Δ is the discriminant of the mixed Hodge module, under the assumption that the latter has an abelian local monodromy group G. In fact, in the Main Theorem (5.1), (C^{2}, 0) has been replaced by an arbitrary normal surface singularity. The mixed Hodge module we consider gives rise to a limit mixed Hodge structure on which G acts [9] and this action is used as input for the formula. The assumption about abelian monodromy is always fulfilled in case the complement of Δ has abelian local fundamental group, e.g., when Δ has normal crossings. We obtain generalizations of the Sebastiani-Thom formula (the case where ϕ = f × g with f and g isolated hypersurface singularities) in Section 11. We also obtain formulas describing the behaviour of the spectral pairs in certain series of singularities, which generalize [11], where the case of Yomdin’s series was treated.

A quick review of mixed Hodge modules and vanishing cycle functors is given in Section 2, which also contains the definition of spectral pairs and their basic properties. The flavour of our result is described in Section 3 by reformulating the case of a 1-dimensional base. The ingredients of the main formula are defined in Section 4, whereas its statement and proof form the content of Section 5. Some illustrative examples are treated in Section 6. Sections 7–10 deal with the application to series of singularities.

The authors thank the MSRI at Berkeley for its hospitality in May 1993, when part of this work was done. The first author thanks the University of Nijmegen for its hospitality in the academic year 1993-94, when this paper was finished.

2. Mixed Hodge Modules and Spectral Pairs

Let X be a (separated and reduced) complex analytic space. In [10] the category MHM(X) of mixed Hodge modules on X is associated with X. This category is stable under certain cohomological functors, for example under H^{j}f^{∗} and H^{j}f^{!} associated with a morphism f of complex analytic spaces, and under H^{j}f_{∗} associated with a projective (or proper Kähler) morphism f. Moreover, if g is a holomorphic function on X and X_{0} = g^{−1}(0), then the vanishing and the nearby cycle functors φ_{g}, ψ_{g} : MHM(X) → MHM(X_{0}) are defined. All these functors are compatible with the corresponding perverse cohomological functors on the underlying perverse sheaves via the forgetful (exact) functor

rat : MHM(X) → P erv(Q_{X})

which assigns to a mixed Hodge module the underlying perverse sheaf (with Q coefficients). (For the definition of the functors φ_{g} and ψ_{g} at the level of the constructible sheaves, see [2].)

The vanishing and nearby cycle functors have a functor automorphism T_{s} of finite order. It is provided by the Jordan decomposition T = T_{s} · T_{u} of the monodromy T.

One has the decompositions:

ψ_{g} = ψ_{g,1} ⊕ ψ_{g,≠1} respectively φ_{g} = φ_{g,1} ⊕ φ_{g,≠1}

such that T_{s} is the identity on ψ_{g,1} and φ_{g,1} and has no 1-eigenspace on ψ_{g,≠1} and φ_{g,≠1}. One has the canonical morphisms:

can : ψ_{g} → φ_{g} and V ar : φ_{g} → ψ_{g}(−1),

compatible with the action of T_{s}, such that

can : ψ_{g,≠1} ⟶^{∼} φ_{g,≠1} (1)

is an isomorphism.

Let D^{b}MHM(X) be the derived category of MHM(X) (i.e., the category of bounded complexes whose cohomologies are mixed Hodge modules on X). Let i : Y → X be a closed immersion and j : U → X the inclusion of the complement of Y. Then the cohomological functors are lifted to functors i_{∗}, i^{!}, i^{∗}, j^{∗}, j_{∗}, j_{!}; and we have the functorial distinguished triangles for M ∈ D^{b}MHM(X):

\begin{matrix}→ j_{!}j^{∗}M → & M → i_{∗}i^{∗}M ⟶^{+1} \\ → i_{∗}i^{!}M → & M → j_{∗}j^{∗}M ⟶^{+1} . \end{matrix} (2)

The connection between the two sets of functors is the following. Set X_{0} = g^{−1}(0) and let i : X_{0} → X be the corresponding immersion. Then for M ∈ Ob MHM(X) one has:

\begin{matrix}0 → H^{−1}i^{∗}M → ψ_{g,1}M ⟶^{can} φ_{g,1}M → H^{0}i^{∗}M → 0; \\ 0 → H^{0}i^{!}M → φ_{g,1}M ⟶^{V ar} ψ_{g,1}M(−1) → H^{1}i^{!}M → 0; \end{matrix} (3)

and H^{k−1}i^{∗}M = H^{k}i^{!}M = 0 if k ∉ {0, 1}.

On the other hand, if f : X → Y is a proper morphism and g is a holomorphic function on Y, then for any M ∈ Ob MHM(X) one has:

(4) ψ_{g}H^{j}f_{∗}M = H^{j}f_{∗}ψ_{g∘f}M (and similarly for φ).

Example 2.1. Assume that X is smooth. A mixed Hodge module M ∈ MHM(X) is called smooth if ratM is a local system ([10]).

Example 2.2. The module M is called pure of weight n (or a polarizable Hodge module of weight n) if Gr_{i}^{W} M = 0 for i ≠ n.

The category of smooth polarizable mixed Hodge modules is equivalent to the category of variation of polarizable mixed Hodge structures which are admissible in the sense of [5].

Example 2.3. MHM(point) is the category of polarizable Q-mixed Hodge structures ([10] (3.9)).

If g_{1} and g_{2} are two holomorphic functions such that g_{1}^{−1}(0) intersects g_{2}^{−1}(0) transversally along X_{0}, then

ψ_{g_{1}}ψ_{g_{2}} = ψ_{g_{2}}ψ_{g_{1}} : MHM(X) → MHM(X_{0}) (the same for φ’s).

In this case ψ_{g_{1}}ψ_{g_{2}}M has two commuting monodromies T_{1} and T_{2} induced by the ψ-functors.

Moreover, consider the holomorphic functions g_{1}, …, g_{s} such that s = dim X and the intersection ∩_{i=1}^{s}g_{i}^{−1}(0) is a regular point x ∈ X, and the divisor ∪_{i=1}^{s}g_{i}^{−1}(0) in a neighbourhood of x has normal crossings. Then on ψ_{g_{1}} ⋯ ψ_{g_{s}}M ∈ MHM({x}) the commuting set of monodromies T_{1}, …, T_{s} acts. We make the set of this type of objects more explicit. For the definition of mixed Hodge structures, see [1].

Definition 2.4. For any abelian group G we let MHS(G) denote the category of representations

ρ : G → Aut_{MHS}(H)

for H a mixed Hodge structure. For such ρ we let ρ^{pq} denote the induced representation of G in Aut_{C}(H^{pq}), where H^{pq} = Gr_{p+q}^{W}Gr_{F}^{p} H_{C}.

Example 2.5. Let M be a mixed Hodge module on X, g : X → C holomorphic and x ∈ g^{−1}(0). Then for all j ∈ Z, we have the objects H^{j}i_{x}^{∗}ψ_{g}M and H^{j}i_{x}^{∗}φ_{g}M of MHS(Z), where the action of 1 ∈ Z is the semisimple part of the monodromy. By the monodromy theorem, this is an automorphism of finite order.

Definition 2.6. For ρ : Z → Aut(H) in MHS(Z) with finite order one defines:

h_{λ}^{pq} := multiplicity of t − λ as a factor of the characteristic polynomial of ρ^{pq}(1) (for λ ∈ C);

and

Spp(ρ) = ∑_{α,w} h_{e^{2πiα}}^{[α],w−[α]}(α, w) ∈ Z[Q × Z],

where [α] is the integral part of α. Moreover, for g : X → C holomorphic and x ∈ g^{−1}(0) one defines for a mixed Hodge module M on X:

Spp_{ψ}(M, g, x) := ∑_{j}(−1)^{j}Spp H^{j}i_{x}^{∗}ψ_{g}M;
Spp_{φ}(M, g, x) := ∑_{j}(−1)^{j}Spp H^{j}i_{x}^{∗}φ_{g}M.

These take their values in Z[Q × Z].

Remark 2.7. In [17] the invariant Spp_{St}(g, 0) of spectral pairs was defined for an isolated hypersurface singularity g : (C^{n+1}, 0) → (C, 0). Its relation with the invariants above is as follows:

if Spp_{St}(g, 0) = ∑ n_{α,w}(α, w), then

Spp_{φ}(Q_{C^{n+1}}^{H}[n + 1], g, 0) = ∑_{α∉Z} n_{α,w}(n − α, w) + ∑_{α∈Z} n_{α,w}(n − α, w + 1).

Example 2.8. Let X be a smooth space-germ, Y ⊂ X a reduced divisor, and x ∈ Y. Let V be a polarized variation of Hodge structure on X \ Y such that its underlying representation is abelian and quasi-unipotent. Then one obtains a limit mixed Hodge structure LV at x equipped with a semi-simple action of H_{1}(X \ Y), cf. [9], i.e., an object of MHS(H_{1}(X \ Y)).

If Y has irreducible components Y_{1}, …, Y_{s}, then H_{1}(X \ Y) is free abelian on generators M_{1}, …, M_{s}, where M_{j} is represented by an oriented circle in a transverse slice to Y_{j}.

Lemma 2.9. a) There is an unique way to extend the definition of Spp_{ψ}(M, g, x) to M ∈ Ob D^{b}MHM(X) in such a way that for any distinguished triangle

M′ → M → M′′ ⟶^{+1}

one has

Spp_{ψ}(M, g, x) = Spp_{ψ}(M′, g, x) + Spp_{ψ}(M′′, g, x).

b) For u ∈ O_{X,x}^{∗} one has

Spp_{ψ}(M, ug, x) = Spp_{ψ}(M, g, x).

c) Spp_{ψ}(M, g, x) = ∑_{l} Spp_{ψ}(Gr_{l}^{W} M, g, x) for M ∈ Ob MHM(X).

d) Let T (p, q) : Z[Q × Z] → Z[Q × Z] be the automorphism mapping (α, w) to (α + p, w + p + q). Then

Spp_{ψ}(M ⊗ Q_{X}^{H}(k), g, x) = T (−k, −k)(Spp_{ψ}(M, g, x)).

e) We let H^{j}i_{x}^{∗}M ∈ MHS(Z) with trivial representation. Then

Spp_{φ}(M, g, x) = Spp_{ψ}(M, g, x) + ∑_{j}(−1)^{j}Spp(H^{j}i_{x}^{∗}M).

f) Let c_{n} : Z[Q×Z] → Z[Q×Z] (n ∈ N^{∗}) be the unique map which sends (α, w) to ∑_{k=0}^{n−1}([α] + \frac{\{α\}+k}{n} , w), (here [β] (resp. \{β\} = β − [β]) is the integer part (resp. the fractional part) of β). Then:

Spp_{ψ}(M, f^{n}, x) = c_{n} Spp_{ψ}(M, f, x).

The properties a)–d) also hold with φ instead of ψ.

The proof is left to the reader.

During the paper the notation Q_{X}^{H} means a_{X}^{∗}Q_{pt}^{H}, where a_{X} : X → pt is the constant function (see [10], p. 324).

3. The General Setup

Assume a complex analytic space X and a point x ∈ X are given. The invariant Spp_{ψ}(M, f, x) depends on M and on f in a complicated way. We want to decompose it into one step depending only on M, and a combinatorial step depending mainly on f. To illustrate this, we first treat the case where dim(X) = 1.

Example 3.1. Let X be one-dimensional, x ∈ X and f : X → C non-constant holomorphic with f(x) = 0. Assume that X is irreducible at x. Let M be a mixed Hodge module on X. We will indicate how to compute Spp_{ψ}(M, f, x).

Let σ : X̃ → X be the normalization of X, and let t be a uniformizing parameter at x̃ = σ^{−1}(x). Then f ∘ σ = u · t^{n} for some germ u ∈ O_{X̃,x̃}^{∗} and n ∈ N^{∗}. If N = H^{0}σ^{∗}M, then by (4) and Lemma 2.9 b and f, one has:

Spp_{ψ}(M, f, x) = c_{n} Spp_{ψ}(N , t, x̃).

