The equivariant signature of hypersurface singularities and eta–invariant

András Némethi^{∗}

October 13, 1997

1 Introduction

Let f be an analytic germ of a hypersurface singularity. The associated monodromy action preserves the intersection form of its Milnor fiber. In this paper we study the corresponding equivariant signatures. The guiding principle is the following. Write f as a composed singularity f = p ∘ ϕ, where ϕ (resp. p) is an isolated complete intersection (resp. curve) singularity; express the equivariant signature of f as the equivariant signature of p with coefficients in a (non-degenerate) hermitian flat bundle; identify this with an index of the signature bundle; and when the primary invariant (Chern–class) vanishes, express it in terms of the eta–invariant of the boundary (of the Milnor fiber of p) with coefficients in the corresponding signature bundle. In the realization of this program, we have two basic obstructions: The monodromy action of p (resp. f) is not compact (finite), and the monodromy representation of ϕ (i.e. the candidate for the flat bundle), in general, is degenerate. This second obstruction is solved by introducing the variation map of ϕ. The variation structure (i.e. the degenerate monodromy representation together with the variation map) substitutes perfectly the non-degenerate representations. It turns out that the equivariant signature can be expressed in terms of this variation structure and the geometry of the curve singularity p. It has a sum-decomposition, corresponding to the Jaco–Shalen–Johansson (or splice) decomposition of the link complement of p^{−1}(0); and each term is closely related to the Seifert geometry of the components. (The structure of the fundamental group of the Seifert components will remove the first obstruction too.) The primary invariant (here the first Chern–class) vanishes when the variation structure is abelian or the intersection form of ϕ is definite. The basic application for the first case are the topological series, the coverings exemplify the second case.

∗Partially supported by NSF Grant No. DMS-9203482, and by the Netherlands Organisation for the Advancement of Scientific Research N.W.O.

The equivariant signature has been computed only for a few families of isolated singularities: curve singularities [13], quasi-homogeneous germs [20], suspensions [14]. This paper gives the variable term of the composed topological series (in particular for the Yomdin’s series) (see 5.4 and 5.11), and reduces the general case (see 5.1) to a signature computation of a non– degenerate hermitian flat bundle over the r-punctured 2-dimensional sphere (for which there exists a clear algorithm [8]).

The paper is organized as follows. Section 2 contains the definition of the isometric and variation structures and our eta–invariants associated with them. In the next section we state our general results about the (equivariant) signature of hypersurface singularities. Their proofs can be found in section 4. The last section contains the applications, and some connections with the limit mixed Hodge structures. Acknowledgement. The author would like to express thanks to Professor J. Steenbrink for reading the manuscript and giving constructive suggestions.

2 The eta–invariant of the variation structures

2.1. If U is a finite dimensional vector space then U^{∗} is its dual Hom_{C}(U, C). There is a natural isomorphism θ : U → U^{∗∗} given by θ(u)(φ) = φ(u). We denote by \overline{·} the complex conjugation. If φ ∈ Hom_{C}(U, U′), then \overline{φ} ∈ Hom_{C}(U, U′) is defined by \overline{φ}(x) := \overline{φ(\overline{x})}, and the dual φ^{∗} : U^{′∗} → U^{∗} of φ by φ^{∗}(ψ) = ψ ∘ φ.

A C-linear endomorphism b : U → U^{∗} with \overline{b^{∗} ∘ θ} = ϵb (ϵ = ±1) is called ϵ-hermitian form on U. The automorphisms h : U → U with \overline{h}^{∗} ∘ b ∘ h = b form the orthogonal group Aut(U; b). Any h ∈ Aut(U; b) determines a spectral decomposition (U; b, h) = ⊕_{χ}(U_{χ}; b_{χ}, h_{χ}), where U_{χ} is the generalized χ-eigenspace of h. 2.2. Definition. The eta-invariant η_{b}(h) is defined by the sum ∑_{χ} η_{b}(h)_{χ}, where

η_{b}(h)_{χ} = \{ \begin{matrix}(1 − 2c)signb_{χ} & if χ = e^{2πic}, 0 < c < 1; \\ −ϵ · sign[b_{χ}(h_{χ} − h_{χ}^{−1})] & if χ = 1. \end{matrix}

By convention, the signatura of a (−1)-hermitian form b is signb := sign(ib). In all our cases χ lies on the unit circle S^{1}. For the sake of exactness, we can define η_{χ} = 0 provided that χ ∉ S^{1}.

Notice also the similarity with the function 2((·)) : R → R, used in number theory, defined by 2((c)) = 0 if c ∈ Z, and 2((c)) = 1 − 2{c} if c ∉ Z ({·} denotes the fractional part).

The above eta-invariant is a “good object” only for non–degenerate hermitian forms b (i.e. when b is an isomorphism). In that case η_{b} can be interpreted as the η–invariant of the signature operator of the circle twisted with the signature bundle of a hermitian flat bundle, in the sense of [2] (cf. 3.4. and 4.11). 2.3. Definition. An ϵ-hermitian isometric structure of the group G is a system I = (U; b, ρ) such that b is an ϵ-hermitian non–degenerate form, and ρ : G → Aut(U; b) is a group endomorphism. Given a system I and an element g ∈ G, we define the eta–invariant η_{I}(g) by η_{b}(ρ(g)). 2.4. 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–homeomorphism is a map V : G → Hom(U^{∗}, U) with V (gh) = ρ(g) ∘ V (h) + V (g). 2.5. Definition. An ϵ–hermitian variation structure of the group G is a system V = (U; b, ρ, V) such that b is an ϵ–hermitian (maybe degenerate) form, ρ is a representation of G in Aut(U; b), V is a twisted–homeomorphism with respect to the left action of G via ρ, and they satify the following compatibility conditions for any g ∈ G:

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

(ii) V (g) ∘ b = ρ(g) − I.

In the sequel we identify U^{∗∗} and U by the map θ.

Notice that if b is non–degenerate, then the variation structure V is completely determined by the underlying isometric structure I = (U; b, ρ) (use (ii)). As we will see, the category of variation structures substitutes perfectly the category of isometric structures in those cases when the geometric situation gives a degenerate hermitian form. 2.6. Definition. Let V be an ϵ–hermitian variation structure. Then for any g ∈ G, ρ(g) determines a spectral decomposition (U; b, ρ(g), V (g)) = ⊕_{χ}(U_{χ}; b_{χ}, ρ(g)_{χ}, V (g)_{χ}), where U_{χ} is the generalized χ–eigenspace of ρ(g).

The eta–invariant η_{V}(g) is defined by the sum ∑_{χ} η_{V}(g)_{χ}, where:

η_{V}(g)_{χ} = \{ \begin{matrix}(1 − 2c)signb_{χ} & if χ = e^{2πic}, 0 < c < 1; \\ −sign[(1 + ρ(g)_{χ}^{−1})V (g)_{χ}] & if χ = 1. \end{matrix}

(In the last section we will use the notation η(V; g) = η_{V}(g) too.) 2.7. Remark. If V is a variation structure with non–degenerate b, and I is the underlying isometric structure, then for any g one has: η_{V}(g) = η_{I}(g). To see this, use (2.5.ii) and the fact that for any non–degenerate b and h ∈ Aut(U; b) one has sign[b(h − h^{−1})] = ϵ · sign[(h − h^{−1})b^{−1}]. 2.8. Lemma. Let b be an ϵ–hermitian form, ρ : G → Aut(U; b) a representation and V : G → Hom_{C}(U^{∗}, U) a map. Define U^{e} := U^{∗} ⊕ U, the map

ρ^{e} : G → Aut(U^{e}) by ρ^{e}(g) = ( \begin{matrix}1 & 0 \\ V (g) & ρ(g) \end{matrix} ); and b^{e} = ( \begin{matrix}0 & ϵ \\ 1 & b \end{matrix} ).

Then V = (U; b, ρ, V) is an ϵ–hermitian variation structure if and only if I^{e} = (U^{e}; b^{e}, ρ^{e}) is an ϵ–hermitian isometric structure. Moreover, for any g ∈ G one has:

η_{V}(g) = η_{I^{e}}(g).

The proof is left to the reader. 2.9. In the computation of the equivariant signature we will need a more complicated invariant too.

