The eta–invariant of variation structures. I

András Némethi^{∗}

October 13, 1997

Abstract. The main motivation of the paper is the signature of the Milnor fiber of the germ f_{1}+f_{2}^{q}, where (f_{1}, f_{2}) is an isolated complete intersection singularity. This is computed in terms of a new invariants of the pair (f_{1}, f_{2}), which corresponds to the algebraic version of the eta–invariant, introduced by Atiyah–Patodi–Singer, for a hermitian flat bundle over the circle.

Key words: Singularities, Milnor fiber, signature, eta–invariant.

AMS Subj. Class.: 32S05, 32S25, 32S55, 11E39

1 Introduction

Let f_{1} : (C^{n+1}, 0) → (C, 0) be the germ of an analytic function with one dimensional singular locus Σ. Consider a germ f_{2} so that the pair ϕ = (f_{1}, f_{2}) defines an isolated complete intersection singularity (ICIS). Our goal is to compute the correction term σ(f_{1} + f_{2}^{q}) − σ(f_{1}), where σ(f) denotes the signature of the Milnor fiber of the germ f.

Let G be the local fundamental group of the complement of the discriminant locus of the ICIS. Then ϕ defines a representation ρ : G → Aut(U; b), where U is the middle homology group of the fiber with intersection form b. Moreover, since the geometric monodromy of ϕ at the boundary of the fiber is trivial, for each g ∈ G, we have a variation map V (g) : U^{∗} → U.

Any g ∈ G defines a spectral decomposition of the system (U; b, ρ, V) = ⊕_{χ}(U_{χ}; b_{χ}, ρ_{χ}, V_{χ}); (here U_{χ} are the generalized eigenspaces associated with ρ(g)). Define η(g) = ∑_{χ} η(g)_{χ}, where:

∗Partially supported by NSF grant No. DMS-9203482

η(g)_{χ} = \{ \begin{matrix}(1 − 2a)signb_{χ} & if χ = e^{−2πia}; 0 < a < 1 \\ sign(I + ρ_{1}(g^{−1}))V_{1}(g) & if χ = 1 \end{matrix}

In fact, η is well–defined on the conjugacy classes of G. Let M, respectively L, be the topological standard meridian, respectively the longitude of the discriminant component ϕ(Σ).

The main result of this paper is the following (Theorem 5.2): Theorem. For q >> 0, the following relation holds:

σ(f_{1} + f_{2}^{q}) − σ(f_{1}) = q · η(M) + η(L) − η(L + qM).

In general, the question is the following. Let inv be any invariant of the analytic germs. For example, χ(f) = the Euler–characteristic of the Milnor fiber, ζ(f) = the zeta–function of f, Sp(f) = the spectrum of f, σ(f) = the signature, etc. Question: Can the correction term inv(f_{1} + f_{2}^{q}) − inv(f_{1}) be computed in terms of some information provided by the restriction of ϕ above a small neighbourhood of the link of ϕ(Σ)? If the answer is yes, then what is this correction term?

For χ(f), ζ(f), Sp(f) the answer is yes. For χ(f) the correction term depends on the transversal Milnor numbers of Σ [18], for ζ(f) (resp. Sp(f)) it depends on the monodromy representations (resp. the mixed Hodge module ϕ_{∗}Q_{(C,0)}^{H}) in the neighbourhood of ϕ(Σ) [16, 8] (resp. [14]). (Note, that these invariants are “additive” invariants.)

Our result shows that for σ(f) the answer is again yes. The correction term depends on the monodromy representation and the variation map (we prefer to call this pair variation structure), and it is expressed in terms of some “eta–invariants”. (For another presentation of these eta–invariants and their relations with the spectral pairs, see [9].)

The attentive reader can realize that our eta–invariant is, in some sense, a generalized version of the eta-invariant (associated with the circle and with a hermitian representation) defined in [1]. The clarification of this relation is the subject of another paper.

Our approach is algebraic, in the sections 2–4 we extend Meyer’s result [5] to the case of variation structures. In section 2, we study abstract variation structures (U; b, ρ, V) and in section 3 we determine their Witt group when G = Z^{k}. The object of section 4 is a special 2–cocycle of G which measures the nonadditivity of the signature of two–dimensional manifolds (with boundary) with coefficients in a variation structure. This generalizes the Meyer’s cocycles [5], which were defined for non-degenerate representations. (See 2.6 for a more detailed explication.) In the classical case, for abelian representations, the Meyer’s cocycle is the co-boundary of the classical eta– invariant–cocycle. This fact is generalized in section 4 for the degenerate case. Section 5 contains the main application: the computation of the correction term of the signature in the case of the Yomdin series.

(In this paper, the signature of a (−1)-hermitian form b is signb = sign(i · b).)

2 Hermitian variation structures

2.1. If U is a finite dimensional C–vector space, we denote its dual Hom(U, C) by U^{∗}. There is a natural identification θ : U → U^{∗∗} given by θ(u)(φ) = φ(u).

Any C–linear endomorphism b : U → U^{∗} with \overline{b^{∗} ∘ θ} = ϵb defines an ϵ– hermitian form on U. (Here ϵ = ±1 and \overline{·} is the complex conjugation.) We will use the notation B : U ⊗ U → C, B(u, v) = b(u)(\overline{v}), too. The automorphisms h : U → U with \overline{h}^{∗} ∘ b ∘ h = b form the orthogonal group Aut(U; b). Any group endomorphism ρ : G → Aut(U; b) is called a b–representation of the group G.

Any representation ρ : G → Aut(U) defines a left action of G on Hom(U^{∗}, U) by g ∗ φ = ρ(g) ∘ φ. Then, by definition, a twisted–homomorphism is a map V : G →Hom(U^{∗}, U) with V (gh) = ρ(g) ∘ V (h) + V (g). 2.2. Definition. An ϵ–hermitian variation structure of the group G is a system V = (U; b, ρ, V) with the following properties:

a) b is an ϵ–hermitian form on (the finite dimensional space) U,

b) ρ is a b–representation of G,

c) V is a twisted–homomorphism (with respect to the left action of G), with the following properties:

(i) \overline{θ^{−1} ∘ V (g)}^{∗} = −ϵV (g)\overline{ρ(g)}^{∗}, and

(ii) V (g) ∘ b = ρ(g) − I.

If b is non-degenerate then V is called non-degenerate too. 2.3. It is not difficult to verify that an ϵ–hermitian variation structure satisfies also the following supplementary Properties.

a.) For any g ∈ G, one has b ∘ V (g) = \overline{ρ(g)}^{∗,−1} − I;

b.) Consider the right action of G on Hom(U^{∗}, U) defined by φ ∗ g = φ ∘ \overline{ρ(g)}^{∗,−1}. Then V is a twisted–homomorpism with respect to the right action, too, i.e. for any g and h ∈ G, one has V (gh) = V (g)∘\overline{ρ(h)}^{∗,−1} +V (h).

c.) If gh = hg, then ρ(h) ∘ V (g) ∘ \overline{ρ(h)}^{∗} = V (g). (Write V (gh) = V (hg) and use 2.2.c and 2.3.b) 2.4. Definition. Two ϵ–hermitian structures (U; b, ρ, V) and (U′; b′ρ′, V′) of the group G are isomorphic if there exists an isomorphism φ : U → U′ such that b = φ̄^{∗}b′φ, ρ(g) = φ^{−1}ρ′(g)φ, and V (g) = φ^{−1}V′(g)(φ̄^{∗})^{−1} for any g ∈ G.