Moreover, Spp_{ψ}(N , t, x̃) = Spp(LM, T_{s}) with LM the limit mixed Hodge structure of N at x̃ (observe that the restriction of N to a punctured neighbourhood of x̃ is an admissible variation of mixed Hodge structure) and T_{s} is the semi-simple part of the monodromy T. Hence Spp_{ψ}(M, f, x) = c_{n} Spp(LM). Here LM depends only on M, and n depends only on f. This means that the computation of Spp_{ψ}(M, f, x) goes in two steps. The first one, the computation of LM as an object of MHM(Z), does not involve f. In the second step, only the multiplicity n of x̃ as a zero of f ∘σ matters.

We are going to generalize the previous example to the two-dimensional case. The first step, passage from M to LM, is possible if M has an abelian monodromy group, which satisfies the condition of Example 2.8, and gives rise to an object LM of MHM(G), where G = H_{1}(X \ Y) and Y is the critical locus of M. The second step involves identification of the relevant discrete invariants of the function f : X → C at x. We will use the decorated resolution graph Γ of f with respect to Y, to be defined in Example 4.5. We will also define a map (see Definition 4.4)

Spp_{Γ} : MHM(G) → Z[Q × Z]

with the property that

Spp_{ψ}(M, f, x) = Spp_{Γ}(LM),

provided that V = j^{∗}M is a polarized variation of Hodge structure, and M = j_{∗}j^{∗}M, where j : X \ Y → X is the inclusion.

This is the main result of the paper.

4. The Definition of Spp_{Γ}

In this section X is a two-dimensional analytic space, Y ⊂ X is a reduced divisor, x ∈ Y a normal singularity of X. Assume that (X, Y) is contractible onto x. Let S(Y) be the set of irreducible components of Y at x.

Definition 4.1. A decorated graph for (Y, x) is a finite connected graph Γ, without edges connecting a vertex to itself, with set of vertices V and set of edges E and the following data and conditions:

a) V = D ⊔ S with D, S non-empty and an injection S(Y) ↪ S; b) a map e : D → Z such that the matrix A on D × D given by

A(d, d′) = \{ \begin{matrix}e(d) & if d = d′ ; \\ 0 & if d ≠ d′ and (d, d′) ∉ E; \\ 1 & if d ≠ d′ and (d, d′) ∈ E \end{matrix}

is negative definite;

c) a map g : D → N;

d) a map m : S → N taking at least one non-zero value.

e) For any d ∈ D, let V_{d} = {v ∈ V | dist(v, d) = 1} be the set of neighbors of d in Γ. Let Z^{V} be the free abelian group generated by {[v]}_{v∈V}. Define the group G(Γ) as the quotient of Z^{V} by the subgroup generated by the following relations:

e(d)[d] + ∑_{v∈V_{d}}[v] = 0 (d ∈ D)

or

∑_{d′∈D} A(d, d′)[d′] + ∑_{v∈V_{d}\D}[v] = 0 (d ∈ D).

Let l be the composition Z^{S} ↪ Z^{V} → G(Γ), and let m : Z^{S} → Z be the linear extension of m (i.e., m[s] = m(s)). Then we assume that m can be extended to G(Γ), i.e., there exists m_{0} : G(Γ) → Z such that m_{0} ∘ l = m.

Our maps are summarized in the following diagram:

[figure: 0; Z^{S}; Z; Z^{V}; G(Γ); coker(A); →^{m}; ⟶; ⟶; ↑; l]

Notice that cokerA is a finite group of order det A, therefore if A is unimodular l is an isomorphism and the assumption in e) is automatically satisfied.

Definition 4.2. For v ∈ V we define: m_{v} = m_{0}([v]), δ_{v} = #V_{v}, and M_{v} ∈ G(Γ) as the image of [v] by the natural projection. For d ∈ D we denote g_{d} := g(d).

It is a well-known fact that all the entries of the matrix −A^{−1} are strictly positive if A is a matrix as in Definition 4.1.b. In particular, m_{d} > 0 for any d ∈ D.

Definition 4.3. Fix a character χ : G(Γ) → C^{∗} of finite order. Let β_{v} ∈ [0, 1) be such that exp 2πiβ_{v} = χ([v]).

For d ∈ D we define Spp_{d}(χ) as follows.

For each v ∈ V_{d} and k ∈ {0, …, m_{d} − 1} define:

R_{d}^{kv} = \{−β_{v} + \frac{m_{v}}{m_{d}} (k + β_{d})\};

α_{d}^{k} = \{\frac{k + β_{d}}{m_{d}} \}.

We let R_{d}^{k} = ∑_{v∈V_{d}} R_{d}^{kv}, and δ_{d}^{k} = #{v : R_{d}^{kv} ≠ 0}. Then Spp_{d}^{k}(χ) :=

\{ \begin{matrix}−(α_{d}^{k}, 0) + (δ_{d} − 1) · (α_{d}^{k} + 1, 2) + g_{d}(α_{d}^{k}, 1) + g_{d}(α_{d}^{k} + 1, 1)  if R_{d}^{k} = 0 \\ (g_{d} + R_{d}^{k} − 1)(α_{d}^{k}, 1) + (g_{d} + δ_{d}^{k} − R_{d}^{k} − 1)(α_{d}^{k} + 1, 1) + (δ_{d} − δ_{d}^{k})(α_{d}^{k} + 1, 2) \\ else, \end{matrix}

and

Spp_{d}(χ) := ∑_{k=0}^{m_{d}−1} Spp_{d}^{k}(χ).

For e = (v, w) ∈ E we define Spp_{e}(χ) as follows. Let m_{e} := g.c.d.(m_{v}, m_{w}). The system of equations:

\{ \begin{matrix}\{β_{v}\} = \{m_{v}γ_{e}/m_{e}\} \\ \{β_{w}\} = \{m_{w}γ_{e}/m_{e}\} \end{matrix}

either has a solution γ_{e} ∈ R/Z or has not. We define Spp_{e}(χ) by:

Spp_{e}(χ) := \{ \begin{matrix}∑_{k=0}^{m_{e}−1}(\{\frac{k+γ_{e}}{m_{e}} \}, 0) − (\{\frac{k+γ_{e}}{m_{e}} \} + 1, 2) & if γ_{e} exists \\ 0 & otherwise \end{matrix}

Finally, we let

Spp_{Γ}(χ) := ∑_{d∈D} Spp_{d}(χ) + ∑_{e∈Ẽ} Spp_{e}(χ),

where Ẽ:= E ∩ (D × D).

R_{d}^{k} ∈ N, therefore Spp_{Γ}(χ) ∈ Z[Q × Z].

Definition 4.4. Let ρ ∈ MHS(G(Γ)). The representation ρ^{pq} splits into a direct sum of characters

ρ^{pq} = ⊕_{i=1}^{d(p,q)}χ_{i}^{pq}, d(p, q) = dim H^{p,q}.

We define

Spp_{Γ}(ρ) := ∑_{p,q} ∑_{i=1}^{d(p,q)} T (p, q)Spp_{Γ}(χ_{i}^{pq}).

Example 4.5. Let X be a two-dimensional complex analytic space with normal singularity at x. Let Y ⊂ X be a (reduced) divisor such that the pair (X, Y) is contractible to x and X \Y is smooth. As in Section 4, S(Y) is the set of irreducible components of Y at x.

We now consider a holomorphic function p : X → C and construct a decorated graph Γ, the decorated resolution graph of p with respect to (Y, x). The point x ∈ X is an isolated singular point of the reduced curve p^{−1}(0) ∪ Y. We let S denote the set of branches of p^{−1}(0) ∪ Y at x; then S(Y) ⊂ S. Let π : U → X be an embedded good resolution of p^{−1}(0) ∪ Y. Then D := π^{−1}(Y ∪ p^{−1}(0)) is a union of smooth curves on the two-dimensional complex manifold U. Let E = π^{−1}(x) and let D be the set of irreducible components of E, and V = D ⊔ S. We assume that D ≠ ∅. For v ∈ V we let D_{v} be the corresponding irreducible component of E if v ∈ D and the strict transform of the corresponding local irreducible component of Y ∪ p^{−1}(0) for v ∈ S. The edges of Γ are pairs (v, w) for which v ≠ w and E ∩D_{v} ∩D_{w} ≠ ∅. We let g(d) = the genus of D_{d} and e(d) = D_{d} ·D_{d} for d ∈ D. The matrix A as defined in (4.1.b) is then the intersection matrix of the components of E, which is negative definite. Finally we let m(v) be the order of zero of p along D_{v} for v ∈ S, or even for v ∈ V. Then m vanishes on the relations (4.1.e) because the divisor π^{∗}(p^{−1}(0)) on U is linearly equivalent to zero, hence has zero intersection product with each D_{d} (d ∈ D). The induced map with source G(Γ) is m_{0}.

Each M_{v} (v ∈ V) can be represented in H_{1}(X\p^{−1}(0)∪Y) by an oriented circle in a transversal slice to D_{v}. They generate the subgroup G(Γ) of H_{1}(X \ p^{−1}(0) ∪ Y). Actually, there exists an exact sequence

0 → G(Γ) →^{i} H_{1}(X \ p^{−1}(0) ∪ Y) → H_{1}(E) → 0.

Since H_{1}(E) is a torsion free group, the above sequence splits.

Notice that for any s ∈ S we have exactly one d_{s} ∈ D such that (s, d_{s}) ∈ E.

5. The Main Result

Assumption: In this section, X is a two-dimensional complex analytic space, x ∈ X a normal point on X, and Y ⊂ X a reduced divisor with x ∈ Y . Assume that X \ Y is smooth and connected. Let V be a polarized variation of Hodge structure on X \ Y such that its underlying representation ρ is abelian and quasiunipotent. Consider K := im ρ ⊂ Aut(H) and its torsion subgroup T. If K/T ≠ 0, we assume that there exist w_{j} ∈ O_{X} (j = 1, …, s), such that Y = ∪_{j=1}^{s}Z(w_{j}), and w = (w_{1}, …, w_{s}) : X \ Y → (C^{∗})^{s} induces an epimorphism w_{∗} : H_{1}(X \ Y) → Z^{s} which fits in the following commutative diagram:

[figure: π_{1}(X \ Y); K; Z^{s}; K/T; w_{∗}; ρ; ρ_{f}]

For such a V, the limit mixed Hodge structure LV ∈ MHS(H_{1}(X \Y)) exists by [9] (cf. Example 2.8). Let M = j_{∗}V where j : X \ Y → X is the natural inclusion.

Let p : X → C be a holomorphic function. Let Γ be a decorated resolution graph of p with respect to (Y, x) (cf. Example 4.5) and Spp_{Γ}(LV) the invariant defined in Definition 4.4 via the composed map G(Γ) ↪ H_{1}(X \ p^{−1}(0) ∪ Y) → H_{1}(X \ Y). Our key result is:

Theorem 5.1. Let X and M be as above. Then:

Spp_{ψ}(M, p, x) = Spp_{Γ}(LV).

Recall that the spectrum Spp_{ψ}(M, p, x) of a mixed Hodge module M with irreducible one-dimensional support Y is zero if p|Y ≡ 0; otherwise it can be computed as follows. Since M is a polarizable admissible variation of Hodge structure on Y \ {x}, it has a limit mixed Hodge structure LM. The topological information from p is the degree deg(p|Y) of the map p|Y : Y → C. Then Spp_{ψ}(M, p, x) = c_{deg(p|Y)}(Spp(LM)) (cf. Example 3.1).

If Y_{s} is one of the irreducible components of the critical locus Y of M and Γ is a decorated resolution graph of p with respect to (Y, x), then deg(p|Y_{s}) = m(d_{s}) where (s, d_{s}) ∈ E.

Theorem 5.2. Let X and M be as in the Assumption. Let Y = ∪_{s∈S(Y)}Y_{s} be the irreducible decomposition of (Y, x), i_{Y_{v}} : Y_{v} → X and j : X \ Y → X the natural inclusions. Then:

Spp_{ψ}(M, p, 0) = Spp_{Γ}(Lj^{∗}M) + ∑_{\begin{matrix}s∈S(Y) \\ p|Y_{s}≢0 \end{matrix}} ∑_{k}(−1)^{k}c_{m(d_{s})} Spp(LH^{k}i_{Y_{s}}^{!}M).

Proof. Use (2), Lemma 2.9, and Theorem 5.1.