Let I be an isomorphic structure of the group G, λ ∈ S^{1}, m ∈ N^{∗} and o, g ∈ G such that [ρ(o), ρ(g)] = 0. Let (U′; b′, ρ′(o), ρ′(g)) be the direct summand of the system (U; b, ρ(o), ρ(g)), where U′ is the generalized λ^{m}– eigenspace of ρ(o). Define A′ = ^{m} \sqrt{ρ′(o)/λ^{m}} by I + ∑_{n≥1} C_{1/m}^{n}((ρ′(o)/λ^{m}) − 1)^{n}, where C_{1/m}^{n} = \frac{1}{n!} · \frac{1}{m}(\frac{1}{m} − 1) ⋯ (\frac{1}{m} − n + 1).

For any real polynomial P one has \overline{P (ρ′(o)/λ^{m})}^{∗}·b′ = b′·P ((ρ′(o)/λ^{m})^{−1}), hence A′ ∈ Aut(U′; b′). In particular, for any s = Z one has a well–defined invariant

η_{I}(λ; o, m; g, s) := η_{b′}(ρ′(g) · (λA′)^{s}).

2.10. In the following we define the corresponding invariant for variation structures. Let V be a variation structure of the group G, λ ∈ S, m ∈ N^{∗} and o, g ∈ G such that og = go. Let H be the free abelian (multiplicative) group generated by o and g in G. Then V induces on the generalized λ^{m}– eigenspace U′ of ρ(o) an abelian variation structure V′ = (U′; b′, ρ′, V′) of the group H. Similarly as above, we define A′ = ^{m} \sqrt{ρ′(o)/λ^{m}}.

Now assume that λ^{m} = 1. Then ρ^{′e}(h) = ( \begin{matrix}I & 0 \\ V′(h) & ρ′(h) \end{matrix} ) for any h ∈ H,

and ^{m} \sqrt{ρ^{′e}(o)} = ( \begin{matrix}I & 0 \\ B′ & A′ \end{matrix} ), where B′ = ∑_{n≥1} C_{1/m}^{n} (ρ′(o) − 1)^{n−1}V′(o).

Let H̃ be the free abelian group generated by õ and g. We regard H as a subgroup of H̃ by the identification (õ)^{m} = o. Then ρ̃^{e}(õ) = ^{m} \sqrt{ρ^{′e}(o)} and ρ̃^{e}(g) = ρ^{′e}(g) define a representation ρ̃^{e} : H̃ → Aut(U^{′e}; b^{′e}). Via Lemma (2.8), ρ̃^{e} comes from a variation structure Ṽ = (U′; b′, ρ̃, Ṽ) of the group H̃, such that its restriction to H is exactly V′ and ρ̃(õ) = A′.

The above construction is compatible with the spectral decomposition of V′ with respect to ρ′(g). Let ⊕_{χ}(U′_{χ}; b′_{χ}, ρ′_{χ}, V′_{χ}) be this spectral decomposition of V′, ⊕_{χ}A′_{χ} and ⊕_{χ}Ṽ_{χ} the corresponding decompositions of A′ and Ṽ.

For any s ∈ Z define η_{V}(λ; o, m; g, s) = ∑_{χ} η_{V,χ}, where

η_{V,χ} = \{ \begin{matrix}(1 − 2c)signb′_{χ} & if χλ^{s} = e^{2πic}, 0 < c < 1; \\ −ϵ · signb′_{χ}[ρ′_{χ}(g)(λA′_{χ})^{s} − ρ′_{χ}(g)^{−1} (λA′_{χ})^{−s}] & if χλ^{s} = 1, but λ^{m} ≠ 1 or χ ≠ 1; \\ −sign[1 + ρ′_{1}(g)^{−1}(A′_{1})^{−s}]Ṽ_{1}(g õ^{s}) & if λ^{s} = λ^{m} = χ = 1. \end{matrix}

Consider also:

η(λ; m; s) = \{ \begin{matrix}2((sc)) & if λ = e^{2πic} and λ^{m} = 1 \\ 0 & if λ^{m} ≠ 1. \end{matrix}

2.11. Lemma. With the above notation, one has:

η_{V}(λ; o, m; g, s) = η_{I^{e}}(λ; o, m; g, s) + signb · η(λ; m; s).

The proof is left to the reader.

3 The main result

3.1. Let ϕ : (X , 0) → (S, 0) ⊂ (C^{2}, 0) be a “good representative” of an n–dimensional isolated complete intersection singularity (ICIS) with discriminant space (Δ, 0) ⊂ (C^{2}, 0). Let ∗ be a base point ∗ ∈ S − Δ, and G = π_{1}(S − Δ, ∗). The cohomological information about the Milnor fiber F is concentrated in the natural map H_{c}^{n}(F) → H^{n}(F) (we will work only with complex coefficients). Denote U := H_{c}^{n}(F). If we identify the group H^{n}(F) with the dual U^{∗} = Hom_{C}(U, C), then the above map can be identified with the hermitian intersection form b : U → U^{∗}; (i.e. b(v, w) = ∫ v ∧w̄). Therefore b satisfies \overline{b}^{∗} = (−1)^{n}b. The monodromy representation in the orthogonal group of b is denoted by ρ : G → Aut(U; b).

If b is degenerate, the isometric structure (U; b, ρ) is not sufficient to determine the signature of a total space of a fibering with fiber F and monodromy representation induced by ρ. The needed supplementary information lies in the variation map V : G → Hom_{C}(U^{∗}, U) (defined by a fixed trivialization of ϕ|_{∂X} : ∂X → (C^{2}, 0)). It is remarkable that the system V = (U; b, ρ, V) constitutes an (−1)^{n}–hermitian variation structure, in the sense of (2.5) (see, [1, p. 11 and ch. 2]). 3.2. Let p : (C^{2}, 0) → (C, 0) be an analytic germ so that p = p_{0}^{l}, where l is a positive integer and p_{0} is irreducible; (cf. 4.14).

Our goal is to compute the equivariant signatures σ_{λ}(f) of the hypersurface singularity f = p ∘ ϕ : (X , 0) → (C, 0).

σ_{λ}(f) is defined as follows. The monodromy defines a spectral decomposition (⊕_{λ}H_{n+1}(F)_{λ}, b_{λ}) of the pair (H_{n+1}(F), b), where F is the Milnor fiber of f, and b is the (hermitian) intersection form on H_{n+1}(F). Then σ_{λ}(f) := sign(i^{(n+1)^{2}}b_{λ}).

Consider the splice diagram Γ(p, Δ) of the multilink determined by p^{−1}(0)∪ Δ ⊂ (C^{2}, 0), where the multiplicity of p^{−1}(0) is l, and the other multiplicities are zero. Each node of the diagram represents a Seifert component Σ(a_{1}, …, a_{r}; S_{1}, …, S_{r}) with multilink (Σ; S_{1}, …, S_{r}; m_{1}, …, m_{r}). Notice that the multiplicities m_{i} can be determined from Γ(p, Δ) by the corresponding splicing conditions. (For the splice geometry, see [5].) Let M_{i} resp. L_{i} be the topological standard meridian resp. longitude of the link component S_{i}. In the description of the geometry of Σ, the following numbers are helpful:

q_{i} = a_{1}a_{2} ⋯ a_{r}/a_{i}, i = 1, …, r;

m = ∑_{i=1}^{r} m_{i}q_{i};

m′_{i} = − ∑_{j≠i} m_{j}q_{j}/a_{i}; n_{i} = gcd(m_{i}, m′_{i}) > 0, i = 1, …, r.

Consider a set of integers (β_{i})_{i} so that ∑_{i=1}^{r} β_{i}q_{i} = 1. In particular β_{i}q_{i} ≡ 1 (modulo a_{i}). (Recall that a_{i}, …, a_{r} are pairwise coprime integers.) Let δ_{i} be so that β_{i}q_{i} + δ_{i}a_{i} = 1, and s_{i} = m′_{i}β_{i} + m_{i}δ_{i} = (m_{i} − β_{i}m)/a_{i}.

Recall that the quotient Σ/S^{1} = B of Σ by its free S^{1}–action is an r– punctured 2–sphere.

3.3. Theorem. By the above notations, the equivariant signature σ_{λ}(f) is a sum over the nodes of Γ(p, Δ): σ_{λ}(f) = ∑ σ_{λ}(Σ, ρ), where:

(a) For a Seifert component Σ(a_{1}, …, a_{r}) (with m = m(Σ) as above) one has:

σ_{λ}(Σ, ρ) = σ_{λ}(Σ) + (−1)^{n}σ(B, ρ_{λ}), where

σ_{λ}(Σ) = signb·∑_{i=1}^{r} η(λ; m; s_{i}) (i.e. σ_{λ}(Σ) is σ(F)–times the contribution of the Seifert component Σ in the equivariant signature σ_{λ}(p) of the curve singularity p); and

σ(B, ρ_{λ}) is the signature of a flat bundle, provided with a non-degenerate hermitian form, over the r–punctured 2–sphere B = Σ/S^{1}. The hermitian flat bundle is well–determined by the system (U; b, ρ, V), the geometry of Σ, and the complex number λ (see 4.7-4.9). In particular, it can be computed by Meyer’s method [8] (see the end of 3.4.).