The set of isomorphism classes of ϵ–hermitian variation structures is denoted by HV_{ϵ}(G). The natural direct sum determines a semigroup structure on it. 2.5. Examples.

1. If b is non-degenerate then V (g) = (ρ(g) − I)b^{−1}, i.e. the semigroup of the non-degenerate variation structures is equivalent to the semigroup HI_{ϵ}(G) of ϵ–hermitian isometric structures (systems (U; b, ρ) with non-degenerate form b and with axioms (a) and (b)).

2. If V (g) is an isomorphism, then ρ(g) = −ϵV (g)(\overline{θ^{−1} ∘ V (g)^{∗}})^{−1}, and b = −V (g)^{−1}−ϵ(\overline{θ^{−1} ∘ V (g)^{∗}})^{−1}. In particular, if G = Z, then the sub–semigroup HV_{ϵ}^{s}(Z) = {V ⊂ HV_{ϵ}(Z); V (1) isomorphism} is equivalent to the semigroup of the sesqui-linear forms over C.

3. An important element in HV_{ϵ}^{s}(Z), provided by an isolated hypersurface singularity f, is V(f) generated by (U; b, h(1), V (1)) = (middle homology of the Milnor fiber of f; intersection form, monodromy, variation map). Notice that the variation map V (1) of f can be identified with the inverse of the Seifert form (up to a sign) [3], in particular, V (1) is an isomorphism.

4. Consider an isolated complete intersection singularity f : (C^{n+1}, 0) → (C^{2}, 0); (n > 1). Let ϕ : (X , 0) → (S, 0) be a “good representative” of f with discriminant locus Δ ⊂ S. Consider a base-point ∗ ∈ S − Δ. The homology of the fiber F = ϕ^{−1}(∗) is concentrated in U = H_{n−1}(F, C). Identify its dual U^{∗} with H_{n−1}(F, ∂F; C), and extend the real intersection form to a hermitian form b : U → U^{∗}. The monodromy representation ρ : G = π_{1}(S − Δ, ∗) → Aut(U; b), and the variation map V : G → Hom(U^{∗}, U) constitute a system V(ϕ) = (U; b, ρ, V) which is our basic example of (−1)^{n−1}–hermitian variation structure. 2.6. Remarks. Even for very simple groups, the classification of the variation structures is non-trivial. In the case G = Z, the elements of two important sub-semigroups of HV_{ϵ}(Z) are classified. Namely, the non-degenerate structures (i.e. the isometric ones) are classified by Milnor [6], the semigroup HV_{ϵ}^{s}(Z) is determined in [10]. (For the semi–ring structure of HV_{ϵ}^{s}(Z), see [11].)

But in the general case, without any supplementary assumption about the variation structures, even in the case G = Z, the classification is not known (for the author). (For the type of anomalies, which may occur already in the case G = Z, see example 2.7.9 in [10].)

The classification of the elements of HV_{ϵ}^{s}(Z) gives a nice description and organization of the topological invariants of an isolated singularity f. For this, and for the relation of V(f) to the spectral pairs of f, see [9, 10].

In the case of G = Z^{k}, (k ≥ 2), even the underlying problem of classification of endomorphisms ρ : Z^{k} → GL_{n}(C) is open. 2.7. Geometric examples. Relation to the signature and motivations for further definitions. a.) Let (B, ∂B) be a two–dimensional, connected manifold with boundary. Assume that p : X → B is a locally trivial fibration with fiber F, a (4k − 2)– dimensional manifold without boundary.

By a result of Meyer [5], the signature σ(X) of X can be computed by the isometric structure (U; b, ρ) associated with p. More precisely: let b be the (−1)–symmetric hermitian (intersection) form on U = H_{2k−1}(F, C), and let ρ : π_{1}(B) → Aut(U; b) be the monodromy representation of p. Then the twisted cohomology group H^{1}(B, ∂B; ρ) carries a symmetric hermitian form with signature σ(ρ). Meyer proved that σ(X) = σ(ρ).

Moreover, even if ∂F ≠ ∅ (in this case X is a manifold with corners), but b is non–degenerate, the above identity still holds. b.) If B and F both have boundaries, and if the intersection forms on H_{1}(B) resp. on H_{2k−1}(F) are both degenerate, then the signature σ(X) can not be reduced to σ(ρ). The form b and the representation ρ have no control over the geometric behaviour of the fibration in the neighbourhood of the boundary of the fibers ( ∂F, in this algebraic sense, corresponds to kerb).

Assume that the restriction of p : ∂X ∩ p^{−1}(B − ∂B) → B − ∂B is a trivial fibration (cf. 2.5.4). Then, we can associate with p a variation structure V(p) = (U; b, ρ, V) (where V is the variation map of p). Now, V preserve a lot of information from the geometry of the fibration in the neighbourhood of the boundary. It turns out that the variation structure V(p) takes the role of the (non–degenerate) isometric structures in the case when b is degenerate. c.) Assume again that b is non–degenerate. Then there exist some methods for the computation of σ(ρ). The question is: can these methods be generalized to the case of variation structures (i.e. to the non–degenerate case)?

The first remark is that σ(ρ) vanishes on the hyperbolic structures, therefore it depends only on the class of (U; b, ρ) in the corresponding Witt group.

Here arises the first question: when is a variation structure “hyperbolic”, and how can the definition of Witt groups be extended to the case of variation structures. This problem is solved in section 3, and the Witt group of G = Z^{k} is computed. (In our case, the signature will not vanish on the hyperbolic structures, but some other invariants will vanish, see section 4.) d.) Meyer computed σ(ρ), when b is non–degenerate, using his “Meyer cocycles” [5]. This gives a motivation to find the corresponding “Meyer cocycles”, when b is degenerate, in terms of the variation structure. This construction is given in section 4. e.) By the index–theory, σ(ρ) can be computed as follows (b is non–degenerate again). The representation ρ is equivalent to a flat bundle Γ. Since b is (−1)– symmetric, there is a natural complex structure (Γ, J) on Γ. The signature bundle sign(Γ) is defined by (Γ, J)^{∗} − (Γ, J). Let c_{1} be its first Chern class (this is, in fact, a pull–back of the cohomology class given by the Meyer cocycles). Then σ(ρ) = σ(B, sign(Γ)), and by [1]

(∗) σ(B, sign(Γ)) = ∫_{B} c_{1} − \frac{1}{2}η(∂B, sign(Γ)).