5.1. The proof of Theorem 5.1. The proof is divided into three steps.

Step 1. Let π : U → X be a resolution of p^{−1}(0) ∪ Y as in Example 4.5.

Set N = H^{0}π^{∗}M and N_{E} = i_{E}^{∗}ψ_{p∘π}N, where i_{E} : E = π^{−1}(0) → U is the natural inclusion. Now it is clear that H^{0}π_{∗}N = M modulo terms with support in {0}, and supp H^{j}π_{∗}N ⊂ {0} if j ≠ 0. Therefore, ψ_{p}H^{j}π_{∗}N = ψ_{p}M if j = 0 and = 0 if j ≠ 0. Hence, by (4),

H^{j}π_{∗}ψ_{p∘π}N = 0 if j ≠ 0. (5)

Consider the following isomorphisms: i_{0}^{∗}ψ_{p}M = i_{0}^{∗}ψ_{p}H^{0}π_{∗}N =^{(4)} i_{0}^{∗}H^{0}π_{∗}ψ_{p∘π}N =^{(5)}

i_{0}^{∗}π_{∗}ψ_{p∘π}N =^{(∗)} π_{∗}i_{E}^{∗}ψ_{p∘π}N = π_{∗}N_{E}.

The relation (∗) follows from i_{0}^{∗}π_{∗} = π_{∗}i_{E}^{∗}. For this, notice, that π_{∗} = π_{!} because π is proper ([10] (4.3.3)), and then use [loc. cit.] (4.4.3).

For each d ∈ D, let D̃_{d} = D_{d} \ ∪_{d′∈D∩V_{d}}D_{d′} and k_{d} : D̃_{d} ↪ D_{d} its inclusion. For each e = (d, d′) ∈ Ẽ denote by i_{e} : D_{d} ∩ D_{d′} → E the natural inclusion. Then by (2) one has the following distinguished triangle:

→ ⊕_{e∈Ẽ}(i_{e})_{∗}i_{e}^{!}N_{E} → N_{E} → ⊕_{d∈D}(k_{d})_{∗}k_{d}^{∗}N_{E} →

By the additivity of the functor Spp, we obtain:

Spp(i_{0}^{∗}ψ_{p}M) = Spp(π_{∗}N_{E}) = ∑_{e∈Ẽ} Spp(i_{e}^{!}N_{E}) + ∑_{d∈D} Spp(π_{∗}(k_{d})_{∗}k_{d}^{∗}N_{E}). (6)

(Everywhere, the action is the natural monodromy provided by ψ.)

Fix d ∈ D. Let D′_{d} = D̃_{d} \ St(p^{−1}(0) ∪ Y) = D_{d} \ ∪_{v∈V_{d}}D_{v}, and j_{d} : D′_{d} ↪ D_{d} denotes the natural inclusion.

Lemma 5.3. The restriction to D′_{d} of the module N′_{E} = (k_{d})_{∗}k_{d}^{∗}N_{E} is smooth (i.e., rat j_{d}^{∗}N′_{E} is a local system). Moreover, it satisfies

N′_{E} = (j_{d})_{∗}(j_{d})^{∗}N′_{E}. (7)

Proof. By construction: j_{d}^{∗}N′_{E} = j_{d}^{∗}N_{E} = j_{d}^{∗}ψ_{p∘π}N .

Since ratN restricted to U \ D is a local system, and D′_{d} is a smooth divisor in U \ ∪_{v∈V_{d}}D_{v}, the sheaf rat j_{d}^{∗}ψ_{p∘π}N is a local system, too.

The obstruction to the isomorphism (7) lies in the points P ∈ D_{d} ∩ (∪_{v∈S}D_{v}) = D̃_{d} \ D′_{d}.

Take P = D_{d}∩D_{v} such that p|Y_{v} ≢ 0 (v ∈ S). Then the assumption M = j_{∗}j^{∗}M and (2) give ψ_{p∘π}i_{D_{v}}^{!}N = 0. Now (3) and the commutativity of the vanishing cycle functors complete the argument in this case.

If P = D_{d} ∩ D_{v} such that v ∈ S and p|Y_{v} ≡ 0, then see [11] (4.7).

Notice that j_{d}^{∗}N′_{E} = j_{d}^{∗}N_{E}, and by Lemma 5.3, (k_{d})_{∗}k_{d}^{∗}N_{E} = (j_{d})_{∗}j_{d}^{∗}N_{E}. Since the isomorphism H^{•}(D_{d}, (j_{d})_{∗}j_{d}^{∗}N_{E}) = H^{•}(D′_{d}, j_{d}^{∗}N_{E}) is compatible with the mixed Hodge structures, one has:

Spp(π_{∗}(k_{d})_{∗}k_{d}^{∗}N_{E}) = Spp(H^{•}(D′_{d}, j_{d}^{∗}N_{E})).

Step 2. The identity Spp(H^{•}(D′_{d}, j_{d}^{∗}N_{E})) = Spp_{d}(LV).

By the additivity of Spp (see 2.13.a) it is enough to prove

∑_{l} Spp(H^{•}(D′_{d}, Gr_{l}^{W} j_{d}^{∗}N_{E})) = Spp_{d}(LV). (8)

Since j_{d}^{∗}N_{E} is smooth (Lemma 5.3), V_{d,l} := Gr_{l}^{W} j_{d}^{∗}N_{E} is a polarizable variation of Hodge structure on D′_{d}.

Lemma 5.4. The representation associated with the local system ratV_{d,l} is abelian and quasi-unipotent.

Proof. D′_{d} has a neighbourhood homeomorphic to D′_{d} × {disc}.

The importance of this lemma appears in the following:

Lemma 5.5. Any polarizable variation of Hodge structure on a quasi-projective smooth curve C, whose underlying local system has a monodromy representation which is abelian and quasi-unipotent, is locally constant.

Proof. Since the monodromy representation on C is semi-simple ([1], (4.2.6)), it follows, that it is a direct sum of one-dimensional representations, which are finite. Hence a finite cover C̃ of C has trivial local and global monodromies. Therefore a global marking C̃ → D can be defined in the moduli space of Hodge structures. This by Griffiths’ theorem [4] can be extended to the smooth closure of C̃. But this extended map is trivial by the rigidity theorem [13].

Consider now the limit mixed Hodge structure LV ∈ MHS(H_{1}(X \ Y)). Its representation defines a locally constant abelian variation M_{∞} on X \ Y. In [9], among other facts, the following is proved:

Lemma 5.6. Let (C, x) ⊂ (X, x) be a curve with C ∩ Y = {x}. Let L(M|C), respectively L(M_{∞}|C) be the limit mixed Hodge structures at x of the restrictions of M, respectively of M_{∞}, to C. Then Gr^{W} L(M|C) = Gr^{W} L(M_{∞}|C).

Now, if we replace in the above construction M by M_{∞}, then we obtain a variation V_{∞,d,l} instead of V_{d,l} Lemma 5.7. The variations of Hodge structure V_{d,l} and V_{∞,d,l} on D′_{d} are isomorphic.

Proof. Both are abelian variations by Lemma 5.4, with flat Hodge bundles by Lemma 5.5. By construction, the underlying representations are the same. We have only to show that in a fixed point P ∈ D′_{d} the stalks are isomorphic (by an isomorphism, which is compatible with the representations).

Let C be a transversal slice to D′_{d} at a point P ∈ D′_{d} and t a uniformizing parameter of (C, P). Then (V_{d,l})_{P} ≃ Gr_{l}^{W} ψ_{t^{m_{d}}} (M|C) ≃ ⊕_{i=1}^{m_{d}}Gr_{l}^{W} L(M|C). Similar isomorphisms holds for the other variation, therefore the result follows from Lemma 5.6. The compatibility follows from the naturality of the constructions.

Therefore, (8) is equivalent to

∑_{l} Spp(H^{•}(D′_{d}, V_{∞,d,l})[1]) = Spp_{d}(LV). (9)

Notice that both sides of (9) depend only on the limit mixed Hodge structure LV.

Now, (LV, ρ) ∈ MHS(H_{1}(X \ Y)) splits in a direct sum ρ = ⊕_{p,q} ⊕_{i=1}^{d(p,q)} χ_{i}^{p,q}, d(p, q) = dim LV^{p,q}. The construction of V_{∞,d,l} preserves this decomposition, therefore V_{∞,d,l} = ⊕_{p+q=l} ⊕_{i=1}^{d(p,q)} V_{∞,d}^{p,q,i}. We have to show that

Spp(H^{•}(D′_{d}, V_{∞,d}^{p,q,i})[1]) = Spp_{d}(χ_{i}^{p,q}). (10)

In the sequel we omit the indices p, q and i. Moreover, we can assume that χ is of type (p, q) = (0, 0).

Lemma 5.8. The variation V_{∞,d} is m_{d}-dimensional. It has a direct sum decomposition ⊕_{k=0}^{m_{d}−1}V_{d}^{k} in one-dimensional locally constant variations of C-Hodge structure (of the same type (0, 0)), such that the monodromy of V_{d}^{k} around the points (P_{d} ∩ P_{v})_{v∈V_{d}} is exp(−2πiR_{d}^{kv}). The monodromy action on V_{d}^{k} given by the vanishing cycle functor is exp(2πiα_{d}^{k}).

Proof. The verification is local in small neighbourhoods of the points D_{d} ∩D_{v} (v ∈ V_{d}). Here, in a suitable coordinate system ρ ∘ π = x^{m_{d}}y^{m_{v}}. The verification is left to the reader.

Proof of (10). By Lemma 5.8, we have to verify only:

Spp(H^{•}(D′_{d}, V_{d}^{k})[1]) = Spp_{d}^{k}(χ). (11)

In order to compute the left hand side of (11), we have to compute the dimensions h^{pq} of the spaces Gr_{p+q}^{W}Gr_{F}^{p} H^{•}(D′_{d}, V_{d}^{k}). Then h_{λ}^{pq} = h^{pq} if λ = exp(2πiα_{d}^{k}), and = 0 otherwise.

Let Ω^{•}(log Σ_{d}) be the complex of meromorphic differentials on D_{d} with at worst logarithmic poles along Σ_{d} = ∪_{v∈V_{d}}(D_{d} ∩D_{v}), and let V denote Deligne’s canonical extension of V_{d}^{k} ⊗_{C} O_{D′_{d}}. Now, H^{∗}(D′_{d}, V_{d}^{k}) = H^{∗}(D_{d}, K^{•}), where K^{•} is the complex {∇ : V → Ω^{1}(log Σ_{d})⊗V}. The Hodge filtration of this complex is Deligne’s “filtration bête” σ_{≥p}, therefore the first term of the Hodge spectral sequence is F E_{1}^{pq} = H^{q}(D_{d}, K^{p}). This spectral sequence degenerates at E_{1}.

If R_{d}^{k} = 0, then V = O_{D′_{d}}. Hence E_{1}^{00} = C, E_{1}^{01} = C^{g_{d}}, and E_{1}^{10} = C^{g_{d}+δ_{d}−1}. This case corresponds to the trivial flat bundle, therefore we recover exactly the mixed Hodge structure of D_{d} \ {δ_{d} points}. In particular, H^{0} = C has type (0, 0), E_{1}^{01} has type (0, 1), C^{g_{d}} ⊂ E_{1}^{10} has type (1, 0), and C^{δ_{d}−1} = E_{1}^{10}/C^{g_{d}} has type (1, 1).

Assume that R_{d}^{k} ≠ 0. Then _{F} E_{1}^{pq} = 0 if p + q ≠ 1 because degV = −R_{d}^{k} < 0 and deg(O(−Σ) ⊗ V^{∗}) = R_{d}^{k} − δ_{d} < 0. Then by the Riemann-Roch Theorem, dim E_{1}^{01} = g_{d} + R_{d}^{k} − 1, and dim E_{1}^{10} = g_{d} + δ_{d} − R_{d}^{k} − 1. This means that the only nontrivial cohomological group is H^{1} = H^{1}(D′_{d}, V_{d}^{k}). In order to compute its Hodge numbers h^{pq}, we need the weight filtration too.