(b)Assume that the variation structure over Σ is abelian (i.e. ρ(gh) = ρ(hg) and V (gh) = V (hg) for any pair g, h ∈ π_{1}(Σ)). Then σ_{λ}(Σ, ρ) depends only on the restriction of the variation structure over the tubular neighbourhoods of the link components {S_{i}}_{i}. More precisely:

σ_{λ}(Σ, ρ) = ∑_{i=1}^{r} η_{V}(λ; L_{i}^{a_{i}}M_{i}^{q_{i}}, m; L_{i}^{β_{i}}M_{i}^{−δ_{i}}, s_{i}).

3.4. In general, the relation (3.3.b) is not true, i.e. σ_{λ}(Σ, ρ) can not be computed only by the “boundary information”. In order to explain this, we present in short the basic idea which guided us to the results of this paper.

Assume that the flat bundle Γ over (B, ∂B) (B as above) is given by a representation ρ_{B} : π_{1}(B) → O, where O = O(p, q) (resp. O = Sp(2p, R)). Then there exists a non-flat splitting Γ = Γ^{+} ⊕ Γ^{−} in vector bundles on which the form is respectively positive and negative definite, (resp. there exists a complex structure (Γ, J) on Γ). Let c_{1} be the first Chern class of the signature bundle sign(Γ) = Γ^{+} − Γ^{−}, (resp. sign(Γ) = (Γ, J)^{∗} − (Γ, J) ). (For a suitable construction, sign(Γ) is flat near the boundary, therefore c_{1} has compact support.) Then by [3] σ(B, ρ_{B}) = σ(B, sign(Γ)) (here the first term is defined similarly as the signature in (4.4), and the second term is the signature of B with coefficients in the signature bundle); and by [2]

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

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

If the representation is abelian, then sign(Γ) has a flat realization, in particular c_{1} = 0. In this case, the above formula is equivalent essentially to the formula of (3.3.b). In general, the obstruction for the validity of (3.3.b) lies in ∫ c_{1}.

On the other hand, for any non-degenerate representation ρ_{B}, the signature σ(B, ρ_{B}) can be computed essentially by the Wall’s formula [21] too. We identify (B, ∂B) with a two-dimensional disc (with boundary) excepting (r−1) open discs. This can be cutted in r−1 pieces A_{1}, …, A_{r−1}, so that the interior of any of them is diffeomorphic to the annulus {z ∈ C; 1 < |z| < 2}. Since the signature σ(A_{i}, ρ_{B}|A_{i}) = 0, by Wall’s formula σ(B, ρ_{B}) is a sum of Wall’s type correction terms (which can be explicitly computed by the representation). For the details, see [8]. (In fact, this ends the computational part of our results.) Unfortunately, this algorithm, even in simple particular cases, can be very complicated. This fact emphasizes the value of (3.3.b) 3.5. We can expect that the signature σ(f) = ∑_{λ} σ_{λ}(f) has a simpler expression. Indeed, one has:

3.6. Theorem. The signature σ(f) is a sum σ(f) = ∑_{nodes} σ(Σ, ρ) over the nodes of Γ(p, Δ), where:

(a) For a Seifert component Σ, the term σ(Σ, ρ) is the signature of a hermitian flat bundle over the Milnor fiber of Σ. Its representation is the natural restriction of ρ^{e} : G → Aut(U^{e}; b^{e}), which is provided by the variation structure V as in (2.8).)

(b) If the variation structure above Σ is abelian, then:

σ(Σ, ρ) = ∑_{1=1}^{r} n_{i} · η_{V}(L_{i}^{m_{i}/n_{i}}M_{i}^{m′_{i}/n_{i}}).

In the sum ∑ σ(Σ, ρ), the terms corresponding to a common edge determined by neighboring (abelian) nodes, cancel out.

3.7. Assume in (3.4) that the representation is symmetric and definite. Then the split Γ = Γ^{+} (or Γ = Γ^{−}) is flat, hence c_{1} = 0. In particular, the signature can again be recovered from the boundary informations. Our version is:

3.8. Addendum. Assume that the hermitian form b of the ICIS ϕ is non– degenerate, symmetric and definite. Then σ_{λ}(Σ, ρ) (resp. σ(Σ, ρ)), for any Seifert component Σ, can be computed by the formula 3.3.b (resp. 3.6.b). In particular, σ depends only on the monodromy representation over the boundary components of p^{−1}(0).

Moreover, in this case, the result is true for arbitrary p (i.e. without the restriction p = p_{0}^{l}).

The particular case of (branched) coverings will be discussed in (4.16). The next section contains the proof of the theorems and the addendum. In

section 5, we present some particular cases and corollaries. The basic example is the Yomdin’s series. For this, we prove a quasi–periodicity property too. In the case of variation structures given by hypersurface singularities, we relate the eta–invariants to the Hodge–theoretical invariants.

4 The proof of 3.3, 3.6 and 3.8.

4.1. It is convenient to replace the variation structure (U; b, ρ, V) of G with a non–degenerate representation. This can be realized geometrically as follows. Let F̃ be a 2n–dimensional oriented real manifold with boundary ∂F̃ = −∂F (∂F with inverse orientation). Using the fixed trivialization of ϕ|_{∂X}, we construct X^{e} = X ∪_{∂X} (F̃ × S) with the natural extension ϕ^{e} : X^{e} → (S, 0) such that ϕ^{e}|_{F̃×S} is the second projection. We prefer to close (F, ∂F) with another copy F̃ of F with inverse orientation. In this case, the extended fiber is F^{e} = F ∪_{∂F} (−F). Consider the maps:

β_{1} : H^{n}(F) → H^{n}(F^{e}), induced by the map F^{e} → F determined by the identifications F → F and F̃ → F; and

β_{2} : H^{n}(F^{e}, F̃) → H^{n}(F^{e}), induced by (F^{e}, ∅) → (F^{e}, F̃).

Then (β_{1}, β_{2}) : U^{∗} ⊕ U → H^{n}(F^{e}) is an isomorphism. By this isomorphism, the non–degenerate intersection form b^{e} (of F^{e}) on U^{e} = U^{∗} ⊕ U, and the monodromy representation ρ^{e} : G → Aut(U^{e}; b^{e}) of G, induced by ϕ^{e}, is given by:

b^{e} = ( \begin{matrix}0 & (−1)^{n} \\ 1 & b \end{matrix} ) , resp. ρ^{e}(g) = ( \begin{matrix}I & 0 \\ V (g) & ρ(g) \end{matrix} ) for any g ∈ G.

4.2. We choose an open tubular neighbourhood T of Δ − {0}, a base space S of ϕ, and δ > 0 so that

p : (p^{−1}(S_{δ}^{1}) ∩ S, p^{−1}(S_{δ}^{1}) ∩ S − T) → S_{δ}^{1} = {z ∈ C : |z| = δ}

is a topological locally trivial fibration (LTF) of pair of spaces. Denote: P = p^{−1}(S_{δ}^{1}) ∩ S, P^{•} = P − T, Y = ϕ^{−1}(P), and Y^{•} = ϕ^{−1}(P^{•}). Then: (P, P^{•}) → S_{δ}^{1} is a LTF with fiber (F_{p}, F_{p}^{•}); (Y, Y^{•}) is a LTF over (P, P^{•}) with fiber F; and (Y, Y^{•}) → S_{δ}^{1} is a LTF with fiber (E, E^{•}). Corresponding to the extension F^{e} = F ∪ (−F), we have the corresponding spaces (Y^{e}, Y^{•e}) and (E^{e}, E^{•e}). The corresponding geometric monodromies of the fibrations p, p ∘ ϕ, and p ∘ ϕ^{e} act on F_{p}, E, E^{e}, and E^{•e}. The intersection forms on the generalized eigenspaces have the equivariant signatures σ_{λ}(p), σ_{λ}(ϕ, p), σ_{λ}(ϕ^{e}, p) , resp. σ_{λ}(ϕ^{•e}, p). The signature of F is denoted by σ(F). 4.3. Lemma.