Here the last term is the eta–invariant associated with the signature operator of ∂B and the signature bundle sign(Γ).

Assume that the representation ρ is abelian. Then c_{1} = 0, i.e. the primary invariant ∫ c_{1} vanishes, and σ(ρ) can be computed by the eta–invariant associated with ρ|∂B.

This motivates one more question: assume that b is degenerate and the variation structure is abelian. Can the signature σ(X) be computed only by the restriction of the variation structure on ∂B? What is the corresponding expression of this new “eta–invariant” in terms of this restriction?

Our basic, new definition is exactly this new “eta–invariant”. Our main application (theorem 5.2) is, in fact, a result of type (*).

3 The Witt group of variation structures

3.1. Definition. A hermitian variation structure is hyperbolic if there exists a kernel K ⊂ U, i.e. a subset K such that

(a) dimK = \frac{1}{2}dimU,

(b) K ⊂ K^{⊥} = {x| B(x, y) = 0 for any y ∈ K},

(c) For any g ∈ G one has ρ(g)(K) ⊂ K and V (g)(K^{∗}) ⊂ K where K^{∗} = \{φ ∈ U^{∗}| \overline{φ}(K) = 0\}. 3.2. Examples.

1. If b is non-degenerate then V is hyperbolic if and only if the isometric structure (U; b, ρ) is hyperbolic (i.e. there exists a ρ–invariant K with K = K^{⊥}).

2. Consider −V = (U; −b, ρ, −V). Then V ⊕ (−V) is hyperbolic with kernel K = ΔU = {(x, x)| x ∈ U}. 3.3. By (3.2.2), the semigroup W V_{ϵ}(G) = (HV_{ϵ}(G)/{hyperbolic structures}, ⊕) is a group. It is called the Witt group of the variation structures of G.

Our first goal is the computation of W V_{ϵ}(Z^{k}). For any χ ∈ Hom(Z^{k}, C^{∗}), we define the generalized eigenspace U_{χ} = {x ∈ U| (ρ(g) − χ(g))^{N}x = 0 for some N and any g ∈ G}. Ĝ = Hom(Z^{k}, S^{1}) denotes the group of characters.

The verification of the following lemma is left to the reader. 3.4. Lemma. Let G = Z^{k}. Then:

(a) There is a direct sum decomposition:

(U; b, ρ, V) = (U′; b′, ρ′, V′) ⊕ ⨁_{χ∈Ĝ}(U_{χ}; b_{χ}, ρ_{χ}, V_{χ}),

where U′ = ⊕_{χ∉Ĝ}U_{χ}. Moreover, (U′; b′, ρ′, V′) is hyperbolic.

(b) (U; b, ρ, V) is hyperbolic if and only if (U_{χ}; b_{χ}, ρ_{χ}, V_{χ}) is hyperbolic for any χ ∈ Ĝ, in particular

W V_{ϵ}(G) = ⨁_{χ∈Ĝ} W V_{ϵ}(G)_{χ}.

Here W V_{ϵ}(G)_{χ} is the Witt group of variation structures V with ρ(g) − χ(g)I nilpotent for any g ∈ G.

These Witt groups are given by the following proposition: 3.5. Proposition.

W V_{ϵ}(Z^{k})_{χ} = \{ \begin{matrix}Z & if χ ∈ Ĝ − \{1\} \\ Z_{2} & if χ = 1. \end{matrix}

The generators are (C; ±i^{(1−ϵ)/2}, χ, ±(χ − 1)i^{(ϵ−1)/2}) if χ ≠ 1, and (C; 0, 1, 0) if χ = 1. Proof. If χ ≠ 1, then W V_{ϵ}(Z^{k})_{χ} is the Witt group of the corresponding isometric structures. Thus the isomorphism is given by V ↦ signature(i^{(1−ϵ)/2}b_{χ}).

Assume that χ = 1. Fix a non-zero element g_{0} ∈ Z^{k}. Consider the weight filtration W^{•} on U_{1}, centered at zero, induced by g_{0}; i.e. the unique filtration W^{•} with the following properties:

(a) W^{k} ⊂ W^{k+1}; N(W^{k}) ⊂ W^{k−2},

(b) N^{k} : Gr_{W}^{k} U_{1} → Gr_{W}^{−k}U_{1} is an isomorphism, where Gr_{W}^{i} U_{1} = W^{i}/W^{i−1} and N = log ρ(g_{0}). Set m ∈ N such that W^{m} = U_{1}. The dual filtration W_{•}^{∗} is given by W_{k}^{∗} = \{φ ∈ U_{1}^{∗}| \overline{φ}(W^{k−1}) = 0\}. Then W_{−m}^{∗} = U_{1}^{∗}; W_{k}^{∗} ⊂ W_{k−1}^{∗}. Moreover, \overline{N}^{∗}W_{k}^{∗} ⊂ W_{k+2}^{∗} and (\overline{N}^{∗})^{k} : Gr_{−k}^{W^{∗}}U_{1}^{∗} → Gr_{k}^{W^{∗}}U_{1}^{∗} is an isomorphism (here \overline{N}^{∗} = log \overline{ρ(g_{0})}^{∗}).

Now, since ρ(g_{0}) ∈ Aut(U_{1}; b_{1}) one has B_{1}(W^{k}, W^{l}) = 0 for k + l < 0. Indeed, take x ∈ W_{k} and y ∈ W_{l}. Consider a maximal s ≥ 0 (resp. t ≥ 0) such that x = N^{s}x′ (resp. y = N^{t}y′). Then N^{2s+k+1}x′ = 0 resp. N^{2t+l+1}y′ = 0. Therefore B(x, y) = B(N^{s}x′, N^{t}y′) = 0 if either s + t ≥ 2s + k + 1 or s+t ≥ 2t+l+1. If both inequalities fail, then s+t ≤ 2s+k and s+t ≤ 2t+l. By taking their sum: 0 ≤ k + l which is a contradiction. The vanishing of B(W^{k}, W^{l}) for k+l < 0 is equivalent to b_{1}(W^{k}) ⊂ W_{−k}^{∗} for any −m ≤ k ≤ m.

We want similar restrictions for V (g) where g ∈ G is an arbitrary element of G. Since G is abelian, ρ(g)V (h)+V (g) = V (gh) = V (hg) = V (h)\overline{ρ(g)}^{∗,−1}+ V (g). In particular, for any g ∈ G one has N ∘V (g) = −V (g)\overline{N}^{∗}. By a similar argument as above: V (g)(W_{k}^{∗}) ⊂ V_{−k}.

Consider a base in U_{1} (and a dual base in U_{1}^{∗}) compatible with the filtration W^{•}. This gives the splittings W^{k} = W^{k−1} ⊕ W_{c}^{k}. Our maps have the following block-decomposition:

b = ( \begin{matrix}0 & & b^{m} \\ & · \\ b^{−m} & & ∗ \end{matrix} ) , ρ = ( \begin{matrix}ρ^{m} & & & & ∗ \\ & . & . & . \\ 0 & & & & ρ^{−m} \end{matrix} ) , V = ( \begin{matrix}∗ & & V^{m} \\ & · \\ V^{−m} & & 0 \end{matrix} ) .