The weight filtration of the complex K^{•} is W_{−1}K^{•} = 0, W_{0}K^{•} = {∇ : V → im ∇}, and W_{1}K^{•} = K^{•}. The extension W_{0} is a resolution of j_{∗}V_{d}^{k} and (by [20]) H^{1}(D_{d}, j_{∗}ratV_{d}^{k}) is pure of weight 1. On the other hand R^{1}j_{∗}ratV_{d}^{k} = C^{δ_{d}−δ_{d}^{k}} and it induces in H^{1} a quotient of weight 2. Therefore, the only nonzero Hodge numbers are: h^{10}, h^{01} and h^{11}. More precisely: h^{11} = δ_{d} − δ_{d}^{k}, h^{01} = g_{d} + R_{d}^{k} − 1, and h^{10} = g_{d} + δ_{d}^{k} − R_{d}^{k} − 1. Now the expression for the spectral pairs readily follows.

Example 5.9. Assume that δ_{d} = 1. Then R_{d}^{k} = 0 for any k. Therefore Spp_{d}^{k} = −(α_{d}^{k}, 0) + g_{d}(α_{d}^{k}, 1) + g_{d}(α_{d}^{k} + 1, 1) for any k.

Example 5.10. Assume that g_{d} = 0, δ_{d} = 2 and V_{d} = {v, w}. Then R_{d}^{k} = R_{d}^{kv} + R_{d}^{kw} ∈ {0, 1}. Using the obvious fact that for x, y ∈ R with x + y ∈ Z one has {x} + {y} = 1 if x ∉ Z, and = 0 if x ∈ Z, we obtain that R_{d}^{k} = 1 if β_{v} − \frac{m_{v}}{m_{d}} (k + β_{d}) ∉ Z and = 0 otherwise.

Lemma 5.11. Let l_{d} = g.c.d.(m_{d}, m_{v}) = g.c.d.(m_{v}, m_{w}). The following assertions are equivalent:

a) There exists k_{0} ∈ Z such that β_{v} − \frac{m_{v}}{m_{d}} (k_{0} + β_{d}) ∈ Z.

b) There exists γ_{d} ∈ R/Z such that

\{ \begin{matrix}\{β_{v}\} = \{m_{v}γ_{d}/l_{d}\} \\ \{β_{d}\} = \{m_{d}γ_{d}/l_{d}\}. \end{matrix}

c) There exists γ_{d} ∈ R/Z such that

\{ \begin{matrix}\{β_{v}\} = \{m_{v}γ_{d}/l_{d}\} \\ \{β_{w}\} = \{m_{w}γ_{d}/l_{d}\}. \end{matrix}

(Notice that b) or c) determines γ_{d} uniquely.)

Moreover, if one of these conditions hold, then:

β_{v} − \frac{m_{v}}{m_{d}} (k + β_{d}) ∈ Z ⇔ \frac{(k − k_{0})m_{v}}{m_{d}} ∈ Z.

Proof. Use the relation [v] + [w] + e_{d}[d] = 0 (cf. Definition 4.1.e).

Therefore, the lemma implies that if g_{d} = 0 and δ_{d} = 2, then

Spp_{d}(χ) = \{ \begin{matrix}∑_{k=0}^{l_{d}−1} −(\{\frac{k+γ_{d}}{l_{d}} \}, 0) + (\{\frac{k+γ_{d}}{l_{d}} \} + 1, 2) & if γ_{d} exists \\ 0 & otherwise. \end{matrix}

We will see later that the expression Spp_{e}(χ) (e ∈ E′) is exactly of this type.

Step 3. The computation of Spp(i_{e}^{!}N_{E}).

Fix a node e = (v, w) ∈ Ẽ. Let us modify the resolution by a blowing up at the point P = D_{v} ∩ D_{w}. The new resolution is π′ : U′ → U ⟶^{π} B. Define D′ and E′ similarly as in the first case. Then D′ = D ⊔ {D_{e}}, E′ = (E \ {P }) ⊔ {P_{v}, P_{w}}, where D_{e} is the new exceptional divisor, and P_{v} = D_{e} ∩ D_{v}, P_{w} = D_{e} ∩ D_{w}, where the strict transforms of D_{v} and D_{w} are denoted by the same symbols. Set E′ = (π′)^{−1}(0), and consider the modules N′ = H^{0}π^{′∗}M and N′_{E} = H^{0}i_{E′}^{∗}ψ_{p∘π′}N′.

Lemma 5.12. The mixed Hodge structures i_{P}^{!} N_{E}, i_{P_{v}}^{!}N′_{E} and i_{P_{w}}^{!}N′_{E} are isomorphic in a way compatible with their monodromy action.

Proof. In a neighbourhood of P (resp. of P_{v}) N_{E} = ψ_{p∘π}N (resp. N′_{E} = ψ_{p∘π′}N′). On the other hand, H^{∗}(i_{P}^{!} ψ_{p∘π}N) = H_{c}^{∗}(F_{p∘π}, N), where F_{p∘π} is the Milnor fiber of p ∘ π in P. Similarly, H^{∗}(i_{P_{v}}^{!}ψ_{p∘π′}N′) = H_{c}^{∗}(F_{p∘π′}, N′). But, for a suitable representatives, one has an inclusion of F_{p∘π′} into F_{p∘π} (induced by the blowing up map) which is a homotopy equivalence, and it identifies the sheaves N and N′. Moreover, it preserves the monodromy action too. Since rat is an exact functor, the lemma follows.

Now, if we write (6) for the resolutions π and π′, we obtain that

Spp(i_{P_{v}}^{!}N′_{E}) + Spp(i_{P_{w}}^{!}N′_{E}) + Spp_{D_{e}}(χ′) = Spp(i_{P}^{!} N_{E}).

Using Lemma 5.12 one has Spp(i_{e}^{!}N_{E}) = −Spp_{D_{e}}(χ′). Now, the result follows from the computation of Example 5.10.

6. Examples

6.1. Abelian coverings. Consider a connected normal surface singularity (X, x) and a covering ϕ : (X, x) → (C^{2}, 0) which is ramified over Δ. Let Δ = ∪_{v=1}^{S(Δ)}Δ_{v} be the irreducible decomposition of Δ. Consider a germ: p : (C^{2}, 0) → (C, 0). We are interested in spectral pairs Spp_{ψ}(Q_{X}^{H}[2], f, x), where f is the composed map f = p ∘ ϕ. The pair (ϕ, (X, x)) is uniquely determined by the nonramified covering X^{∗} = ϕ^{−1}(B \ Δ) → B \ Δ [16] (here (B, Δ) is a good representative).

In particular, the mixed Hodge module M = H^{0}ϕ_{∗}Q_{X}^{H}[2] on (C^{2}, 0) is completely determined by the exact sequence

1 → π_{1}(X^{∗}) → π_{1}(B \ Δ, ∗) ⟶^{τ} G → 1.

The critical locus of M is contained in Δ and the representation of M|_{B\Δ} is the induced representation by τ of the regular representation ρ_{G} of G.

Now, (4) assures, that H^{•}i_{x}^{∗}ψ_{f}Q_{X}^{H}[2] = H^{•}i_{0}^{∗}ψ_{p}M, therefore

Spp_{ψ}(Q_{X}^{H}[2], f, x) = Spp_{ψ}(M, p, 0).

Suppose, that G is abelian. Then Theorem 5.2 can be applied. The variation M|_{B\Δ} is a flat variation of Hodge structure. It can be identified with (Q^{|G|}, F, W), where Gr_{F}^{p} Q^{|G|} = Gr_{l}^{W} Q^{|G|} = 0 if p ≠ 0 or l ≠ 0. The underlying abelian representation is

ρ^{ab} = ρ_{G} ∘ τ^{ab} : H_{1}(B \ Δ, Z) → GL(|G|, Q)

where τ^{ab} : H_{1}(B \ Δ, Z) → G is induced by τ. Actually, this is the description of LM, too. In particular Spp_{Γ}(LM) is well-defined.

Let v ∈ S(Δ). The variation i_{Δ_{v}}^{!}M has the following description: If j : B \ Δ → B is the natural inclusion, then R^{0}j_{∗}j^{∗}ϕ_{∗}C_{X} = ϕ_{∗}C_{X} and for any point P_{v} ∈ Δ_{v} − {0} one has dim(R^{1}j_{∗}j^{∗}ϕ_{∗}C_{X})_{P} = dim(R^{0}j_{∗}j^{∗}ϕ_{∗}C_{X})_{P} = a_{v}, where above P_{v} there lie exactly a_{v} points of X. The above spaces can be recovered from the representation ρ^{ab} via the expression C_{v}^{|G|} = ker(ρ^{ab}(M_{v}) − 1) ⊂ C^{|G|}. Let d_{v} ∈ D be the unique vertex such that (v, d_{v}) ∈ E. Then C_{v}^{|G|} is a sub-Hodge structure of C^{|G|} with the automorphism ρ^{ab}([M_{d_{v}}]).

The above discussion shows that M → j_{∗}j^{∗}M is one-to-one. In particular, i_{∗}H^{k}i_{Δ_{v}}^{!}M = 0 if k ≠ 1. The exact sequence (3) shows that H^{1}i_{Δ_{v}}^{!}M is of type (1, 1). Its restriction to Δ_{v} − {0} is a locally constant variation; it can be identified with C_{v}^{|G|}(−1) with monodromy ρ^{ab}([M_{d_{v}}]). Similarly as above, its limit LH^{1}i_{Δ_{v}}^{!}M has the same presentation too.

Finally, notice that degree(f|Δ_{v}) = m(d_{v}).

Proposition 6.1.

Spp_{ψ}(Q_{X}^{H}[2], f, x) = Spp_{Γ}(C^{|G|}, ρ^{ab}) − ∑_{\begin{matrix}v∈S(Δ) \\ p|Δ_{v}≢0 \end{matrix}} c_{m(d_{v})}Spp(C_{v}^{|G|}(−1), ρ^{ab}([M_{d_{v}}])).

Example 6.2. The case of Hirzebruch-Jung singularities. Let (z_{1}, z_{2}) be a local coordinate system in (C^{2}, 0). We make the above formula more explicit in the case when Δ = {z_{1}z_{2} = 0}.

Let A_{n,q} = C^{2}/Z_{n} be the cyclic quotient singularity, where the action of G = Z_{n} is given by (u_{1}, u_{2}) ↦ (ζu_{1}, ζ^{p}u_{2}), ζ = exp(2πi/n) and pq ≡ 1(mod n). Then ϕ : A_{n,q} → C^{2} given by z_{i} = u_{i}^{n} (i = 1, 2) defines a covering which is ramified over {z_{1}z_{2} = 0}.

In this case, the covering transformation group is G = Z_{n}, and

τ^{ab} : H_{1}(B \ Δ, Z) = Z^{2} → Z_{n}

is τ^{ab}(e_{1}) = ˆ1, τ^{ab}(e_{2}) = ˆ−q.

On the other hand, for any v ∈ S(Δ) one has C_{v}^{|G|} = C and the transformation ρ^{ab}([M_{d_{v}}]) is the identity. Therefore:

Spp_{ψ}(Q_{X}^{H}[2], p ∘ ϕ, x) = Spp_{Γ}(C^{n}, ρ^{ab}) − ∑_{\begin{matrix}v∈S(Δ) \\ p|Δ_{v}≢0 \end{matrix}} ∑_{k=0}^{m_{dv} −1} (1 + \frac{k}{m_{d_{v}}} , 2) .

Example 6.3. Take p(z_{1}, z_{2}) = z_{1} + z_{2} in Example 6.2. For x ∈ R take δ(x) = 0 if x ∈ Z, and = 1 if x ∉ Z. Then:

Spp_{ψ}(Q_{X}^{H}[2], p ∘ ϕ, x) = −(0, 0) + ∑_{i=1}^{n−1} (2 − \{\frac{−i}{n}\} − \{\frac{qi}{n}\} , 2 − δ (\frac{1 − q}{n} i)) .

Remark 6.4. Above, we computed the spectral pairs of a composed function f. By our general result, we can compute Spp_{ψ}(Q_{X}^{H}[2], f, x) of an arbitrary function f : (X, x) → (C, 0), provided that we know the decorated resolution graph of f (cf. the second part of this section).