(1) σ_{λ}(ϕ, p) = σ_{λ}(ϕ^{e}, p) + σ_{λ}(p) · σ(F),

(2) σ_{λ}(ϕ^{e}, p) = σ_{λ}(ϕ^{•e}, p). Proof. (1) By Wall’s theorem [21, 8]: σ_{λ}(ϕ^{e}, p) = σ_{λ}(ϕ, p) + σ_{λ}(F̃ × F_{p}) + σ_{λ}(K_{1}, K_{2}, K_{3}), where K_{i} are kernels in W = H_{n}(Z), for Z = ∂F_{p} × ∂F. For simplicity we assume that l = 1. Consider the natural inclusions k_{i} : Z → Y_{i}, (i = 1, 2, 3), where Y_{1} = F_{p} × ∂F, Y_{2} = ∂F_{p} × F̃, and Y_{3} = ϕ^{−1}(∂F_{p}). Then K_{i} = ker k_{i∗}, where k_{i∗} : H_{n}(Z) → H_{n}(Y_{i}) is the induced map. It is easy to see that K_{1} = K_{2} if n ≥ 2, and K_{1} ∩ K_{2} + K_{1} ∩ K_{3} = K_{1} ∩ (K_{2} + K_{3}) if n = 1. Therefore, in both cases σ(K_{1}, K_{2}, K_{3}) = 0.

(2) follows by Novikov additivity and by the fact that σ(E^{e} − E^{•e}) = 0. (Actually, when n ≥ 2 then H_{n+1}(E^{e} − E^{•e}) = 0; if n = 1 then the intersection form is trivial.) 4.4. Consider the fibration p : P^{•} → S_{δ}^{1} with fiber F_{p}^{•}. The representation ρ^{e} defines a flat bundle on P^{•}, respectively on a fixed fiber F_{p}^{•}. The twisted cohomology group H^{1}(F_{p}^{•}, ∂F_{p}^{•}; ρ^{e}) carries a (−1)^{n+1}–hermitian form: H^{1}(F_{p}^{•}, ∂F_{p}^{•}; ρ^{e})⊗H^{1}(F_{p}^{•}, ∂F_{p}^{•}; ρ^{e}) → H^{2}(F_{p}^{•}, ∂F_{p}^{•}; ρ^{e}⊗ρ^{e}) → H^{1}(F_{p}^{•}, ∂F_{p}^{•}; C) = C, where the first map is the cup product and the second is the coefficient map induced by b^{e}. Since the representation is defined on the space P^{•}, the monodromy action of p induces an action m^{e} on H^{1}(F_{p}^{•}, ∂F_{p}^{•}; ρ^{e}). The corresponding equivariant signatures are denoted by σ_{λ}(ρ^{e}, p). By a result of Meyer [8]: σ_{λ}(ϕ^{•e}, p) = (−1)^{n}σ_{λ}(ρ^{e}, p). 4.5. The geometric monodromy of F_{p}^{•} can be obtained by pasting the monodromy maps on the Milnor fibers F_{Σ} of the Seifert components Σ along boundary circles. Therefore, by the Novikov additivity we have: σ_{λ}(ρ^{e}, p) = ∑_{nodes} σ_{λ}(F_{Σ}, ρ^{e}|F_{Σ}), where the last term is the equivariant signature of (H^{1}(F_{Σ}, ∂F_{Σ}; ρ^{e}|F_{Σ}); b_{Σ}^{e}, m_{Σ}^{e}) (defined similarly as σ_{λ}(ρ^{e}, p) in (4.4)) of the Milnor fiber of Σ with the induced representation (denoted by ρ^{e}|F_{Σ}).

Moreover, again by the Novikov additivity, we can replace in each Seifert component the boundary vertices into arrowhead vertices with zero multiplicity. 4.6. Let Σ(a_{1}, …, a_{r}) be a Seifert component with arrowhead vertices only and multiplicities (m_{1}, …, m_{r}). Define q_{i}, m′_{i}, β_{i}, δ_{i}, (i = 1, …, r) and m as in (3.2). A geometric realization of Σ is:

Σ = {z ∈ S^{2r−1} ⊂ C^{r} : ∑_{i=1}^{r} a_{ij}z^{a_{i}} = 0; j = 1, …, r − 2; (a_{ij})_{ij} generic}.

The S^{1}–action is given by t∗(z_{1}, …, z_{r}) = (t^{q_{1}}z_{1}, …, t^{q_{r}}z_{r}). q : Σ → Σ/S^{1} = B is the quotient map. φ : Σ → S^{1}, φ(z_{1}, …, z_{r}) = z_{1}^{m_{1}} ⋯ z_{r}^{m_{r}}/|z_{1}^{m_{1}} ⋯ z_{r}^{m_{r}}| defines the Milnor fibration with Milnor fiber F_{Σ}. The S^{1}–action induces a regular m–fold cyclic covering F_{Σ} → B. The monodromy of φ is the generator ˆ1 of the covering transformation group Z_{m}.

Set d = gcd(m_{1}, …, m_{r}). Then F_{Σ} has exactly d components, which are cyclically permuted by the monodromy. If we replace m_{i} by m_{i}/d, then a_{i}, q_{i}, β_{i}, δ_{i} (i = 1, …, r) remain the same, but m resp. s_{i} will change in m/d resp s_{i}/d. Analysing the formula (3.3.b) with respect to this modification (use 2.11 and 2.9), we deduce that we can assume d = 1.

Set G_{Σ} = π_{1}(Σ, ∗), H = π_{1}(F_{Σ}, ∗). Then a part of the geometry of Σ is summarized in the diagram with exact rows and columns:

00  H H=  0Z G_{Σ} π_{1}(B) 0  0Z Z Z_{m} 0  00 ⟶  ⟶  ⟶  ⟶ c_{∗} ⟶  ⟶  ⟶  ⟶ φ_{∗} ⟶ ↑  ↑ ↑ ↑ ↑  ↑ ↑ ↑ ↑ k_{∗} q_{∗}  1 ↦ m 1 ↦ ˆ−1

The m–covering induces the first row, the Milnor fibration the second row, the S^{1}–action the first column; o = k_{∗}(1) corresponds to a generic S^{1}–orbit. Moreover, since S^{1} → Σ → B is topologically trivial, the exact sequence 0 → Z → G_{Σ} → π_{1}(B) → 0 splits in G_{Σ} = Z × π_{1}(B). In particular, o is in the center of G_{Σ}.

We will need an explicit description of the maps. A basic reference is [5]. Let T_{i} be a tubular neighbourhood of the special fiber S_{i}, and choose the topological longitude L_{i} and meridian M_{i} in T_{i}. The quotient T_{i}/S^{1} can be identified with one of the boundary components ∂_{i}B of B. Then φ_{∗}(L_{i}^{x}M_{i}^{y}) = −xm′_{i} + ym_{i}, and q_{∗}(L_{i}^{x}M_{i}^{y}) = xq_{i} − ya_{i} ∈ π_{1}(∂_{i}B) = Z. On the other hand, c_{∗} is well-defined on the conjugacy classes [∂_{i}B] of G, and it is determined by these values: c_{∗}(∂_{i}B) = ŝ_{i}.

The S^{1}–orbit o in T_{i} is represented by L_{i}^{a_{i}}M_{i}^{q_{i}}.

The element L_{i}^{β_{i}}M_{i}^{−δ_{i}} is a lift of [∂_{i}B] , i.e. q_{∗}(L_{i}^{β_{i}}M_{i}^{−δ_{i}}) = 1_{Z} ∈ π_{1}(∂_{i}B). Notice that 1_{Z} projects in ˆ−1, where ˆ1 represents the monodromy transformation. Therefore φ_{∗}(L_{i}^{β_{i}}M_{i}^{−δ_{i}}) = −s_{i}. Since ∑_{i} s_{i} = (1− ∑_{i} β_{i}q_{i})m/ ∏_{j} a_{j} = 0, the map s_{∗} : π_{1}(B) → Z given by s_{∗}([∂_{i}B]) = −s_{i} is well-defined.