(The dimension of the i–block is n_{i} = dimGr_{W}^{i} U_{1}, where −m ≤ i ≤ m.) Since the weight filtration can be characterized completely by kernels and images of powers of N, the commutativity of G implies that the weight filtration W^{•} is ρ–invariant. Consider −Gr_{W}^{0} V_{1} = (Gr_{W}^{0} U_{1}; [−b^{0}], [ρ^{0}], [−V^{0}]) and the direct sum H = V_{1} ⊕ (−Gr_{W}^{0} V_{1}). Define K ⊂ U_{1} ⊕ Gr_{W}^{0} U_{1} by K = (W_{−1} ⊕ {0}) ⊕ {(x, [x])| x ∈ W_{c}^{k}}. Then K is a kernel (use the above matrix forms) and defines a hyperbolic structure on H. Consequently, in the Witt group, V_{1} can be replaced with Gr_{W}^{0} V_{1}. The representation ρ^{0} of this latter structure has the property ρ^{0}(g_{0}) =identity. Using induction over the generators of G, we obtain that the Witt group is generated by structures with ρ(g) = 1 for any g ∈ G. Now, since \overline{b}^{∗} = ϵb, \overline{V}^{∗} = −ϵV and V ∘ b = 0, we deduce that, in fact, it is generated by the structures (C; ±i^{(1−ϵ)/2}, 1, 0), (C; 0, 1, ±i^{(1+ϵ)/2}) and T = (C; 0, 1, 0). But the direct sum of any of these with T , is hyperbolic, in particular W V (G)_{1} is generated by T and 2T = 0. On the other hand T is not zero in the Witt group because the dimension of a hyperbolic space is even. □ 3.6. Remark. In the classical theory the following statement is true: If an isometric structure I = (U; b, ρ), (b non-degenerate) is zero in the Witt group W U(G), then I is hyperbolic. In our case this is not true. Take for example V = 2(C; 1, 1, 0). Moreover, this example shows that the signature does not vanish on the set of hyperbolic structures, that is, the set of hyperbolic structures is too large apparently. This is true, but exactly for this reason it will be helpful in our discussion. In fact, we want to define some cocycles on W V_{ϵ}(G) (as secondary invariants) rather than elements in the cohomology of G (i.e. primary characteristic classes).

4 The cocycle associated to the nonadditivity of the signature

4.1. Let V = (U; b, ρ, V) be an ϵ–hermitian variation structure of G. Then b defines an ϵ–hermitian non–degenerate form Φ on U^{∗} ⊕ U by

Φ((φ, u), (ψ, v)) = ϵ\overline{ψ}(u) + φ(\overline{v}) + b(u)(\overline{v}).

Any g ∈ G defines two maps s_{r}(g), s_{l}(g) : U^{∗} ⊕ U → U^{∗} ⊕ U defined by

s_{l}(g)(φ, u) = (φ, ρ(g)u − ρ(g)V (g^{−1})φ),

s_{r}(g)(φ, u) = (\overline{ρ(g)}^{∗,−1}φ, −V (g)φ + u).

By a straightforward verification (use 2 × 2 blocks computation), we obtain that s_{l} resp. s_{r} are representations of the group G in the orthogonal group O(Φ) of Φ. 4.2. For any g ∈ G, define K_{g} = {(φ, u) ∈ U^{∗} ⊕ U : V (g)φ = u}. It is not hard to verify that K_{g} is a Φ–kernel, i.e. K_{g} = K_{g}^{⊥} (the latter one is the Φ–orthogonal). Moreover, for any g and h:

(4.3) s_{l}(h)K_{g} = K_{hg} and s_{r}(h)K_{g} = K_{gh^{−1}}.

4.4. Any three kernels K_{i}; (i = 1, 2, 3) in U^{∗} ⊕ U define an (−ϵ)–hermitian form [17]. We recall this construction. On the space K_{1} ∩ (K_{2} + K_{3}) = {x_{1} ∈ K_{1}| there exist x_{2} ∈ K_{2}, x_{3} ∈ K_{3} with x_{1} + x_{2} + x_{3} = 0} define the sesqui– linear form Ψ(x_{1}, x′_{1}) = Φ(x_{1}, x′_{2}) (where x′_{1} + x′_{2} + x′_{3} = 0 and x′_{2} ∈ K_{2}, x′_{3} ∈ K_{3}). Then Ψ is (−ϵ)–hermitian with kernel Ker = K_{1} ∩ K_{2} + K_{1} ∩ K_{3}. We define σ(V; K_{1}, K_{2}, K_{3}) as the signature of the induced non-degenerate form on K_{1,2,3} = K_{1} ∩ (K_{2} + K_{3})/Ker multiplied by ϵ. (if there is no danger of confusion then it is denoted by σ(K_{1}, K_{2}, K_{3})). If K_{i} = K_{j} for some pair (i, j), then K_{1,2,3} = 0, hence σ(K_{1}, K_{2}, K_{3}) = 0. 4.5. Lemma.

(a) σ(K_{τ(1)}, K_{τ(2)}, K_{τ(3)}) = sign(τ)σ(K_{1}, K_{2}, K_{3}) for any permutation τ ∈ S_{3}; (here sign(τ) ∈ {±1} is the sign of τ).

(b) σ(K_{1}, K_{2}, K_{3}) − σ(K_{0}, K_{2}, K_{3}) + σ(K_{0}, K_{1}, K_{3}) − σ(K_{0}, K_{1}, K_{2}) = 0 for any kernels K_{i}; (i = 1, 2, 3, 4).

(c) If o ∈ O(Φ) is an orthogonal automorphism, then σ(o(K_{1}), o(K_{2}), o(K_{3})) = σ(K_{1}, K_{2}, K_{3}). In particular, σ(K_{hg_{1}}, K_{hg_{2}}, K_{hg_{3}}) = σ(K_{g_{1}h}, K_{g_{2}h}, K_{g_{3}h}) = σ(K_{g_{1}}, K_{g_{2}}, K_{g_{3}}). Proof. (a) follows from [17], (b) from [13]. If o ∈ O(Φ) then o induces an isometry of the corresponding Ψ forms. This with (4.2)-(4.3) proves (c). □ 4.6. By (4.5), σ(V; K_{1}, K_{2}, K_{3}) defines a homogeneous cocycle of the group G in Z (Z is considered with the trivial G–action). The corresponding nonhomogeneous cocycle is σ(g, h) = σ(K_{e}, K_{g}, K_{gh}) (e is the neutral element of G). This cocycle is a coboundary if there exists a function f : G → Z such that σ(g, h) = f(g) + f(h) − f(gh); i.e. if σ(K_{e}, K_{g}, K_{h}) = f(g) + f(g^{−1}h) − f(h). The semigroup morphism

Θ_{I} : HV_{ϵ}(G) → H^{2}(G, Z), Θ_{I}(V) = σ(V; ·, ·, ·)

is not trivial in general. For example, if G is the mapping class group Γ_{g} and

V = (C^{2g}; ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) , ρ, (ρ − I)b^{−1})

where ρ is given by Γ_{g} → Sp(2g, C) (”passing to the homology”), then Θ_{I}(V) is the generator of H^{2}(G, Z) = Z [4]. 4.7. A kernel K̃ ⊂ U^{∗} ⊕U is called V–invariant if for any g ∈ G s_{l}(g)K̃ = K̃. For example, if K ⊂ U is a kernel of the variation structure V (see 3.1), then K̃ = K^{∗} ⊕ K is an invariant kernel in U^{∗} ⊕ U.