6.2. The case of the trivial mixed Hodge module. Let (X, x) be a normal surface singularity and f : (X, x) → (C, 0) an analytic germ. Assume that M = Q_{X}^{H}[2]. The limit LM is the one-dimensional mixed Hodge structure Q^{H} with trivial action and Gr_{l}^{W} Q^{H} = Gr_{F}^{p} Q^{H} = 0 if l ≠ 0 or p ≠ 0. Denote the set of spectral pairs merely by Spp_{ψ}(f). By Theorem 5.2 one has: Spp_{ψ}(f) = Spp_{Γ}(Q^{H}), where Γ is the decorated resolution graph of f.

Let h = rank H_{1}(Γ) be the number of independent cycles of Γ. Alternatively, h = rank H_{1}(E) − rank H_{1}(Ẽ), where Ẽ is a smooth model for E = π^{−1}(x).

By a computation, we obtain an “almost symmetric” form of Spp_{ψ}(f):

Proposition 6.5. Let g = ∑_{d∈V} g_{d}, R_{d}^{k} = ∑_{v∈V_{d}}{k · m_{v}/m_{d}}, and R^{∗}(n) = {1, … n − 1}. Then:

Spp_{ψ}(f) = (h − 1)(0, 0) + (h + #S − 1)(1, 2) + g(0, 1) + g(1, 1)

+ ∑_{s∈S} ∑_{k∈R^{∗}(m(e_{s}))}(1 + \frac{k}{m(e_{s})}, 2) + ∑_{e∈Ẽ} ∑_{k∈R^{∗}(m_{e})}((\frac{k}{m_{e}} , 0) + (2 − \frac{k}{m_{e}} , 2))

− ∑_{d∈D} ∑_{\begin{matrix}k∈R^{∗}(m_{d}) \\ R_{d}^{k}=0 \end{matrix}}((\frac{k}{m_{d}} , 0) + (2 − \frac{k}{m_{d}} , 2))

+ ∑_{d∈D} ∑_{\begin{matrix}k∈R^{∗}(m_{d}) \\ R_{d}^{k}≠0 \end{matrix}}(R_{d}^{k} − 1) ((\frac{k}{m_{d}} , 1) + (2 − \frac{k}{m_{d}} , 1))

+ ∑_{d∈D} ∑_{k∈R^{∗}(m_{d})} g_{d}((\frac{k}{m_{d}} , 1) + (2 − \frac{k}{m_{d}} , 1)).

Here m(e_{s}) = g.c.d.(m(s), m(d_{s})), where d_{s} is the unique vertex in D with (s, d_{s}) ∈ E.

Example 6.6. Let (X, x) = {z^{2} = (x + y^{2})(x^{2} + y^{7})} ⊂ (C^{3}, 0) and f(x, y, z) = z. Then the decorated resolution graph is the following:

[figure: 1; 1; (−3); (−1); (−3); (−3); 4; 9; 3; 4]

In parentheses are the numbers e_{d}, the others are the multiplicities. All exceptional divisors are rational. The spectral pairs are:

Spp_{ψ}(f) = ∑_{k=1}^{8}(2k/9, 1) + (2/3, 1) + (4/3, 1) + 2 · (1, 2).

Remark 6.7. Using (3) one has: Spp_{φ}(f) = Spp_{ψ}(f) + (0, 0).

7. Topological Series of Curve Singularities

Let p : (C^{2}, 0) → (C, 0) be a curve singularity with irreducible decomposition p = ∏_{j=1}^{r} p_{j}^{m_{j}}, and Δ ⊂ (C^{2}, 0) a reduced one-dimensional analytic space-germ with irreducible decomposition ∪_{i=1}^{s}Δ_{i}. Let Γ be the decorated resolution graph of p^{−1}(0) ∪ Δ. Let S(Δ) (resp. S(p)) be the subset of S corresponding to the strict transforms St(Δ_{i}) (resp. St(p_{j}^{−1}(0))). Recall that the multiplicity of a vertex v ∈ V is the multiplicity of p ∘ π on the divisor D_{v}. In particular, the multiplicities of the vertices v ∈ S are: {m_{j}}_{j=1}^{r} corresponding to s ∈ S(p), and m_{s} = 0 corresponding to s ∈ S(Δ) \ S(p) In the sequel we use the index notation j ∈ {1, …, r} for the set S(p).

The schematic graph of Γ, together with multiplicities, is:

[figure: 0; 0; ⋮; ⋮; ⋮; ⋮; ⋮; m_{r}; m_{1}; S(Δ) \ S(p); D; S(p); l_{r}; l_{1}] (12)

The vertices indexed by D are in the box. Those indexed by S are drawn as arrows. The arrows from the left-hand side corresponds to S(Δ) \ S(p), those from the right-hand side to S(p).

For each j ∈ S(p), let d_{j} ∈ D be the unique vertex which is adjacent to j. The corresponding exceptional divisor (D_{d_{j}}) is denoted by E_{j} and its multiplicity is l_{j} = m_{d_{j}}. The intersection point D_{j} ∩ E_{j} is denoted by P_{j}.

Definition 7.1. The topological series of curve singularities belonging to p, relative to Δ, consist of all curve singularities p′ such that there are decorated resolution graphs Γ of p^{−1}(0) ∪ Δ, and Γ′ of (p′)^{−1}(0) ∪ Δ respectively, such that Γ has form (12), and Γ′ has the following form:

[figure: Γ_{r}; Γ_{1}; 0; 0; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; l_{r}+m_{r}; l_{1}+m_{1}; l_{r}; l_{1}]

The index sets D′, S′ and E′ are defined similarly as D, S and E. For j ∈ {1, …, r}, the new index set of exceptional divisors (resp. of strict transforms or arrows) of Γ_{j} is denoted by D_{j} (resp. S_{j}). Therefore D′ = D ⊔ (⊔_{j}D_{j}), S′ = (S \ S(p)) ⊔ (⊔_{j}S_{j}), and E′ = E ⊔ (⊔_{j}E_{j}) ⊔ (⊔_{j}{e_{j}}), where E_{j} are the edges in Γ_{j}, and e_{j} is the new edge which joins Γ and Γ_{j}.

It can happen that the box Γ_{j} is empty. In that case, instead of Γ_{j} we have just an arrow as in the case of Γ.

7.1. Geometric meaning. Let U_{j} be a small neighbourhood of P_{j} = D_{j} ∩ E_{j}. Fix a coordinate system (x, y) in U_{j} so that: U_{j} ∩D_{j} = {y = 0}, U_{j} ∩E_{j} = {x = 0} and p ∘ π|U_{j} = x^{l_{j}}y^{m_{j}}.

Definition 7.1 has the following meaning: p′ ∘ π|U_{j} = p̃_{j}(x, y)x^{l_{j}}, where p̃_{j} is a curve singularity at P_{j} such that:

a) the line {x = 0} is not in the tangent cone of {p̃_{j} = 0},

b) the multiplicity of p̃_{j} at P_{j} is m_{j}.

If p̃_{j} = ∏_{k=1}^{r_{j}} p_{jk}^{m_{jk}} is the irreducible decomposition of p̃_{j}, then ∑_{k} m_{jk}(x, p_{jk})_{P_{j}} = m_{j}. Here (·, ·)_{P_{j}} denotes the intersection multiplicity at the point P_{j}.

We define the numbers {ϵ_{j}}_{j=1}^{r} as follows. If j ∈ S(p) \ S(Δ), then the index set S_{j} is exactly {1, …, r_{j}}; we set ϵ_{j} = 0. If j ∈ S(p) ∩ S(Δ) (i.e., {p_{j} = 0} = Δ_{s_{j}}), then one of the following conditions must hold:

1. St(Δ_{s_{j}}) = {p_{jk} = 0} for some k (i.e., p_{jk}(x, y) = y), or

2. St(Δ_{s_{j}}) ⊄ {p̃_{j} = 0}. In the first case, S_{j} = {1, …, r_{j}}, and we set ϵ_{j} = 0. In the second case, S_{j} = {1, …, r_{j}}∪{s_{j}}, and we take ϵ_{j} = 1. This shows, that the index set S′(Δ)\S′(p′) (which is crucial in the main theorem 5.2) is (S(Δ) \ S(p)) ⊔ ⊔_{j:ϵ_{j}=1}{s_{j}}.

It is clear, that the diagram Γ_{j} is a part of the decorated resolution graph Γ̃_{j} of p′ ∘ π|U_{j} (relative to {xy = 0}). Actually, the resolution graph Γ̃_{j} can be obtained from Γ_{j} by adding an arrow corresponding to the strict transform of {x = 0} = E_{j} ∩ U_{j}. The arrow takes the place of the new edge e_{j}, which in Γ′ joins Γ and Γ_{j}. Notice, that the multiplicity of (the strict transform of) E_{j} is l_{j}.

[figure: ⋮; ⋮; l_{j}+m_{j}; Γ̃_{j} :; l_{j}; e_{j}] (13)

Remark 7.2. A compatible definition of the topological series of curve singularities can be given using Eisenbud-Neumann diagrams [3] (see [14] in the case Δ = ∅, and [6] in the general case).

7.2. Topologically trivial series.

Definition 7.3. [6] We say that p′ is an element of the “topologically trivial series” belonging to p, relative to Δ, if for any j ∈ {1, …, r} either p̃_{j}(x, y) = y^{m_{j}}a(x, y) or p̃_{j}(x, y) = (y + x^{b_{j}}b(x, y))^{m_{j}}a(x, y) where a, b ∈ O_{U_{j}}^{∗} and b_{j} ≥ 1.

In the definition, “topologically trivial” means that the topological types of the germs p and p′ are the same, but the positions of their zero set with respect to Δ are different.

In the first case (p̃_{j} = y^{m_{j}}a) one has ϵ_{j} = 0 and the topological modification, even relative to Δ, is trivial. In the second case ϵ_{j} = 1, and the diagram Γ_{j} is as follows:

[figure: ⋮; …; l_{j}; l_{j}+m_{j}; l_{j}+(b_{j}−1)m_{j}; l_{j}+b_{j}m_{j}; E′; m_{j}; 0; Γ; ](14)

7.3. Intrinsic invariants. Notice that the choice of the graphs Γ and Γ′ is not unique. Therefore, the numbers b_{j}, l_{j} (or even the cycle [M_{d_{j}}]) have no intrinsic, geometric meaning. In our computations, we wish to express the spectral pairs in terms of some intrinsic invariants. Actually, this paragraph makes the connection between the set of numerical data used in the decoration of the resolution graph and those used in the Eisenbud-Neumann diagrams.

Fix j ∈ {1, …, r} such that ϵ_{j} = 1.

We define

n_{j} := ∑_{i∈S(p)\{j}} −A^{−1}(d_{j}, d_{i})m_{i},

where A is the intersection matrix of Γ as in Definition 4.1.

Let η_{j} be the multiplicity of p_{j} on E_{j}, i.e., p_{j} ∘ π|U_{j} = x^{η_{j}}y · c(x, y), c ∈ O_{U_{j}}^{∗}. Then the relation [d_{j}] = ∑_{v∈S} −A^{−1}(d_{j}, d_{v})[v] gives a relation between the set of multiplicities of p_{j} ∘ π on the components of D, namely: η_{j} = −A^{−1}(d_{j}, d_{j}). On the other hand, the same relation applied for the set of multiplicities of p gives: l_{j} = n_{j} + η_{j} · m_{j}. Therefore:

b_{j}m_{j} + l_{j} = a_{j}m_{j} + n_{j}, where a_{j} := b_{j} + η_{j}. (15)

Notice that S(p) = S(p′). Order the irreducible factors of p′ in such a way that p′ = ∏_{j}(p′_{j})^{m_{j}}, and Z(p′_{j}) in U_{j} is {y + x^{b_{j}}b(x, y) = 0}

Lemma 7.4. The numbers a_{j}, n_{j} and a_{j}m_{j} + n_{j} have the following expression in terms of intersection multiplicities at the origin:

a_{j} = m(p_{j}, p′_{j}); n_{j} = m(p_{j}, ∏_{i≠j}(p′_{i})^{m_{i}});

a_{j}m_{j} + n_{j} = m(p_{j}, p′).

Proof. m(p_{j}, p′_{j}) = deg(p_{j}|Z(p′_{j})) = deg(x^{η_{j}}y|y + x^{b_{j}} = 0) = b_{j} + η_{j} = a_{j}. The others follow by similar argument.