The geometric meaning of s_{∗} is the following. A trivialisation of S^{1} → Σ → B is a map r : B → Σ such that q ∘ r = 1_{B}. The relation r(∂_{i}B) = L_{i}^{β_{i}}M_{i}^{−δ_{i}} gives the correspondence between trivializations and sets of integers (β_{i})_{i} with ∑_{i} β_{i}q_{i} = 1. If r is given, then s_{∗} = φ_{∗} ∘ r_{∗} is induced by s = φ ∘ r. The map r_{∗} : π_{1}(B) → G_{Σ}, r_{∗}(ĝ) = g is characterized by: g is the unique element so that q_{∗}(g) = ĝ and φ(g) = s_{∗}(ĝ). 4.7. The representation ρ^{e}|Σ : G_{Σ} → Aut(U^{e}; b^{e}) determines a flat bundle. The induced flat bundle V on F_{Σ} is given by the induced representation ρ^{e}|F_{Σ}. The quotient map determines its direct image flat bundle R^{0}(q|F_{Σ})_{∗}V. The monodromy representation ρ_{B}^{e} : π_{1}(B) → Aut((U^{e}; b^{e})^{⊕m}) of this bundle is the following. Set ĝ ∈ π_{1}(B). Let g = r_{∗}(ĝ) be its lift as above. Denote n(ĝ) = −s_{∗}(ĝ) − m · [−s_{∗}(ĝ)/m], and ḡ = go^{[−s_{∗}(ĝ)/m]}; (where [·] is the integer part). Then: ρ_{B}^{e}(ĝ)(u_{0}, …, u_{m−1}) =

(ρ^{e}(oḡ)u_{m−n(ĝ)}, …, ρ^{e}(oḡ)u_{m−1}, ρ^{e}(ḡ)u_{0}, …, ρ^{e}(ḡ)u_{m−n(ĝ)−1}).

The monodromy m_{B}^{e}, (which commutes with im(ρ_{B}^{e})), is given by:

m_{B}^{e}(u_{0}, …, u_{m−1}) = (ρ^{e}(o)u_{m−1}, u_{0}, …, u_{m−2}).

4.8. Using again Meyer’s result now for ϕ^{e} and q ∘ ϕ^{e}, we obtain the isomorphism of the following structures:

(H^{1}(F_{Σ}, ∂F_{Σ}; ρ^{e}|F_{Σ}); b_{Σ}^{e}, m_{Σ}^{e}) ≃ (H^{1}(B, ∂B; ρ_{B}^{e}); b_{B}^{e}, (m_{B}^{e})_{∗}).

The monodromy (m_{B}^{e})_{∗} is induced by a bundle morphism m_{B}^{e} which is the identity on the base B, and commutes with ρ_{B}^{e}. Therefore, the representation splits in the orthogonal sum ⊕_{λ}(ρ_{B}^{e})_{λ} with

(ρ_{B}^{e})_{λ} : π_{1}(B) → (U^{e}; b^{e})_{λ}^{⊕m} := {the generalized λ–eigenspace of m_{B}^{e}}.

Hence: 4.9 Lemma. If λ is an eigenvalue of m_{B}^{e}, then σ_{λ}(F_{Σ}, ρ^{e}|F_{Σ}) = σ(B, (ρ_{B}^{e})_{λ}). For other values of λ, both terms are zero.

Since σ_{λ}(p) = ∑_{nodes} σ_{λ}(Σ) [13], we finished the proof of (3.3.a) (with the notation ρ_{λ} = (ρ_{B}^{e})_{λ}). 4.10. Consider the representation (ρ_{B}^{e})_{λ}. Let I_{λ}^{e} denotes the corresponding isometric structure on B. Notice that if ρ^{e}|Σ is abelian, then (ρ_{B}^{e})_{λ} is abelian too. 4.11. Lemma. [2, 15] Assume that the representation ρ^{e}|Σ is abelian. Then:

(−1)^{n}σ(B, (ρ_{B}^{e})_{λ}) = ∑_{i=1}^{r} η_{I_{λ}^{e}}(∂_{i}B).

Therefore: σ_{λ}(Σ, ρ) = σ_{λ}(Σ) + ∑_{i=1}^{r} η_{I_{λ}^{e}}(∂_{i}B). 4.12. Let U_{λ^{m}}^{e} be the generalized λ^{m}–eigenspace of ρ^{e}(o). Then (U^{e})_{λ}^{⊕m} = \{(x, (λA)^{−1}x, …, (λA)^{−m+1}x) : x ∈ U_{λ^{m}}^{e}\}, where A = ^{m} \sqrt{ρ^{e}(o)/λ^{m}}. By this identification of (U^{e})_{λ}^{⊕m} with U_{λ^{m}}^{e}, the form (b^{e})_{λ}^{⊕m} corresponds to m · b_{λ^{m}}^{e}. Moreover, if g = r_{∗}(ĝ) ∈ G_{Σ} is a lift of ĝ so that φ_{∗}(g) = s_{∗}(ĝ), then (ρ_{B}^{e})_{λ}(ĝ) becomes (λA)^{−s_{∗}(ĝ)} · ρ^{e}(g). Now, o is L_{i}^{a_{i}}M_{i}^{q_{i}}, and [∂_{i}B] is lifted in l_{i} = L_{i}^{β_{i}}M_{i}^{−k_{i}} and the proof of the theorem can be finished by a straightforward verification (cf. 2.11 and 2.9).

This ends the proof of 3.3. 4.13. The proof of the theorem 3.6 is similar to the one of 3.3. In this non-equivariant case, σ(Σ, ρ) is the signature of ρ^{e} over F_{p}^{•}. Since F_{p}^{•} ∩ T_{i} consists of n_{i} copies of the (m_{i}/n_{i}, m′_{i}/n_{i})–cable of S_{i}, the (b) part follows. 4.14. Remark. If p is an arbitrary germ (i.e. it does not satisfy the restriction p = p_{0}^{l}), then only (4.3.1) fails. More precisely, in (4.3.1) appears a Wall–type correction term which depends on the global variation structure (i.e. it can not be localized over the boundary components). 4.15. The proof of 3.8. In this case it is not necessary to replace ρ by ρ^{e} because ρ is non–degenerate. In particular, we do not need (4.3), therefore we can work with arbitrary p, (cf. 4.14). Since ρ is definite, so is (ρ_{B})_{λ}. Now apply [2] or [15]. 4.16. Coverings. Consider a normal surface singularity (X, x) and a covering ϕ : (X, x) → (C^{2}, 0), which is branched over Δ. Set X^{∗} = ϕ^{−1}(S − Δ). The exact sequence associated with the non-ramified covering is: 1 → π_{1}(X^{∗}) → π_{1}(S − Δ) →^{τ} C → 1. Let |C| be the cardinality of C, U = C^{|C|} and b the natural hermitian form b(∑ a_{i}e_{i}, ∑ b_{j}e_{j}) = ∑ a_{i}b̄_{i}. Let ρ_{C} : C → Aut(U; b) be the regular representation of C. Then the representation ρ : π_{1}(S − Δ) = G → Aut(U; b) is in fact the composition ρ_{C} ∘ τ. Therefore, we can apply (3.8). By the above discussion, τ and the splice geometry of Γ(p, Δ) determine the equivariant signatures.

5 Applications

5.1. The interested reader can apply the main theorem in the geometric cases which appear in [11].

For example, the (equivariant) signature of a hypersurface germ with one or zero–dimensional singular locus can be determined as follows. Let f be as above. Take f_{2} so that the pair ϕ = (f, f_{2}) forms an ICIS (take, for example, a generic linear function). Set p(c, d) = c, then f = p ∘ ϕ. By our results, σ_{λ}(f) can be computed by the variation structure of ϕ (=polar variation structure of f), and by the splice–geometry of the polar curve. 5.2. The basic object of this section is the variable term of the topological series. Let f = p ∘ ϕ be a composed singularity such that ϕ is an ICIS and p = p_{0}^{l}, where p_{0} is an irreducible curve singularity. Let f′ = p′ ∘ ϕ (where p′ = (p′_{0})^{l′} and p′_{0} is irreducible) be an element of the topological series of composed singularities belonging to f [11]. By the very definition, the splice diagram Γ(p′, Δ) arrises as splice of the diagram Γ(p, Δ) and of another diagram Γ. By the first part of the theorem, the variable term σ_{λ}(f′) − σ_{λ}(f) is a sum ∑ σ_{λ}(Σ, ρ) over the nodes of Γ. The representations and variations over these Seifert components are induced by the abelian structure over the torus along which the splicing is done. In particular, the variable term is given by the formula from the second part of the theorem. In the case of the Yomdin’s series we give the explicit presentation. 5.3. Assume that f_{1} is a germ with one–dimensional singular locus and the pair (f_{1}, f_{2}) is an ICIS. Let p(c, d) = c and p′(c, d) = c + d^{a}. The number a is so large that Γ(p′, Δ), as a splice of Γ(p, Δ) and the Seifert diagram Γ = Γ(Σ; a, 1, 1), satisfies the semi–algebraicity condition [11]. Then the correction term σ_{λ}(f_{1} + f_{2}^{a}) − σ_{λ}(f_{1}) is given by this Seifert diagram Γ(Σ; a, 1, 1) with multilink structure (0, 1, 0), and the variation structure of ϕ above the splicing–torus.