If K̃ is an invariant kernel, then 4.5.(b)-(c) gives for K̃, K_{e}, K_{g} and K_{h}:

(4.8) σ(K_{e}, K_{g}, K_{h}) = σ(K̃, K_{e}, K_{g}) + σ(K̃, K_{e}, K_{g^{−1}h}) − σ(K̃, K_{e}, K_{h}).

In particular, σ(V) is a coboundary. (Moreover, f(V; g) = σ(V; K̃, K_{e}, K_{g}) is independent of the choice of the invariant kernel K̃; to see this, consider 4.5(b) for K̃_{1}, K̃_{2}, K_{e} and K_{g}.)

Therefore Θ_{I} induces a map (denoted in the same way):

Θ_{I} : W V_{ϵ}(G) → H^{2}(G, Z).

4.9. Set V ∈ HV_{ϵ}(G). Any g ∈ G defines a morphism Z → G by n ↦ g^{n}. This induces a structure g^{∗}V = (U; b, g^{∗}ρ, g^{∗}V) ∈ HV_{ϵ}(Z) where g^{∗}ρ(n) = ρ(g^{n}) and g^{∗}V (n) = V (g^{n}). This is an abelian structure, hence

g^{∗}V = ⨁_{χ}(g^{∗}V)_{χ} = (U′; b′, ρ′, V′) ⊕ ⨁_{χ∈Ẑ}(U_{χ}; b_{χ}, ρ_{χ}, V_{χ}),

where U′ = ⊕_{χ∉Z^{}}ˆ U_{χ} (cf. 3.4). We define η : G → Z by

η(V; g) = η(g^{∗}V) = ∑_{χ} η(g^{∗}V)_{χ} , where

η(g^{∗}V)_{χ} = \{ \begin{matrix}0 & if χ ∉ Ẑ \\ (1 − 2a)sign(b_{χ}) & if χ(g) = e^{−2πia}; 0 < a < 1 \\ sign[(I + ρ_{1}(g^{−1}))V_{1}(g)] & if χ = 1 \end{matrix}

(Notice that b_{χ} is ϵ–hermitian, and (I +ρ(g^{−1}))V (g) is (−ϵ)–hermitian form.) By definition η(g) depends only on the variation g^{∗}V ∈ HV_{ϵ}(Z).

The first result of this section is: 4.10. Proposition. Suppose that V is hyperbolic, i.e. it has a kernel K ⊂ U. If K̃ = K^{∗} ⊕ K denotes the corresponding invariant kernel in U^{∗} ⊕ U, then

σ(K̃, K_{e}, K_{g}) = η(g).

In particular, σ(K̃, K_{e}, K_{g}) depends only on g^{∗}V. Proof. In order to simplify the notations, we will assume that the eigenvalues of g lie on the unit circle; the proof in the general case is the same.

The element g defines a decomposition ⊕_{χ}(U_{χ}; b_{χ}, ρ_{χ}, V_{χ}) of g^{∗}V. Notice, that U_{χ}^{∗} = \{φ ∈ U^{∗}| \overline{φ}(U_{χ′}) = 0 if χ′ ≠ χ\}. In particular, the kernel K_{g} has a decomposition ⊕_{χ}(K_{g})_{χ} , where (K_{g})_{χ} = {(φ, u) ∈ U_{χ}^{∗} ⊕ U_{χ}| V_{χ}φ = u}. On the other hand, K_{e} = U^{∗} obviously has a decomposition K_{e} = ⊕_{χ}U_{χ}^{∗} = ⊕_{χ}(K_{e})_{χ}, too.

Since the kernel K ⊂ U is ρ(g)–invariant, one has K = ⊕_{χ}K_{χ}, where K_{χ} = K ∩ U_{χ}. Obviously K_{χ} ⊂ K_{χ}^{⊥}. If χ ≠ 1 then b_{χ} is non-degenerate and V_{χ} is an isomorphism. Therefore K_{χ} is a kernel for χ ≠ 1. Now, from a dimension computation we get this for χ = 1 too. In particular, K̃_{χ} = K_{χ}^{∗} ⊕ K_{χ} is a kernel in U_{χ}^{∗} ⊕ U_{χ} and

σ(K̃, K_{e}, K_{g}) = ∑_{χ} σ(K̃_{χ}, (K_{e})_{χ}, (K_{g})_{χ}).

Therefore, we can assume that U = U_{χ} for some χ. Consider χ ≠ 1. Then signb_{χ} = 0 because b_{χ} is non-degenerate and (U_{χ}; b_{χ}) is hyperbolic. On the other hand, Ψ_{χ} is the zero form (because V (g)^{−1}(K) = K^{∗}).

Assume that χ = 1. We want to compute σ(K̃, K_{e}, K_{g}) = σ(K_{g}, K̃, K_{e}). Consider the map Λ : K_{g} ∩ (K̃ + K_{e}) → V (g)^{−1}(K) given by

{(α, v)| V α = v, α = −φ − ψ, φ ∈ K^{∗}, v = −k, where k ∈ K} ↦ α.

Λ defines an isomorphism, so we can transfer the form Ψ on V (g)^{−1}(K). Notice that we can choose φ = 0. Therefore Ψ((α, v), (α′, v′)) = Φ((α, v), (φ′, k′)) = Φ((α, v), (0, −V (g)α′)) = −α(\overline{V (g)α′}) − b(V (g)α)(\overline{V (g)α′}). Since V (g)α and V (g)α′ are in K, and K ⊂ K^{⊥}, the latter term vanishes. In conclusion, Ψ on V (g)^{−1}(K) has the form Ψ(α, α′) = −α(\overline{V (g)α′}).

Consider the sesqui-linear form Ψ_{e} = −\frac{1}{2}(Ψ−ϵ\overline{Ψ}^{∗}) = −\frac{1}{2}(I +\overline{ρ(g)}^{−1})\overline{V (g)} on U^{∗}. Obviously, Ψ_{e} is (−ϵ)–hermitian, and it extends Ψ.