8. Topological Series of Plane Singularities with Coefficients in a Mixed Hodge Module

Let M ∈ MHM(C^{2}, 0) be mixed Hodge module with singular locus Δ such that V = M|_{B\Δ} is a polarisable variation of Hodge structure. Here (B, Δ) are some representatives as usual.

Consider a curve singularity p and let p′ be an element of the topological series belonging to p, relative to Δ. Our purpose is to compare Spp_{ψ}(M, p, 0) and Spp_{ψ}(M, p′, 0). Since p′ is a modification of p in the neighbourhoods {U_{j}}_{j=1}^{r}, we expect that the difference of their spectral pairs depends only on the germs p̃_{j} : (U_{j}, P_{j}) → (C, 0) and the restriction of H^{0}π^{∗}M on ∪_{j=1}^{r}U_{j}. Since the restriction on U_{j} has abelian representation over U_{j} − {xy = 0}, by our general principle, we expect that we need only the corresponding limit mixed Hodge structures at the points {P_{j}}_{j} (and the graphs of the germs {p̃_{j}}_{j}).

8.1. Limit mixed Hodge structures. Let π : U → B be the resolution of p^{−1}(0) ∪ Δ as in Section 7. Let π^{∗}V be the lifting of V on U \ D and V_{j} its restriction to U_{j} \ D (j ∈ S(p)). If j ∈ S(p) \ S(Δ), then G_{j} := H_{1}(U_{j} \ D, Z) = Z, and it is generated by e_{1} = [M_{d_{j}}]. If j ∈ S(p) ∩ S(Δ), then G_{j} = Z^{2}, and it is generated by e_{1} = [M_{d_{j}}] and e_{2} = [M_{j}].

In both cases the variation V_{j} has abelian representation, therefore its limit mixed Hodge structure LV_{j} exists ([13]). Set L_{j}V := Gr^{W} LV_{j} ∈ MHS(G_{j}). Its representation is denoted by ρ_{j}.

8.2. Intrinsic meaning. We show, that L_{j}V can be recovered in a more direct way, without using the resolution π.

Let Z_{j} = {p_{j} = 0} ⊂ (C^{2}, 0). Let σ_{j} : (C, 0) → (Z_{j}, 0) be the normalization of Z_{j}, and let x be a local uniformization of (C, 0). Then H_{j} := Gr^{W} ψ_{x}H^{0}σ_{j}^{∗}ψ_{p_{j}}M ∈ MHM({0}) is a mixed Hodge structure with two semi-simple monodromy actions: T_{j}^{h} (called the horizontal monodromy) is induced by ψ_{p_{j}} and T_{j}^{v} (called the vertical monodromy) is induced by ψ_{x}.

Now, if π is the resolution as above, then π|_{St(Z_{j})} : St(Z_{j}) → Z_{j} is the normalization of Z_{j}. Moreover, if (x, y) are the coordinates in U_{j} as in Section 7.1, then x is a local coordinate in St(Z_{j}). By our notation (cf. Section 7.3) p_{j} ∘π|U_{j} = x^{η_{j}}y c(x, y), where c ∈ O_{U_{j}}^{∗}. Then by the properties of Section 2, H_{j} = Gr^{W} ψ_{x}ψ_{p_{j}∘π}N, where N = H^{0}π^{∗}M. Obviously, this depends only on N |_{U_{j}\{xy=0}} = V_{j} hence H_{j} = Gr^{W} ψ_{x}ψ_{x^{η_{j}} yc}V_{j}.

Notice that LV_{j} = ψ_{x}ψ_{y}V_{j}, and this limit depends on the choice of the coordinates (x, y). The ambiguity is modulo the action of exp(σ_{C}), where σ_{C} is the complex monodromy cone. Therefore L_{j}V = H_{j}.

By a local computation:

T_{j}^{h} = ρ_{j}([M_{j}]) and T_{j}^{v} = ρ_{j}([M_{d_{j}}] − η_{j}[M_{j}]). (16)

Notice, that if j ∈ S(p) \ S(Δ), then T_{j}^{h} = ρ_{j}([M_{j}]) is the identity and the only essential action is T_{j}^{v} = ρ_{j}([M_{d_{j}}]) (cf. Section 8.1).

8.3. Further computations. Similarly as in Section 8.2, we can consider the mixed Hodge structure H̃_{j} = Gr^{W} ψ_{x}H^{0}σ_{j}^{∗}φ_{p_{j}}M ∈ MHM({0}) with semi-simple monodromies T̃_{j}^{h} (induced by φ_{p_{j}}) and T̃_{j}^{v} (induced by ψ_{x}).

Let H_{j} = H_{j,≠1} ⊕ H_{j,1} be the eigenspace decomposition given by T_{j}^{h}; similarly H̃_{j} = H̃_{j,≠1} ⊕ H̃_{j,1} provided by T̃_{j}^{h}.

On the other hand, if ϵ_{j} = 1 (e.i. Z_{j} = Δ_{s_{j}}, but p′|_{Δ_{s_{j}} ≢0}), then j ∈ S(Δ) \ S(p′); and to apply Theorem 5.2 we need the limit mixed Hodge structures V_{j}^{k} = Gr^{W} LH^{k}i_{Δ_{s_{j}}}^{!} M, too. Their monodromy is denoted by T_{j}^{k}.

We have the following relations between our limit mixed Hodge structures and their semi-simple actions:

Proposition 8.1. Let j ∈ S(p) ∩ S(Δ). Then:

can : (H_{j,≠1}; T_{j}^{h}, T_{j}^{v}) ˜→(H̃_{j,≠1}; T̃_{j}^{h}, T̃_{j}^{v}) is an isomorphism in MHS(Z^{2}), and

0 → (V_{j}^{0}, T_{j}^{0}) → (H̃_{j,1}, T̃_{j}^{v}) ⟶^{V ar} (H_{j,1}, T_{j}^{v})(−1) → (V_{j}^{1}, T_{j}^{1}) → 0

is an exact sequence in MHS(Z).

Proof. Use (1), (2) and (3) and the fact that the vanishing cycle functor is exact.

9. The Spectral Pairs of Series of Plane Singularities

9.1. General formula. Let M ∈ MHM(C^{2}, 0) be a polarizable mixed Hodge module with critical locus Δ. For simplicity, we assume that M restricted to the complement of Δ is pure (but this is not essential because of Lemma 2.9.c). Let p : (C^{2}, 0) → (C, 0) be a curve singularity, and p′ a germ in the topological series belonging to p relative to Δ.

Fix j ∈ S(p). In Section 8.1 we constructed a representation (H_{j}, ρ_{j}) ∈ MHS(G_{j}). If Γ̃_{j} denotes the decorated resolution graph of p′ ∘ π|U_{j} : (U_{j}, P_{j}) → (C, 0) (relative to {xy = 0}), and e_{j} its edge which joins Γ and Γ_{j} (cf. Section 7.1), then SppΓ̃_{j}(ρ_{j}) and Spp_{e_{j}}(ρ_{j}) are defined (cf. Definitions 4.3–4.4). (The latter one is ∑_{p,q,i} T (p, q)Spp_{e_{j}}(χ_{i}^{pq}), with the notations of Definition 4.4.) We let

Spp_{Γ̃_{j},e_{j}}(ρ_{j}) := SppΓ̃_{j}(ρ_{j}) + Spp_{e_{j}}(ρ_{j}).

Theorem 9.1. Let ϵ_{j} and s_{j} be as in Section 7.1. Then one has:

Spp_{ψ}(M, p′, 0) − Spp_{ψ}(M, p, 0) = ∑_{j∈S(p)} C_{j}, where

C_{j} = Spp_{Γ̃_{j},e_{j}}(ρ_{j}) + ϵ_{j} ∑_{k}(−1)^{k}c_{deg(p′|Δ_{s_{j}})}Spp(V_{j}^{k}, T_{j}^{k}).

Notice also that deg(p′|Δ_{s_{j}}) is equal to the intersection multiplicity m_{P_{j}}(y, p′ ∘ π|U_{j}).

Proof. Using (2) and Lemma 2.9 we can assume that M = j_{∗}V, where j : B \Δ → B is the inclusion. Consider the resolution π′ corresponding to the graph Γ′. Notice that in the first step of Section 5.1 we did not use the commutativity assumption. Therefore, the relation (6) holds for both p and p′. If d ∈ D (or e ∈ Ẽ) then in a neighbourhood of D_{d} (or of P_{e}, respectively) the functions p′ ∘π and p∘π differ only by an invertible function. Therefore, their contributions are equal. Using again the first step of Section 5.1 applied to p′ ∘ π|U_{j}, one has:

Spp(i_{0}^{∗}ψ_{p′}M) − Spp(i_{0}^{∗}ψ_{p}M) = Spp_{e_{j}}(ρ_{j}) + Spp(i_{P_{j}}^{∗}ψ_{p′∘π}π^{∗}M).

Now, since G_{j} is abelian, the result follows from Theorem 5.1.

We end this subsection with the following remark. By Lemma 2.9.e, one has:

Spp_{φ}(M, p′, 0) − Spp_{φ}(M, p, 0) = Spp_{ψ}(M, p′, 0) − Spp_{ψ}(M, p, 0)

for any M, p and p′.

9.2. The case of topologically trivial series. In the sequel δ denotes the characteristic map of R \ Z, i.e., δ : R → {0, 1} is defined by δ(x) = 0 if x ∈ Z and = 1 otherwise.

Fix a character χ : Z^{2} → C^{∗} of finite order. Choose ω and λ such that χ(e_{1}) = exp(2πiω) and χ(e_{2}) = exp(2πiλ). Given a triple (n, m, a) of integers (where n ≥ 0, m > 0 and a > 0), we define in Z[Q × Z] the elements C_{ψ}(χ) and C_{φ}(χ) as follows.

For any k ∈ {0, …, n + am − 1} we let

ξ_{k} = \frac{k + λ + aω}{n + am} .

Then define C_{ψ}(χ) :=

\begin{matrix}∑_{k=0}^{n+am−1}(1 − \{−ω\} + \{ξ_{k}\} + \{−ω + mξ_{k}\} & − \{mξ_{k}\}, \\ 2 − δ(ω) − δ(mξ_{k}) + δ(−ω + mξ_{k})), \end{matrix}

and define C_{φ}(χ) :=

∑_{k=0}^{n+am−1}({ω} + {ξ_{k}} + {−ω + mξ_{k}} − {mξ_{k}}, δ(ω) − δ(mξ_{k}) + δ(−ω + mξ_{k})).

For ρ ∈ MHS(Z^{2}) define

C_{ψ}(ρ) := ∑_{p,q} ∑_{i=1}^{d(p,q)} T (p, q)C_{ψ}(χ_{i}^{pq})

(and similarly C_{φ}(ρ)), where the characters χ_{i}^{pq}’s are as in Definition 4.4.

Example 9.2. 1) Assume that m = 1. Since ω and λ are defined modulo an integer, we can take some representatives ω ∈ [0, 1) and λ ∈ [nω, nω + 1). Then

C_{φ}(χ) = ∑_{k=0}^{n+a−1}(ξ_{k}, δ(ω) − δ(ξ_{k}) + δ(−ω + ξ_{k})).

2) Assume that m = 1 and n = 0. Then

C_{φ}(χ) = ∑_{k=0}^{a−1} (ω + \frac{k + λ}{a}, δ(ω) − δ (ω + \frac{k + λ}{a}) + δ (\frac{k + λ}{a})) .

Now we return to our representations (H_{j}, ρ_{j}) and (H̃_{j}, ρ̃_{j}), (j ∈ S(p)). Recall that in G_{j} = Z^{2} one has: e_{1} = [M_{d_{j}}] and e_{2} = [M_{j}]. We define the representation ρ_{j}^{∗} : Z^{2} → Aut_{MHS}(H_{j}) by the following change of bases: ρ_{j}^{∗}(e_{1}) = ρ_{j}(e_{2}) and ρ_{j}^{∗}(e_{2}) = ρ_{j}(e_{1} − η_{j}e_{2}). Then ρ_{j}^{∗}(e_{1}) = T_{j}^{h} and ρ_{j}^{∗}(e_{2}) = T_{j}^{v} by (16). Similarly we define ρ̃_{j}^{∗}.

The relation between the above invariants is given in the following lemma.