Let L resp. M be the longitude resp. the meridian of Δ_{1} = {c = 0} ⊂ Δ. Then the pair L, M generates Z^{2} = π_{1}(splicing–torus). By the splice– geometry, we can make the identifications M_{1} = L, M_{2} = 1_{Z^{2}}, M_{3} = M. Since [M_{i}]_{i} generates H_{1}(Σ), using linking number argument, we get: L_{1} = M, L_{2} = LM^{a}, L_{3} = L.

We list the needed invariants:

m_{i} a_{i} q_{i} m′_{i} n_{i} β_{i} δ_{i} s_{i} L_{i}^{β_{i}}M_{i}^{−δ_{i}} L_{i}^{m_{i}/n_{i}}M_{i}^{m′_{i}/n_{i}}
i = 1 0 a 1 −1 1 1 0 −1  M  L^{−1}
i = 2 1 1 a 0 1 0 1 1  1_{Z^{2}}  LM^{a}
i = 3 0 1 a −a a 0 1 0  M^{−1}  M^{−1}

and m = a, and o = LM^{a}. 5.4. Corollary. The Yomdin’s series case.

(1) Let V = (U; b, ρ, V) be the variation structure of ϕ = (f_{1}, f_{2}). Let L and M be the longitude and the meridian of {c = 0}. Then:

σ_{λ}(f_{1} + f_{2}^{a}) − σ_{λ}(f_{1}) =

η_{V}(λ; LM^{a}, a; M, −1) + η_{V}(λ; LM^{a}, a; 1_{Z^{2}}, 1) + η_{V}(λ; LM^{a}, a; M^{−1}, 0).

(2) With the same notations:

σ(f_{1} + f_{2}^{a}) − σ(f_{1}) = η_{V}(LM^{a}) − a · η_{V}(M) − η_{V}(L).

5.5. Above we used η_{V}(g^{−1}) = −η_{V}(g). In our expressions we preserved the multiplicative notation of Z^{2} = π_{1}(torus); by this 1_{Z^{2}} denotes its neutral element. 5.6. Example. Let f_{1}(z_{1}, z_{2}, z_{3}) = z_{1}^{2} + z_{2}^{2}; f_{2}(z_{1}, z_{2}, z_{3}) = z_{3}. Then (U; b) = (C; 0), ρ(M) = 1_{C} and V (M) = −1, and the variation structure induced on the subgroup generated by L is trivial (i.e. ρ(L) = 1_{C} and V (L) = 0). Then V (M^{a}) = −a. Since σ_{λ}(f_{1}) = 0 for any λ, one has: σ(z_{1}^{2} + z_{2}^{2} + z_{3}^{a}) = η(M^{a}) − a · η(M) = 1 − a; and σ_{λ}(z_{1}^{2} + z_{2}^{2} + z_{3}^{a}) = −1 for any λ with λ^{a} = 1 but λ ≠ 1.

This example emphasizes the importance of the variation map: the intersection form is zero, the monodromy representation is trivial, and the whole contribution comes from the variation map. 5.7. The invariants in the surface case. If f defines an isolated singularity, we denote by μ(λ) the dimension of the generalized λ–eigenspace U_{λ}(f) of the monodromy (acting on H_{c}^{n}(Milnor fiber)); μ_{±}(λ) denotes the maximal dimension of a positive resp. negative definite subspace of (U_{λ}(f); b_{λ}(f)); μ_{0}(λ) is the dimension of ker b_{λ}(f). Set μ_{0} = ∑_{λ} μ_{0}(λ), μ_{±} = ∑_{λ} μ_{±}(λ), μ = μ_{0} + μ_{+} + μ_{−}.

Obviously, μ_{0}(λ) = 0 if λ ≠ 1; in particular μ_{0} = μ_{0}(1).

Our theorems determine σ(f_{1} + f_{2}^{a}) and σ_{1}(f_{1} + f_{2}^{a}) in terms of the invariants of f_{1} and V. On the other hand, the invariants μ and μ(1) can be computed by the zeta–function formula (see, [11, 18]) as follows:

μ(f_{1} + f_{2}^{a}) = −1 + e(f_{1}) + a · ∑_{j} d_{j}μ_{j}^{0},

μ(1)(f_{1} + f_{2}^{a}) = −1 + e_{1}(f_{1}) + dim ker((LM^{a} − I)^{n+1}), where e(f_{1}) is the Euler–characteristic of the Milnor fiber of f_{1}, e_{1}(f_{1}) is the order of t − 1 in the zeta function of f_{1}, and {d_{j}, μ_{j}^{0}}_{j} are the usual local invariants of Sing f_{1}^{−1}(0) (see 5.8.).

Now assume that f_{1}, f_{2} : (C^{3}, 0) → (C, 0). Then μ_{−}(1)(f_{1} + f_{2}^{a}) = 0 (see, for example, [12]). Therefore, the invariants μ_{0}, μ_{+} and μ_{−} can be computed from σ, σ_{1}, μ and μ(1). In particular, we obtain an expression of the geometric genus p(f_{1} + f_{2}^{a}) by Durfee’s formula: 2p = μ_{+} + μ_{0} [4]. Moreover, either by Laufer’s formula [7]: μ = K^{2} + s − μ_{0} + 12p, or by the signature formula of Durfee [4], we obtain the behaviour of the resolution invariant (K^{2} + s)(f_{1} + f_{2}^{a}) as well.(Here, for a given resolution, K denotes its canonical class and s is the number of irreducible exceptional divisors.) 5.8. Localization. Since the geometric obstruction (for the triviality of the family f_{1} + f_{2}^{a}) is concentrated in the singular locus S of f_{1}^{−1}(0), it is natural to express the correction term in local data provided by the transversal singularities. In this subsection we present this localization procedure in the case of the signature.

We recall the local invariants. Let S_{1}, …, S_{t} be the irreducible components of S. The topological degree of the restriction ϕ|S_{i} : S_{i} → Δ_{1} = {c = 0} is d_{i}. The singular fiber F_{d} = ϕ^{−1}(0, d), (0, d) ∈ S ∩ Δ_{1} has exactly ∑_{i=1}^{t} d_{i} isolated singularities. If z_{i} ∈ S_{i} − {0}, the local Milnor fiber of the ICIS (ϕ^{−1}(ϕ(z_{i})), z_{i}) is denoted by (F_{i}, ∂F_{i}). Set U_{i} = H_{c}^{n}(F_{i}) and μ_{i}^{0} = dim U_{i}. The disjoint union F_{loc} = ∪_{i=1}^{t} ∪_{j=1}^{d_{i}} F_{i} of the Milnor fibers of the singularities on F_{d} is invariant under the geometric monodromies m(L^{x}M^{y}), (x, y ∈ Z). The corresponding algebraic monodromies on the homology group U_{loc} = ⊕_{i=1}^{t} ⊕_{j=1}^{d_{i}} U_{i} have the form:

ρ_{loc}(M) = ⊕_{i=1}^{t}ρ_{loc,i}(M), ρ_{loc}(L) = ⊕_{i=1}^{t}ρ_{loc,i}(L);

ρ_{loc,i}(M) = ( \begin{matrix}M_{i} & & & & 0 \\ & . & . & . \\ 0 & & & & M_{i} \end{matrix} ) , ρ_{loc,i}(L) = ( \begin{matrix}0 & & & & & L_{i} \\ I \\ & . & . & . \\ & & & & I & 0 \end{matrix} ) ,

where M_{i} resp. L_{i} are the horizontal resp. vertical monodromies. The horizontal monodromy M_{i} : U_{i} → U_{i} is, in fact, the monodromy of the transversal singularity at z_{i} ∈ S_{i} − {0}. This latter one is an isolated singularity with (−1)^{n}–hermitian variation structure (U_{i}; b_{i}, M_{i}, V_{i}). This means that M_{i} ∈ Aut(U_{i}; b_{i}) and V_{i} ∈ Hom(U_{i}^{∗}, U_{i}) satisfies \overline{V} _{i}^{∗} = (−1)^{n+1}M_{i}^{−1}V_{i} and V_{i}b_{i} = M_{i} − I. This can be considered a variation structure of the group Z given by ρ_{i}(1) = M_{i} and V_{i}(1) = V_{i}. Their direct sum (over i = 1, …, t; j = 1, …, d_{i}) is denoted by V_{loc} = (U_{loc}; b_{loc}, ρ_{loc}(M), V_{loc}).