Fact. The Ψ_{e}–orthogonal of V (g)^{−1}(K) in U^{∗} is in V (g)^{−1}(K). Proof of Fact. Set α ∈ V (g)^{−1}(K)^{⊥} such that Ψ_{e}(α, α′) = 0 for any α′ ∈ V (g)^{−1}(K). Then α(I + \overline{ρ(g)}^{−1})(k̄) = 0 for any k ∈ K. But, I + ρ(g)^{−1} is an isomorphism and preserves K, thus α ∈ K^{∗}. The inclusion K^{∗} ⊂ V (g)^{−1}(K) ends the proof of Fact. Now, by a well-known result, signΨ_{e} = signΨ_{e}|V (g)^{−1}(K) = signΨ.

The verification ϵsignΨ_{e} = η(g) is easy, considering the two cases ϵ = ±1. (Recall that for ϵ = 1, signb = signb̄, and for ϵ = −1, signb = sign(ib).) □ 4.11. Example. Assume that b_{1} is non-degenerate. Then

η_{1}(g) = sign{[ρ_{1}(g) − ρ_{1}(g)^{−1}]b_{1}^{−1}} = ϵ · sign{b_{1}[ρ_{1}(g) − ρ_{1}(g)^{−1}]}.

The latter equality follows, for example, by a verification on the generators of HI_{ϵ}(Z); (for their description, see for example [9] (3.18)). Notice, that even in this non-degenerate case, η_{1}(g) does not vanish, in general, on hyperbolic spaces. In fact, it vanishes on odd-dimensional indecomposable structures, and it is nonzero on even–dimensional ones. 4.12. Denote the 2–dimensional G–cocycles (resp. coboundaries) by Z^{2}(G, Z) (resp. B^{2}(G, Z)). For arbitrary V ∈ HV_{ϵ}(G), define the following 2–cocycle:

θ(V; g, h) = σ(V; K_{e}, K_{g}, K_{h}) − η(g) − η(g^{−1}h) + η(h).

This defines a map Θ_{II} : HV_{ϵ}(G) → Z^{2}(G, Z). Notice that Θ_{II} factorized in H^{2}(G, Z) is exactly Θ_{I}. Now, Θ_{II} vanishes on hyperbolic structures (by 4.8. and 4.10.). The induced map, still denoted by Θ_{II}, is: W V_{ϵ}(G) → Z^{2}(G, Z).

Notice that there are examples with Θ_{II} ≠ 0 (even with Θ_{I} = 0, for example when G = {free group}). For the interested reader, we sketch out one example. Consider a flat bundle Γ_{ρ} with a non-degenerate flat hermitian form b over an orientable surface S with genus g ≥ 2. Assume that σ(S; Γ_{ρ}) ≠

0. Consider an embedded (open) disk D^{2} ⊂ S. Set G = π_{1}(S − D^{2}) and V = (U; b, ρ) the induced structure associated with G. Then, by the general index–theory, Θ_{II}(V) ≠ 0 (even if we replace η with any f(V) which depends only on g^{∗}V).

The vanishing of Θ_{I} is equivalent to imΘ_{II} ⊂ B^{2}(G, Z); i.e. σ(V; K_{e}, K_{g}, K_{h}) = f(V; g) + f(V; g^{−1}h) − f(V; h) for some f(V). In general, f(V; g) depends on V and g, and not only on g^{∗}V. The main question is: can f(V) be constructed “independently of V”, i.e. so that f(V; g) depends only on g^{∗}V? A candidate for such an f is η. The vanishing of Θ_{II} gives a positive answer to this question with f(V) = η. 4.13. Theorem. If G is a (finitely generated) abelian group then Θ_{II} vanishes. In particular,

σ(V; K_{e}, K_{g}, K_{h}) = η(g) + η(g^{−1}h) − η(h).

Proof. Since we can construct an epimorphism Z^{k} → G, it is enough to verify the theorem for G = Z^{k}. If χ = 1, then W V_{1}(Z^{k}) = Z_{2} is generated by T (3.5), and Θ_{II}T = 0.

Assume that χ ∈ Ĝ − {1}. Then, by (3.5), we have to verify the identity only for V = (C; b, χ, V), where b ≠ 0, b̄ = ϵb and V (g′) = (χ(g′) − 1)b^{−1} for any g′ ∈ G.

If χ(g) = 1, then V (g) = 0 and K_{g} = K_{e}, hence σ(K_{e}, K_{g}, K_{h}) = 0. On the other hand, η(g) = η_{1}(g) = 0 and η(g^{−1}h) = η(h). Similarly, if χ(h) = 1 or χ(g^{−1}h) = 1, both sides of the identity vanish. So, we can assume that 1 ≠ χ(g) = e^{−2πia} ≠ χ(h) = e^{−2πic} ≠ 1, where 0 < a < 1, 0 < c < 1. Then the form Ψ is

\frac{[1 − χ(g)][1 − χ(h)]}{[χ(h) − χ(g)]bϵ}.

Therefore, ϵ · signΨ = sign(sin(c − a)π) · sign[−i/b] = signb · sign(sin(c − a)π). Now, the verification is elementary. □

5 The signature of the Yomdin series

5.1. Let f_{1} : (C^{n+1}, 0) → (C, 0) (n > 1) be a non-isolated singularity with one–dimensional singular locus Σ. Consider a germ f_{2} : (C^{n+1}, 0) → (C, 0) so that the pair (f_{1}, f_{2}) forms an isolated complete intersection singularity. Let ϕ : (X , 0) → (D^{2}, 0) be a “good representative” of (f_{1}, f_{2}). The irreducible components of the discriminant locus Δ are Δ_{1}, …, Δ_{s}. Here Δ_{1} = ϕ(Σ).

Now, ϕ defines two sets of invariants. The first one is the associated (−1)^{n−1}–hermitian variation structure V(ϕ) = (U; b, ρ, V) as in (2.5.4). The second one is a set of η–invariants, as it will be defined in the sequel.

Any g ∈ G defines a structure g^{∗}V(ϕ) ∈ HV_{(−1)^{n−1}}(Z), in particular a number η(g) = η(g^{∗}V(ϕ)) as in (4.9). It is not hard to verify that η(g^{−1}) = −η(g) and η(h^{−1}gh) = η(g). This shows that η is well–defined on the conjugacy classes of G. In particular, η(S^{1}) is well–defined for any (oriented) embedding S^{1} ↪ D^{2} − Δ.

Let T be a closed tubular neighbourhood of Δ_{1} ∩ ∂D^{2} in S^{3} = ∂D^{2}. Let L and M be the standard topological longitude and meridian in ∂T. They can be represented by embeddings S^{1} → ∂T. Moreover, for any q ∈ Z, the homology class [L + qM] ∈ H_{1}(∂T, Z) can also be represented (essentially, in a unique way, up to isotopy) by an S^{1} ↪ ∂T. In particular, the numbers η(L), η(M), and η(L + qM) are well–defined. 5.2. Theorem. For q ∈ N sufficiently large:

σ(f_{1} + f_{2}^{q}) − σ(f_{1}) = q · η(M) + η(L) − η(L + qM).