Lemma 9.3. Let (H_{j}, ρ_{j}), (H̃_{j}, ρ̃_{j}) ∈ MHS(Z^{2}) and (V_{j}^{k}, T_{j}^{k}) be as in Section 8. Define C_{ψ} and C_{φ} using the triplet (n_{j}, m_{j}, a_{j}). Then

C_{ψ}(ρ_{j}^{∗}) + ∑_{k}(−1)^{k}c_{n_{j}+a_{j}m_{j}}Spp(V_{j}^{k}, T_{j}^{k}) = C_{φ}(ρ̃_{j}^{∗}).

Proof. Notice that if χ(e_{1}) ≠ 1 (i.e., ω ∉ Z) then C_{ψ}(χ) = C_{φ}(χ). Otherwise, C_{φ}(χ) = T (−1, −1)C_{ψ}(χ) = c_{n+am}Spp(C, χ(e_{2})). Now use Proposition 8.1.

Now we consider p′, a germ in the topological series belonging to p relative to Δ, as in Section 7.2. From the first step of Section 5.1, it is clear that the correction C_{j} is zero provided that j ∈ S(p)\S(Δ). In the sequel we assume that j ∈ S(p)∩S(Δ). We also assume that p̃_{j} = (y + x^{b_{j}}b(x, y))a(x, y); otherwise C_{j} = 0 again.

We will compute C_{j} in a more direct way. First notice that our relations do not depend on the choice of the resolution graph. In the case of p′, we will take the graph Γ′, but for p we will use the graph of π modified by b_{j} blowing-ups. This graph will be denoted by Γ′′, and its schematic form is the following:

(17)  [figure: ⋮; …; l_{j}; l_{j}+m_{j}; l_{j}+(b_{j}−1)m_{j}; l_{j}+b_{j}m_{j}; E′′; m_{j}; Γ; ]

The arrow corresponds to the strict transform of Z_{j} = Δ_{s_{j}}.

By a similar argument to that in the proof of Theorem 9.1, one has:

C_{j} = Spp_{E′}(ρ_{j}) − Spp_{E}(ρ_{j}) + ∑_{k}(−1)^{k}c_{n_{j}+a_{j}m_{j}}Spp(V_{j}^{k}, T_{j}^{k}), (18)

where the vertices E′′ and E′ are drawn on the corresponding graphs, (cf. (17) and (14)).

Lemma 9.4. Spp_{E′}(ρ_{j}) − Spp_{E}(ρ_{j}) = C_{ψ}(ρ_{j}^{∗}).

Proof. We can assume that ρ_{j} is one dimensional. Let ρ_{j}^{∗}(e_{1}) = exp(2πiω) and ρ_{j}^{∗}(e_{2}) = exp(2πiλ).

In the case of p (graph Γ′′) the invariants are:

– the degree of E′′ is 2;

– the multiplicities of the adjacent vertices are: n_{j} + (a_{j} − 1)m_{j} and m_{j};

– the monodromies around the adjacent vertices are: exp(2πi(λ + (a_{j} − 1)ω)) and exp(2πiω).

In the case of p′ (graph Γ′) the invariants are:

– the degree of E′ is 3;

– the multiplicities of the adjacent vertices are: n_{j} + (a_{j} − 1)m_{j}, m_{j} and 0;

– the monodromies around the adjacent vertices are: exp(2πi(λ + (a_{j} − 1)ω)), 1 and exp(2πiω).

The multiplicity of E′ and E′′ is equal to n_{j} + a_{j}m_{j}, and the monodromy around them is exp(2πi(λ + a_{j}ω)).

The verification is left to the reader.

The main result of this section is the following:

Theorem 9.5. Let M be a pure mixed Hodge module with critical locus Δ, p′ a germ of the topologically trivial series belonging to p = ∏_{j} p_{j}^{m_{j}}, relative to Δ. Let H̃_{j} be the limit mixed Hodge structure of φ_{p_{j}}M with semisimple monodromies T_{j}^{h} (induced by φ_{p_{j}}) and T_{j}^{v} (induced by the monodromy of φ_{p_{j}}M). Then

Spp_{ψ}(M, p′, 0) − Spp_{ψ}(M, p, 0) = ∑_{j:ϵ_{j}=1} C_{φ,j}(H̃_{j}; T_{j}^{h}, T_{j}^{v}),

where C_{φ,j} is C_{φ} with parameters (n_{j}, m_{j}, a_{j}).

Proof. Use Theorem 9.1, (18), Lemma 9.4 and Lemma 9.3.

Corollary 9.6. Let M and p, p′ as in Theorem 9.5. Assume that p is irreducible. Set

(H̃_{j}; T_{j}^{h}, T_{j}^{v}) = ⊕_{p,q} ⊕_{i=1}^{d(p,q)} (H_{i}^{pq}; exp(2πiω_{i}^{pq}), exp(2πiλ_{i}^{pq})),

where H_{i}^{pq} is one-dimensional C-Hodge structure of type (p, q), and ω_{i}^{pq}, λ_{i}^{pq} ∈ [0, 1). The integer a denotes the intersection multiplicity m(p, p′). Then:

Spp_{ψ}(M, p′, 0) − Spp_{ψ}(M, p, 0) =

∑_{p,q,i} T (p, q) ∑_{k=0}^{a−1} (ω_{i}^{pq} + \frac{k + λ_{i}^{pq}}{a} , δ(ω_{i}^{pq}) − δ (ω_{i}^{pq} + \frac{k + λ_{i}^{pq}}{a} ) + δ (\frac{k + λ_{i}^{pq}}{a} )) .

Proof. If p is irreducible, then S(p) has only one element whose multiplicity is one, and n = 0. Now apply Example 9.2.

10. Spectral Pairs of Series of Composed Singularities

10.1. Some invariants of an ICIS with 2-dimensional base space. Let ϕ : (X, x) → (C^{2}, 0) be an analytic germ such that both (X, x) and (ϕ^{−1}(0), 0) are isolated complete intersection singularities, and dim X = n ≥ 2. By [18] there exists a projective extension ϕ̄ : X̄ → (C^{2}, 0) of ϕ such that the central fiber X̄_{0} = ϕ̄^{−1}(0) has only x as singular point and the germ ϕ̄ : (X̄, x) → (C^{2}, 0) is analytically equivalent to ϕ. Fix a good representative ϕ̄ : X̄ → B of ϕ̄ such that a suitable restriction of it is a good representative ϕ : X → B of ϕ. We let C ⊂ X be the critical locus of ϕ and Δ = ϕ(C) the (reduced) discriminant locus with irreducible decomposition ∪_{i=1}^{s}Δ_{i}. Over Δ_{i} lie the irreducible components C_{i1}, …, C_{it_{i}} of C. The degree of the rectriction ϕ : C_{il} → Δ_{i} is d_{il}.

Let σ_{i} : U → Δ_{i} ( resp. σ_{il} : U → C_{il}) be the normalization and z_{i} (resp. z_{il} ) a uniformizing parameter in U.

Let p_{i} be a generator of the ideal of Δ_{i}. Then the support of the mixed Hodge

module φ_{p_{i}∘ϕ̄}QH ; X̄[n] is a subset of X̄_{0} ∪C. For any i ∈ {1, …, s} and l ∈ {1, …, t_{i}}

we define (H̃_{il}; T̃_{il}^{h}, T̃_{il}^{v}) ∈ MHS(Z^{2}) by

H̃_{il} := Gr^{W} ψ_{z_{il}}H^{0}σ_{il}^{∗}φ_{p_{i}∘ϕ̄}QH ; X̄[n].

The (horizontal) monodromy T_{il}^{h} is induced by φ_{p_{i}∘ϕ̄}, the other (vertical) one T_{il}^{v} by ψ_{z_{il}}. By the properties of the vanishing cycle functors, they commute.

The structure (H̃_{il}, T̃_{il}^{h}) has another interpretation too. For this, take a point P ∈ Δ_{i} − {0} and a transversal slice S to Δ_{i} at P. Choose P′ ∈ ϕ^{−1}(P) ∩ C_{il}. Then ϕ : (ϕ^{−1}(S), P′) → (S, P) defines an isolated hypersurface singularity. Its limit mixed Hodge structure on its reduced cohomology is exactly H̃_{il} and T̃_{il}^{h} is the semi-simple part of its monodromy.

If H is a mixed Hodge structure with commuting automorphisms T^{h} and T^{v}, define c_{m}(H) as ⊕_{i=1}^{m}H with automorphisms:

c_{m}(T^{h})(x_{1}, …, x_{m}) = (T^{h}(x_{1}), …, T^{h}(x_{m})), and
c_{m}(T^{v})(x_{1}, …, x_{m}) = (T^{v}(x_{m}), x_{1}, …, x_{m−1}).

The new structure in HMS(Z^{2}) is denoted by c_{m}(H; T^{h}, T^{v}).

Define the mixed Hodge module M ∈ MHM(C^{2}, 0) by M = H^{0}ϕ̄_{∗}QH ; X̄[n]. Then the critical locus of M is Δ and M|_{B\Δ} is pure. In Section 8.3 we associated the set of invariants (H̃_{i}, T̃_{i}^{h}, T̃_{i}^{v})_{i=1}^{s} ∈ MHS(Z^{2}) with such a module, namely H̃_{i} = Gr^{W} ψ_{z_{i}}H^{0}σ_{i}^{∗}φ_{p_{i}}M. The horizontal monodromy T̃_{i}^{h} is induced by φ_{p_{i}}, the vertical one T̃_{i}^{v} by ψ_{z_{i}}.

Lemma 10.1. The following isomorphism holds:

(H̃_{i}; T̃_{i}^{h}, T̃_{i}^{v}) = ⊕_{l=1}^{t_{i}}c_{d_{il}}(H̃_{il}; T̃_{il}^{h}, T̃_{il}^{v}).

Proof. Use (4).

10.2. Topological series of composed singularities. Let ϕ be as above and p : (C^{2}, 0) → (C, 0) an arbitrary analytic germ.

Definition 10.2. The topological series of composed singularities belonging to f = p∘ϕ consists of all composed singularities f′ = p′∘ϕ such that p′ is in the topological series of curve singularities belonging to p relative to the discriminant locus Δ of ϕ.

For simplicity, we will consider only germs f′ = p′∘ϕ, where p′ belongs to a topologically trivial series of p, relative to Δ. For these germs we extend Theorem 9.5 and Corollary 9.6. The interested reader can formulate easily the corresponding result which extends Theorem 9.1 and holds for any p′.

We recall our notations. Let p = ∏_{j=1}^{r} p_{j}^{m_{j}} be the irreducible decomposition of p. Let p′ be as above. We write p′ = ∏_{j=1}^{r}(p′_{j})^{m_{j}}, where we ordered the irreducible factors of p′ as in Lemma 7.4. For any j ∈ S(p) ∩ S(Δ) (with ϵ_{j} = 1) consider the numerical invariants

a_{j} = m(p_{j}, p′_{j}) and n_{j} = m(p_{j}, ∏_{i≠j}(p′_{i})^{m_{i}}).

Theorem 10.3. With the notation Spp_{ψ}(f) = Spp_{ψ}(Q_{X}^{H}[n], f, x) one has:

Spp_{ψ}(f′) − Spp_{ψ}(f) = ∑_{j∈S(p)∩S(Δ)} ϵ_{j} C_{φ,j}(H̃_{j}; T̃_{j}^{h}, T̃_{j}^{v}),

where C_{φ,j} is C_{φ} with parameters (n_{j}, m_{j}, a_{j}).

Remark 10.4. Lemma 10.1 implies that

C_{φ,j}(H̃_{j} : T̃_{j}^{h}, T̃_{j}^{v}) = ∑_{l=1}^{t_{i}} C_{φ}(n_{j}d_{jl}, m_{j}, a_{j}d_{jl})(H̃_{jl}; T̃_{jl}^{h}, T̃_{jl}^{v}).

Proof of Theorem 10.3. First notice, that Spp_{ψ}(f′) − Spp_{ψ}(f) = Spp_{φ}(f′) − Spp_{φ}(f) (by Lemma 2.9.e).