We can construct a representative of the (global) geometric monodromy m(M), acting on F, which is trivial in a complement of a tubular neighbourhood of S − {0}. If i_{∗} : U_{loc} → U resp. i^{∗} : U^{∗} → U_{loc}^{∗} are induced by the natural inclusion i : F_{loc} → F, then V (M) = i_{∗} ∘ V_{loc}(M) ∘ i^{∗}. The geometric monodromy of the longitude L induces a nontrivial action on the boundary D = ∪_{i=1}^{t} ∪_{j=1}^{d_{i}} ∂F_{i}. This obstructs the splitting of the variation operator V (L).

In the sequel, we replace V by its restriction to the group Z^{2}, generated by M and L.

Consider the spectral decomposition ⊕_{χ(M)}(U_{χ}; b_{χ}, ρ_{χ}, V_{χ}) = ⊕_{χ}V_{χ} of V given by the automorphism ρ(M); and similarly the decompositions V_{loc} = ⊕_{χ}V_{loc,χ} and ρ_{loc}(L) = ⊕_{χ}ρ_{loc,χ}(L) given by ρ_{loc}(M).

Assume that χ ≠ 1. Then (U_{χ}; b_{χ}, ρ_{χ}(M), V_{χ}(M)) = V_{loc,χ} and ρ_{χ}(L) = ρ_{loc}(L)_{χ}. Since b_{χ} is nondegenerate, V (L)_{χ} = (ρ_{loc}(L) − 1)_{χ}b_{loc,χ}^{−1}. In particular, the structure V_{loc,χ} can be extended to a structure of the group Z^{2} by V_{loc}(L)_{χ} = V (L)_{χ}. So, the global variation term C_{χ}(a) = η(LM^{a})_{χ} − a · η(M)_{χ} − η(L)_{χ} is equal to the local variation term C_{loc,χ}(a) (defined by the similar formula).

Assume that χ = 1. If the local transversal singularities are nondegenerate, then the variation term associated with χ = 1 vanishes. In the general case, notice that the middle term of C_{1}(a) is local: η(M)_{1} = η(V_{loc,1}; M). Consider the spectral decomposition V_{1} = ⊕_{λ}V_{1,λ} given by ρ(L); (we do not have a local equivalent of this). Then, if λ ≠ 1, then η(V_{1,λ}; LM^{a}) − η(V_{1,λ}; L) = 0. So the real obstruction in the localization procedure is the term η(V_{1,1}; LM^{a}) − η(V_{1,1}; L). 5.9. Lemma. Let V = (U; b, ρ, V) be a variation structure of the group G, and consider g, h ∈ G so that ρ(g) − I and ρ(h) − I are nilpotent. Then there exists a_{0} ≥ 0 such that η_{V}(hg^{a}) is a constant, provided that a > a_{0}. 5.10. Remark. Let V be as above and g ∈ G so that ρ(g) − I is nilpotent and V (g) is an isomorphism. Then for any a > 0, η(g^{a}) = η(g). But, in lemma 5.9, in general, a_{0} can not be = 0. For example, assume that V (g) is an isomorphism and η(g) ≠ 0. Then for any a_{0} > 0 and h = g^{−a_{0}}, by the above remark: η(hg^{a}) = [sign(a − a_{0})] · η(g).

Proof of Lemma 5.9. Let P (z) be the polynomial

P (z) = z + \frac{z(z − 1)}{2!}N + \frac{z(z − 1)(z − 2)}{3!}N^{2} + ⋯,

where ρ(g) = I+N. Now, V (hg^{a}) = V (h)+ρ(h)V (g^{a}) = V (h)+ρ(h)P (a)V (g). Define V (h, g; z) = V (h) + ρ(h)P (z)V (g) and SV (h, g; z) = V (h, g; z) + (−1)^{n+1}\overline{V (h, g; z)}^{∗}. Then, for any m ∈ Z, the set \{z| signSV (h, g; z) = m\} is an algebraic constructible, in particular, there exists a_{0} such that η(hg^{a}) = signSV (h, g; a) is constant, provided that a > a_{0}. 5.11. Corollary. There exists a_{0} so that if a > a_{0} one has:

σ(f_{1} + f_{2}^{a}) − σ(f_{1}) = ∑_{χ(M)_{loc}≠1}[η(V_{loc,χ}; LM^{a}) − a · η(V_{loc,χ}; M) − η(V_{loc,χ}; L)]
−a · η(V_{loc,1}; M) + C(V_{1,1}; L, M),

where C(V_{1,1}; L, M) = lim_{a→∞} η(V_{1,1}; LM^{a}) − η(V_{1,1}; L) is a constant independent of a.

In particular, σ(f_{1} + f_{2}^{a}) − σ(f_{1} + f_{2}^{a_{0}}) can be expressed in local terms. If the local transversal singularities are nondegenerate, then the last two

terms in the above formula vanish. 5.12. Example. The constant C_{1,1} = C(V_{1,1}; L, M), in general, is not zero. Let f_{1}(x, y, z) = x^{3} + y^{3} + λxyz (λ ≠ 0) and f_{2}(x, y, z) = z. Then σ(f_{1}) = 0 [6], and σ(f_{1} + f_{2}^{a}) = −3a + 3. The transversal type is 3A_{1}, in particular ρ_{loc}(M) is the identity. Therefore, in the right hand side of (5.11), the sum is zero, η(V_{loc,1}; M) = 3 and the constant C_{1,1} = 3. 5.13. The (quasi–)periodicity. By the monodromy theorem (i.e. the eigenvalues are roots of unity), and by Lemma 5.9, we obtain: 5.14. Corollary. σ(f_{1} + f_{2}^{a}) is a sum of a linear function and some periodic functions. The periods are given by the eigenvalues of ρ_{loc}(M).

This corollary generalizes the corresponding result for the suspension case, conjectured by Brieskorn, Durfee, and Zagier, and proved by Neumann [14].

The behaviour of the equivariant signature is even more regular. Fix λ. Consider an integer c = c(λ, M) > 0 so that λ^{c} = 1 and the semisimple part of ρ(M) satisfies ρ(M)_{ss}^{c} = 1_{U}. By (5.4.1), we obtain: 5.15. Corollary. σ_{λ}(f_{1} + f_{2}^{a+nc}) = σ_{λ}(f_{1} + f_{2}^{a}) for any n ≥ 0.

Similar results are true for the equivariant Milnor numbers, too. By [11, 18] we get: μ_{λ}(f_{1} + f_{2}^{a+nc}) = μ_{λ}(f_{1} + f_{2}^{a}) for any n ≥ 0. This shows that if the corresponding monodromies are diagonalizable, then the λ–components of the variation structures are isomorphic, provided that λ ≠ 1:

V(f_{1} + f_{2}^{a+nc})_{λ} = V(f_{1} + f_{2}^{a})_{λ}.

In other words, increasing a, the λ–components will not be “thicker”. The variation structure is growing by the appearance of new components which become more and more disperse. This phenomenon can be exemplified easily on the A_{a−1}–singularities z_{1}^{2} + z_{2}^{2} + z_{3}^{a}, described in (5.6).

We expect that this “telescopic phenomenon” is true in general, even at the deeper level of mixed Hodge structures. 5.16. The connection with the mixed Hodge structure.

Let f be an isolated singularity. Similarly as in (3.1), we construct its (−1)^{n}–symmetric variation structure V(f) = (U; b, ρ, V) defined on the group Z. This is essentially given by ρ(1) = M_{f} =the algebraic monodromy, and V (1) = V_{f} =variation map. Since V_{f} is an isomorphism, it determines the whole structure. In fact, it is equivalent to the real Seifert form.

The variation structures of Z with V (1) isomorphism are classified [12]. In the following, we recall briefly the indecomposable elements for which the eigenvalues of ρ(1) are on the unit circle. 5.17. [12] Let J_{k} : C^{k} → C^{k} be the k–dimensional Jordan block with diagonal entries = 1.

For λ ∈ S^{1} − {1}, we have, up to isomorphism, exactly two indecomposable (−1)^{n}–hermitian variation structures with ρ(1) = λJ_{k}:

W_{λ}^{k}(±1) = (C^{k}; b_{±}^{k}, λJ_{k}, (λJ_{k} − I)(b_{±}^{k})^{−1}),

where b_{±}^{k} is characterised by (b_{±}^{k})_{ij} = 0 if i + j ≤ k, and (b_{±}^{k})_{k,1} = ±i^{−n^{2}−k+1}. (This sign convention has Hodge theoretical motivation, in fact it is related to the polarization formula, see 5.21.)