Proof. Step 1. Let (F_{i}, ∂F_{i}), (i = 1, 2) be two copies of (the fiber of ϕ) (F, ∂F). Assume that the orientation of (F_{1}, ∂F_{1}) (resp. of (F_{2}, ∂F_{2})) is the orientation (resp. inverse orientation) of (F, ∂F). Glue F_{1} and F_{2} along ∂F_{1} = −∂F_{2}. The result is Z, an oriented 2(n − 1)–dimensional manifold. Consider the maps β_{1} : H_{n−1}(Z) → H_{n−1}(Z, F_{2}) induced by the injection (Z, ∅) ↦ (Z, F_{2}), and β_{2} : H_{n−1}(Z) → H_{n−1}(F_{1}) induced by the projection Z ↦ F_{1} (given by the identifications F_{i} ↦ F_{1}). Then (β_{1}, β_{2}) : H_{n}(Z) → U^{∗} ⊕ U is an isomorphism. This identifies a cycle φ ∈ U^{∗} = H_{n−1}(F, ∂F) with the cycle φ ∪_{∂φ} (−φ) in Z, where φ is considered in F_{1} and −φ in F_{2}, and the gluing is along ∂φ. A cycle u ∈ U = H_{n−1}(F) is identified with the corresponding cycle imbedded in F_{1}. Now, it is easy to verify that the (−1)^{n−1}–hermitian intersection form on Z, identified on U^{∗} ⊕ U, is exactly Φ. Step 2: Consider the closed unit disc D_{1} in C with its natural orientation and boundary S^{1}. Let S_{+}^{1} = {z ∈ S^{1}|Rez > 0} be a segment in S^{1} with its natural orientation.

Set g ∈ G and consider a representative of its geometric monodromy m_{g} : (F, ∂F) → (F, ∂F) so that m_{g}|∂F = 1_{∂F}. Construct E_{g} = (F, ∂F) × [0, 1]/_{(x,0)∼(m_{g}(x),1)}, and identify [0, 1]/_{∼} with S^{1}. Then E_{g} is a locally trivial fibration of pairs of spaces over S^{1} with fiber (F, ∂F) and characteristic map m_{g}. Since S_{+}^{1} is contractible, we can assume that this fibration has a product structure F × S_{+}^{1} over S_{+}^{1}. Since ∂E_{g} = ∂F × S^{1}, we can construct X_{g} = E_{g} ∪ (∂F × D_{1}) by gluing along K_{g} × S^{1}. Now we cut a window in X_{g}: consider Z_{g} = X_{g} − (F − ∂F) × S_{+}^{1}. This Z_{g} is a (2n − 1)–dimensional manifold with boundary Z; (its corners can be smoothed). Moreover, the kernel ker{H_{n−1}(Z) → H_{n−1}(Z_{g})} is exactly K_{g} ∈ U^{∗} ⊕ U. Step 3. We are in the following geometric situation. (See, for example [16, 7, 10].) There is an embedding j of the closed disc D_{2} = {|z| ≤ 2} in D^{2} (in fact j(D_{2}) = {(c, d) ∈ D_{2}| c + d^{q} = small constant}), so that:

(a) ∂(j(D_{2})) ∩ Δ = ∅, ∂(j(D_{1})) ∩ Δ = ∅, where D_{1} ⊂ D_{2} is the closed unit disc,

(b) ϕ^{−1}(j(D_{2})) (resp. ϕ^{−1}(j(D_{1}))) can be identified with the Milnor fiber of f + f_{2}^{q} (resp. of f),

(c) j(D_{2}) ∩ Δ_{1} is a subset of j(D_{2} − D_{1}), and it contains exactly q points; the small, positively oriented circles in j(D_{2}) around these intersection points correspond to the meridian M of Δ_{1}. Moreover, the oriented boundary ∂j(D_{1}) is the longitude L of Δ_{1}.

In the sequel, we identify D_{2} with j(D_{2}). Consider an embedded path γ : [0, 1] → D_{2} with endpoints γ(0) ≠ γ(1), γ(0), γ(1) ∈ ∂D_{2}, such that it separates the points D_{2} ∩ Δ_{1} from the point of D_{1}. D_{2} − imγ has two connected components, D′ and D′′_{q} with D_{1} ⊂ D′ and D′′_{q} ∩ Δ = D_{2} ∩ Δ_{1}. In particular ϕ^{−1}(D′) is diffeomorphic to the Milnor fiber of f. Applying Wall’s theorem [17] (for the skew-symmetric case, see [5]), we get

σ(f_{1} + f_{2}^{q}) = σ(f_{1}) + signϕ^{−1}(D′′_{q}) + σ(K_{e}, K_{qM}, K_{−L}).

Now, if D(j) is a small disc which contains an intersection point z_{j} ∈ D_{2}∩Δ_{1}, then signϕ^{−1}(D(j)) = 0, because for an arbitrary point z ∈ D(j) − {z_{j}}, the natural map H_{n−1}(ϕ^{−1}(z)) → H_{n−1}(ϕ^{−1}(D(j))) is onto. Cutting ϕ^{−1}(D′′_{q}) q − 1 times by paths as above, we obtain:

signϕ^{−1}(D′′_{q}) = ∑_{k=1}^{q−1} σ(K_{e}, K_{M}, K_{−(q−k)M}).

Since ML = LM, by (3.13), and by η(−g) = −η(g), we obtain the desired relation. □ 5.3. In [9] the equivariant version of 5.2 is proved by a different method. Moreover, also the quasi–periodicity of the expression {qη(M)−η(L+qM)}_{q} is proved. 5.4. Example. The suspension case: Let f : (C^{n}, 0) → (C, 0) be an isolated singularity. Consider f_{1} : (C^{n+1}, 0) → (C, 0) defined by f_{1}(z, z_{n+1}) = f(z). Set f_{2} = z_{n+1} and ϕ = (f_{1}, f_{2}) as above. The singular locus of the ICIS ϕ is Σ = {z_{1} = … = z_{n} = 0}, and the discriminant locus Δ contains only one irreducible component Δ = Δ_{1}, which is smooth. In particular, G = Z and L = 0, and the (generally very rich) structure provided by the variation map and the monodromy representation of ϕ collapses in the variation structure V(f) of f. Therefore, there is a huge qualitative and technical difference between the suspension case and our case of the Yomdin series.

Since σ(f_{1}) = 0, theorem 5.2, for suspensions, gives:

σ(f + z_{n+1}^{q}) = q · η(V(ϕ); M) − η(V(ϕ); qM) = q · η(V(f); 1) − η(V(f); q).

5.5. In the following, we compute the expression q · η(V; 1) − η(V; q) for some (−1)–hermitian variation structure of the group Z. In particular, we will reprove Brieskorn’s signature formula [2] for his singularities z_{1}^{a_{1}} +⋯+z_{n+1}^{a_{n+1}}, (n even).