Define N = i∗ ; X̄_{0}φ_{p∘ϕ̄}QH ; X̄[n]. Let j : X̄_{0}\{x} ↪ X̄_{0} be the natural inclusion. Then j^{∗}N = j^{∗}φ_{p∘ϕ̄}QH ; X̄[n], because φ_{p∘ϕ̄}QH ; X̄[n] is supported in X̄_{0} ∪ C; and Gr^{W} j^{∗}N is a constant variation of mixed Hodge structure with stalk isomorphic to φ_{p}Q_{B}^{H}[2]. Similarly we consider N′ defined by p′. Then Gr^{W} j^{∗}N ≃ Gr^{W} j^{∗}N′. Therefore, by the first distinguished triangle of (2), one has:

Spp_{φ}(f′) − Spp_{φ}(f) = Spp(ϕ̄_{∗}N′) − Spp(ϕ̄_{∗}N).

But ϕ̄_{∗}N = i_{0}^{∗}φ_{p}ϕ̄_{∗}QH ; X̄. Since for j ≠ 0 the module H^{j}ϕ̄_{∗}QH ; X̄ is smooth

Sppφ_{p}H^{j}ϕ̄_{∗}QH ; X̄ = Sppφ_{p′}H^{j}ϕ̄_{∗}QH ; X̄

by Theorem 5.1. Therefore

Spp_{φ}(f′) − Spp_{φ}(f) = Spp(M, p′, 0) − Spp(M, p, 0).

Now apply Theorem 9.5.

Corollary 10.5. Assume that p is irreducible, and {p = 0} = Δ_{j}. Let

(H̃_{jl}; T̃_{jl}^{h}, T̃_{jl}^{v}) = ⊕_{p,q} ⊕_{i=1}^{d(p,q,l)} (H_{i}^{pq,l}; exp(2πiω_{i}^{pq,l}), exp(2πiλ_{i}^{pq,l}))

for any l = 1, …, t_{j}, where H_{i}^{pq,l} is one-dimensional C-Hodge structure of type (p, q), and ω_{i}^{pq,l}, λ_{i}^{pq,l} ∈ [0, 1). Let d_{l} be the degree of ϕ : C_{jl} → Δ_{j} and a = m(p, p′). Then Spp_{ψ}(f′) − Spp_{ψ}(f) =

∑_{l,i,p,q} T (p, q) ∑_{k=0}^{ad_{l}−1}(ω_{i}^{pq,l} + \frac{k + λ_{i}^{pq,l}}{ad_{l}} , δ(ω_{i}^{pq,l}) − δ(ω_{i}^{pq,l} + \frac{k + λ_{i}^{pq,l}}{ad_{l}}) + δ(\frac{k + λ_{i}^{pq,l}}{ad_{l}})).

Proof. Apply Corollary 9.6 and Remark 10.4.

Example 10.6. The Yomdin’s series (see [21, 15, 19, 11]). Let f : (C^{n}, 0) → (C, 0) (n ≥ 2) be an analytic germ with one-dimensional critical locus Σ, which has an irreducible decomposition ∪_{l=1}^{t}Σ_{l}. The mixed Hodge module φ_{f}Q_{X}^{H}[n] is supported on Σ and its restriction on Σ_{l} \ {0} is an admissible variation of mixed Hodge structure. Denote its limit by K̃_{l}. Two natural semi-simple automorphisms act on K̃_{l}: T̃_{l}^{h} induced by φ_{f}, and T̃_{l}^{v} induced by the monodromy of the restriction φ_{f}Q_{X}^{H}[n]|Σ_{l} \ 0.

Let l : (C^{n}, 0) → (C, 0) be a generic linear form. Then the pair ϕ := (f, l) : (C^{n}, 0) → (C^{2}, 0) defines an ICIS, and ϕ(Σ) = Δ_{1} is one of the irreducible components of Δ. Then

(K̃_{l}, T̃_{l}^{h}, T̃_{l}^{v}) = (H̃_{1l}, T̃_{1l}^{h}, T̃_{1l}^{v}).

Notice that f = p∘ϕ where p(z_{1}, z_{2}) = z_{1}. Then for sufficiently large a p′(z_{1}, z_{2}) = z_{1} + z_{2}^{a} is in the topologically trivial series belonging to p, relative to Δ.

Therefore Spp_{ψ}(f + l^{a}) − Spp_{ψ}(f) follows from Remark 10.4.

11. A Generalized Sebastiani-Thom Type Result

If (H_{i}, W_{•}^{i}, F_{i}^{•}) (i = 1, 2) are mixed Hodge structures, then H_{1} ⊗ H_{2} carries a mixed Hodge structure with weight filtration W_{k} = ∑_{p+q=k} W_{p}^{1} ⊗ W_{q}^{2} and Hodge filtration F^{k} = ∑_{p+q=k} F_{1}^{p} ⊗ F_{2}^{q}. This mixed Hodge structure is still denoted by H_{1} ⊗ H_{2}.

A graded mixed Hodge structure is a finite direct sum H^{•} = ⊕_{k}H^{k} of mixed Hodge structures {H^{k}}_{k}. Their category is denoted by GMHS. There is a natural extension of the tensor product to GMHS by H_{1}^{•} ⊗ H_{2}^{•} := ⊕_{k}(⊕_{p+q=k}H_{1}^{p} ⊗ H_{2}^{q}).

There is a natural tensor product MHS(G_{1}) ⊗ MHS(G_{2}) → MHS(G_{1} × G_{2}), which associates with two elements, say ρ_{i} : G_{i} → Aut_{MHS}(H_{i}) (i = 1, 2), an element ρ = ρ_{1} ⊗ ρ_{2} : G_{1} × G_{2} → Aut_{MHS}(H_{1} ⊗ H_{2}) defined by ρ(g_{1}, g_{2}) = ρ_{1}(g_{1}) ⊗ ρ_{2}(g_{2}).

If GMHS(G) denotes the graded version of MHS(G) (i.e., the category of representations ρ^{•} = ⊕_{k}ρ^{k}, where ρ^{k} ∈ MHS(G)), then the above tensor product has an extension

GMHS(G_{1}) ⊗ GMHS(G_{2}) → GMHS(G_{1} × G_{2})

given by ρ_{1}^{•} ⊗ ρ_{2}^{•} = ⊕_{k}(⊕_{p+q=k}ρ_{1}^{p} ⊗ ρ_{2}^{q}).

There is an extension Spp : GMHS(Z) → Z[Q × Z] of the map defined in Definition 2.6 by Spp(ρ^{•}) = ∑_{k}(−1)^{k}Spp(ρ^{k}). Moreover if Γ is as in Section 3, then Spp_{Γ} also extends by the same formula as above.

The map (α, w) ∗ (β, ω) = (α + β, w + ω) extends to a bilinear map Z[Q × Z] ⊗_{Z} Z[Q×Z] → Z[Q×Z]. In general it is not true that Spp(ρ_{1}⊗ρ_{2}) = Spp(ρ_{1})∗Spp(ρ_{2}) but still one has the following result.

Lemma 11.1. Set (H_{i}, ρ_{i}) ∈ MHS(Z) such that ρ_{1} = 1_{H_{1}}. Let c_{n} (n ∈ N^{∗}) be the map defined in Lemma 2.9.f. Then one has:

a) Spp(ρ_{1} ⊗ ρ_{2}) = Spp(ρ_{1}) ∗ Spp(ρ_{2}).

b) If ξ, ζ ∈ Z[Q × Z] such that ξ = ∑(α, ω) satisfies α ∈ Z, then

c_{n}(ξ ∗ ζ) = ξ ∗ c_{n}(ζ).

The easy proof is left to the reader.

In the sequel it is convenient to define the map c_{∞} as the zero map.

Let g : (X, x) → (C, 0) be an analytic germ. Denote ⊕_{j}H^{j}i_{0}^{∗}ψ_{g}Q_{X}^{H}[n] by (H_{g}^{•}, T_{g}^{•}) ∈ GMHS(Z). Here T_{g}^{•} = ρ_{g}^{•}(1) is the semisimple part of the monodromy. The decomposition ψ_{g} = ψ_{g,1} ⊕ ψ_{g,≠1} gives a decomposition H_{g,1}^{•} ⊕ H_{g,≠1}^{•} of H_{g}^{•}. Therefore the spectrum Spp_{ψ}(g) := Spp(H_{g}^{•}, T_{g}^{•}) can be written as the sum of the corresponding spectral pairs Spp_{ψ,1}(g) and Spp_{ψ,≠1}(g). There are similar notations for φ instead of ψ. By Lemma 2.9.e one has:

Spp_{ψ}(g) = Spp_{φ}(g) + (−1)^{n−1}(0, 0).

We introduce the notation:

Spp_{!}(g) = Spp_{φ,1}(g) − T (1, 1)Spp_{ψ,1}(g) = (1 − T (1, 1))Spp_{ψ,1} + (−1)^{n}(0, 0).

Consider an analytic germ p : (C^{2}, 0) → (C, 0) and the space-germ (Δ, 0) given by {cd = 0}, where (c, d) are the local coordinates in (C^{2}, 0). Let Γ be the decorated resolution graph of p with respect to (Δ, 0) (cf. Example 4.5). Notice that G(Γ) = Z^{S} = H_{1}(C^{2} \ ((p^{−1}(0) ∪ Δ)), Z). The group H_{1}(C^{2} \ Δ, Z) = Z^{2} is generated by [M_{c}] and [M_{d}], where M_{c} (resp. M_{d}) is an oriented circle in a transversal slice to {c = 0} (resp. {d = 0}).

Let g : (X, x) → (C, 0) and h : (Y, y) → (C, 0) be two analytic germs which define ICIS’s, and p : (C^{2}, 0) → (C, 0) an arbitrary analytic germ. Define f : (X × Y, x × y) → (C, 0) by f(x, y) = p(g(x), h(y)).

Consider the tensor product H_{g}^{•} ⊗ H_{h}^{•} ∈ GMHS(Z^{2}). By the identification of Z^{2} with H_{1}(C^{2} \ Δ, Z) one has:

ρ^{•}([M_{c}]) = T_{g}^{•} ⊗ id and ρ^{•}([M_{d}]) = id ⊗ T_{h}^{•}.

176 A. Némethi and J.H.M. Steenbrink

Via the map G(Γ) → H_{1}(C^{2} \ Δ, Z), induced by the inclusion, H_{g}^{•} ⊗ H_{h}^{•} becomes an element of GMHS(G(Γ)).

Let r(c) be the intersection multiplicity m_{0}(p, c) if {c = 0} is not a factor of p, and = ∞ otherwise. Symmetrically define r(d). The main result of this section is the following.

Theorem 11.2. Let h, g, p and f be as above. Then:

Spp_{ψ}(f) = Spp_{Γ}(H_{h}^{•} ⊗ H_{g}^{•}) + c_{r(d)}Spp_{ψ}(g) ∗ Spp_{!}(h) + c_{r(c)}Spp_{ψ}(h) ∗ Spp_{!}(g).

Proof. Consider a projective extension ḡ : (X̄, X̄_{0}) → (C, 0) of the germ g such that x ∈ X̄_{0} = ḡ^{−1}(0) is the only singular point of ḡ and ḡ : (X̄, x) → (C, 0) analytically equivalent to g. (For the existence of ḡ, see [18].) Similarly, let h̄ : (Ȳ , Ȳ_{0}) → (C, 0) be a projective extension of h with the above properties. Now use several times the first distinguished triangle (2) applied to the natural stratification of X̄_{0} × Ȳ_{0}, and apply Theorem 5.2. The details are left to the reader.

For the corresponding formula at the zeta function level, see [7, 8].

Example 11.3. Let g and h be as above. Then one has:

a) If Spp_{ψ}(g) = ∑(α, w) and Spp(h) = ∑(β, ω), then

Spp_{ψ}(g(x) · h(y)) = [T (1, 1) − 1] · ∑_{[α]=[β]}([α] + [β] + α, w + ω).

b) For the sum f(x, y) = g(x) + h(y), it is simpler to rewrite Theorem 11.2 in terms of Spp_{St} (see Remark 2.7):

Spp_{St}(f(x) + g(y)) = T (1, 0)Spp_{St}(g) ∗ Spp_{St}(h)

(cf. [12]).

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Spectral Pairs 177

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Department of Mathematics, The Ohio State University, Columbus, OH 43210, U.S.A

nemethi@math.ohio-state.edu

Department of Mathematics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands

steenbri@sci.kun.nl