We start with a one–dimensional structure V(α) ∈ HV_{−1}^{s}(Z), generated by V (1) = e^{πiα}, where α ∈ R. Then by 2.5.2, b = 2i sin πα, and for any n ∈ Z the representation is ρ(n) = e^{2πinα}, and

V (n) = \{ \begin{matrix}ne^{πiα} & if α ∈ Z \\ b^{−1}[ρ(n) − 1] = \frac{sin πnα}{sin πα}e^{πinα} & if α ∉ Z. \end{matrix}

The signature σ(V(α); K_{0}, K_{1}, K_{n+1}), (n ∈ Z), can be computed directly from the definition (see 4.4). If K_{n+1} = K_{0} then this signature is zero, otherwise it is equal to sign\frac{V (1)·V (n+1)}{V (n+1)−V (1)}. This gives for n ≥ 1:

σ(V(α); K_{0}, K_{1}, K_{n+1}) = \{ \begin{matrix}(−1)^{α} & if α ∈ Z \\ sign( sin πnα · sin π(n + 1)α) & if α ∉ Z. \end{matrix}

The right hand side of the above identity has a completely different presentation, too. Define the function h : R → {−1, 0, 1} by h(α) = sign sin πα and F_{n}(α) = ∑_{k=1}^{n−1} h(α + \frac{k}{n}) for n ≥ 2 and F_{1}(α) = 0. 5.6. Lemma. Let n ≥ 1. Then:

F_{n+1}(α) − F_{n}(α) = \{ \begin{matrix}(−1)^{α} & if α ∈ Z \\ sign( sin πnα · sin π(n + 1)α) & if α ∉ Z. \end{matrix}

Proof. The verification is elementary, and it is left to the reader. □ 5.7. Corollary. Fix q ≥ 1. Then:

q · η(V(α); 1) − η(V(α); q) = F_{q}(α).

Proof. q · η(V(α); 1) − η(V(α); q) = ∑_{n=1}^{q−1} σ(V(α); K_{0}, K_{1}, K_{n+1}) = ∑_{n=1}^{q−1}(F_{n+1}(α) − F_{n}(α)) = F_{q}(α). □ 5.8. Example. The signature of the Brieskorn’s singularity. Set f = z_{1}^{a_{1}} + · + z_{n}^{a_{n}} where n is even. The variation map V (f) is the invers of the Seifert form of f (see [3]). This latter one, by a result of Sakamoto [15], is (−1)^{(n−1)n/2}Γ_{a_{1}} ⊗ ⋯ ⊗ Γ_{a_{n}}, where Γ_{a} is an (a − 1)–dimensional form with (Γ_{a})_{ij} = 1 if i = j or i + 1 = j, and = 0 otherwise. Now, if we write Γ_{a} in the base given by the eigenvectors of the monodromy −Γ_{a}^{−1}Γ_{a}^{∗}, then Γ_{a} is diagonal. By a computation we obtain that Γ_{a}^{−1} = ⊕_{k=1}^{a−1}(1 − e^{2πik/a}). Therefore:

V (f) = V (1) = (−1)^{n(n−1)/2}(−2i)^{n} ⨁_{k_{1}=1}^{a_{1}−1} ⋯ ⨁_{k_{n}=1}^{a_{n}−1} e^{πi ∑_{j=1}^{n} k_{j}/a_{j}} · ∏_{j=1}^{n} sin \frac{πk_{j}}{a_{j}}

which is equivalent (as a sesqui-linear form) to ⊕_{k_{1}=1}^{a_{1}−1} ⋯⊕_{k_{n}=1}^{a_{n}−1}V(∑_{j=1}^{n} k_{j}/a_{j}). By 5.7, a_{n+1} ·η(V(f); 1)−η(V(f); a_{n+1}) = ∑_{k_{1}=1}^{a_{1}−1} ⋯ ∑_{k_{n}=1}^{a_{n}−1} F_{a_{n+1}}(∑_{j=1}^{n} k_{j}/a_{j}). Therefore, we obtain Brieskorn’s formula:

σ(z_{1}^{a_{1}} + ⋯ + z_{n+1}^{a_{n+1}}) = ∑_{k_{1}=1}^{a_{1}−1} ⋯ ∑_{k_{n+1}=1}^{a_{n+1}−1} sign sin(π ∑_{j=1}^{n+1} k_{j}/a_{j}).

References

[1] Atiyah, M. F., Patodi, V. K. and Singer, I. M.: Spectral asymmetry and Riemannian geometry. I.II.III. Math. Proc. Cambridge Philos. Soc. 77, (1975) 53-69; 78, (1975) 405-432; 79, (1976) 71-99.

[2] Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten, Invent. Math., 2, (1966) 1-14.

[3] Durfee, A. and Kaufman, L.: Periodicity of branched cyclic covers. Math. Ann. , 218, (1975) 157-174.

[4] Harer, J.: The Second Cohomology Group of the Mapping Class Group of an Orientable Surface. Invent. Math. 72, (1983) 221-239.

[5] Meyer, W.: Die Signatur von Localen Koeffizientensystemen und Faserbündeln. Bonner Math. Schriften, 53 (Bonn, 1972).

[6] Milnor, J.: On Isometries of Inner Product Spaces. Inventiones Math., 8, (1969) 83-97.

[7] Némethi, A.: The Milnor fiber and the zeta function of the singularities of type f = P (h, g), Compositio Math., 79, (1991) 63-97.

[8] Némethi A.: The zeta function of singularities. Journal of Algebraic Geometry, 2, (1993) 1-23.

[9] Némethi, A.: The equivariant signature of hypersurface singularities and eta–invariant, will appear in Topology.

[10] Némethi A.: The real Seifert form and the spectral pairs of isolated hypersurface singularities, will appear in Compositio Math..

[11] Némethi, A.: The semi–ring structure and the spectral pairs of sesqui– linear forms, Algebra Colloq., 1:1 (1994) 85-95.

[12] Neumann, W.: Cyclic suspension of knots and periodicity of signature for singularities. Bulletin AMS, 80, 5, (1974) 977-981.

[13] Neumann, W.: Signature related invariants of manifolds–I. Monodromy and γ–invariants. Topology, Vol. 18, 147-172.

[14] Saito, M.: The Steenbrink’s conjecture, Math. Ann., 289, (1991) 543-572.

[15] Sakamoto, K.: Milnor fiberings and their Characteristic Maps. Proc. Intern. Conf. on Manifolds and Related topics in Topology, (Tokyo, 1973).

[16] Siersma, D.: The monodromy of a series of hypersurface singularities, Comment. Math. Helvetici, 65, (1990) 181-197.

[17] Wall, C.T.C.: Non–Additivity of the Signature. Invent. Math. 7, (1969) 269-274.

[18] Yomdin, I.N.: Complex surfaces with a one dimensional set of singularities. Sibirsk. Mat. Z., 15(5), (1974) 1061-1081.

IMAR, Bucharest, Romania, and The Ohio State University, Columbus, OH 43210-1174, USA. e-mail: nemethi@math.ohio-state.edu