If λ = 1, again, there are exactly two indecomposable (−1)^{n}–hermitian variation structures (up to isomorphism) with ρ(1) = J_{k} and with V = V (1) isomorphism. These structures are degenerate with one–dimensional Ker(b). They are:

W_{1}^{1}(±1) = (C; 0, 1_{C}, ±i^{n^{2}−1}) if k = 1,
W_{1}^{k}(±1) = (C^{k}; b̃_{±}^{k}, J_{k}, Ṽ_{±}^{k}) if k ≥ 2,

where b̃_{±}^{k} is characterized by (b̃_{±}^{k})_{i,j} = 0 if i + j ≤ k + 1 and (b̃_{±}^{k})_{k,2} = ±i^{−n^{2}−k+2}.

5.18. An isometric structure consists of a non–degenerate hermitian form (U; b) (\overline{b}^{∗} = (−1)^{n}b), and a representation ρ : G → Aut(U; b). If G = Z then the structure (U; b, ρ) is given by the pair (b, ρ(1)). In this case the indecomposable ones (with the same eigenvalue–restriction as above) are [9]:

I_{λ}^{k}(±1) = (C^{k}; b_{±}^{k}, λJ_{k}), where λ ∈ S^{1}, k ≥ 1.

Any isometric structure can be extented to a variation structure by V (g) = (ρ(g)−I)b^{−1}. (By this, I_{λ}^{k}(±1) can be identified with W_{λ}^{k}(±1), provided that λ ≠ 1.) 5.19. The following notation will be helpful: s(λ) = 0 if λ ≠ 1 and = 1 if λ = 1. The signatures of the corresponding forms are:

σ(W_{λ}^{k}(±1)) = ±\frac{1 + (−1)^{k+1+s(λ)}}{2} , σ(I_{λ}^{k}(±1)) = ±\frac{1 + (−1)^{k+1}}{2} .

Set λ = e^{2πic} with 0 < c ≤ 1. Then, by a verification:

η(W_{λ}^{k}(±1); 1) = ±(1−2c)\frac{1 + (−1)^{k+1}}{2} , η(I_{λ}^{k}(±1); 1) = ±(1−2c)\frac{1 + (−1)^{k+1+s(λ)}}{2} .

If V = (U; b, ρ, V) is a variation structure of Z, and a ∈ N^{∗}, then we define another structure a^{∗}V = (U; b, a^{∗}ρ, a^{∗}V) of Z by a^{∗}ρ(1) = ρ(a) and a^{∗}V (1) = V (a). By this notations: a^{∗}W_{1}^{k}(±1) = W_{1}^{k}(±1), and for λ ≠ 1 one has a^{∗}W_{λ}^{k}(±1) = W_{λ^{a}}^{k}(±1) if λ^{a} ≠ 1, and a^{∗}W_{λ}^{k}(±1) = I_{1}^{k}(±1) if λ^{a} = 1. Therefore:

η(W_{λ}^{k}(±1); a) = \{ \begin{matrix}η(W_{λ^{a}}^{k}(±1); 1) & if λ = 1 or λ^{a} ≠ 1 \\ η(I_{1}^{k}(±1); 1) & if λ^{a} = 1 and λ ≠ 1. \end{matrix}

5.20. Denote by p_{λ}^{p,q}(f) (where r = p+q−n−s(λ) ≥ 0) the dimensions of the primitive spaces of the mixed Hodge structure of the germ f [19]. Consider the invariant Σpp_{λ,±}(f) = ∑(−1)^{q}p_{λ}^{p,q}, where the sum is over the pairs (p, q) such that r = p + q − n − s(λ) satisfies (−1)^{r} = ±1. Obviously, this can be derived also from the spectral pairs.

The definition of the spectral pairs is as follows.

Let h_{λ}^{p,q} be the Hodge numbers of f, in particular h_{λ}^{p,q} = ∑_{l≥0} p_{λ}^{p+l,q+l} (p+ q ≥ n + s(λ)). Then the collection of the spectral pairs Spp(f) ∈ N[Q × N] is

Spp(f) = ∑_{(α,w)} h_{exp(−2πiα)}^{n+[−α],w+s−n−[−α]} · (α, w).

If we forget the weight filtration, then the information of the equivariant Hodge filtration is codified in the spectral numbers:

Sp(f) = ∑ α ∈ N[Q] (the sum over the spectral pairs (α, w)).

(For the definition of the spectral pairs and the spectral numbers, see [19, 17].)

It is remarkable that the invariant Σpp_{λ,+} is a spectral number invariant. Indeed, consider:

Σp_{λ,±}(f) = #{c|c is a spectral number with e^{−2πic} = λ and (−1)^{[c]} = ±1},

and Σp_{λ}(f) = Σp_{λ,−}(f) − Σp_{λ,+}(f). Then: Σp_{λ}(f) = Σpp_{λ,+}(f) (see, for example, [12]). 5.21. The connection between the Hodge structure and the variation structure is given in the relation [12]:

V(f)_{λ} = ⨁_{{2n≥p+q≥n+s \atop r=p+q−n−s}} p_{λ}^{p,q}(f) · W_{λ}^{r+1}((−1)^{q}).

By (5.19) and (5.20) one has:

σ_{λ}(f) = \{ \begin{matrix}Σpp_{λ,+}(f) & if λ ≠ 1 \\ Σpp_{λ,−}(f) & if λ = 1. \end{matrix}

In particular, σ_{λ} is a spectral number invariant for λ ≠ 1.

In order to compute the eta-invariants, fix λ = e^{−2πic} (notice the negative sign in the exponent) with 0 ≤ c < 1. Then by (5.19) and (5.20): η(V(f); 1)_{λ} = −(1 − 2c)Σpp_{λ,+}(f). On the other hand, η(V(f); a)_{λ} = ∑ p_{λ}^{p,q}η(a^{∗}W_{λ}^{r+1}(−1)^{q}; 1) and by (5.19):

η(V(f); a) = − ∑_{{λ^{a}=1 \atop λ≠1}} Σpp_{λ,−}(f) − ∑_{{λ^{a}≠1 \atop or λ=1}}(1 − 2\{ca\})Σpp_{λ,+}(f).

5.22. 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 = 0}, and the discriminant locus Δ contains only one irreducible component Δ = Δ_{1}, which is smooth. In particular, G = Z and L = 0. Since σ(f_{1}) = 0, theorem (5.4.2), for suspensions, gives:

σ(f + z_{n+1}^{a}) = η(V(f); a) − a · η(V(f); 1).

(This can be derived also from the Sebastiani-Thom type formula of the spectral pairs [16].)

The term η(V(f); 1) is − ∑_{λ}(1 − 2c)Σp_{λ}(f), which is a spectral number invariant of f. On the other hand, the term Σpp_{λ,−} in η(V(f); a) can be computed only by the spectral pairs. From this reason, if the monodromy operator of f has an even dimensional Jordan block with eigenvalue λ ≠ 1, but with λ^{a} = 1, then the correction term depends essentially on the block structure of the operator. 5.23. Example. Set f_{k,l;m,n} : (C^{2}, 0) → (C, 0) defined by:

f_{k,l;m,n} = ((y−x^{2})^{2}−x^{5+k})((y+x^{2})^{2}−x^{5+l})((x−y^{2})^{2}−y^{5+m})((x+y^{2})^{2}−y^{5+n}).

Then the spectral numbers of f_{−1,−1;1,1} and f_{−1,1;−1,1} are the same, but the spectral pairs are different [17]. In the first case one has spectral pairs (−\frac{1}{2}, 2) and (\frac{1}{2}, 0), and in the second case (−\frac{1}{2}, 1) and (\frac{1}{2}, 1), whereas the other pairs are the same. Therefore, the decompositions of the variation structures (Seifert forms) are: in the first case one has W_{−1}^{2}(−1), and in the second case W_{−1}^{1}(+1) ⊕ W_{−1}^{1}(−1), whereas the other indecomposable structures are the same. Take a = 2. Then ∑_{λ^{2}=1,λ≠1} Σpp_{λ,−} in the first case is = −1, and in the second = 0. In particular, σ(f_{−1,−1;1,1} + z^{2}) = σ(f_{−1,1;−1,1} + z^{2}) + 1. 5.24. The equivariant signature σ_{λ}(f + z_{n+1}^{a}) (λ ≠ 1) depends only on the spectrum of f + z_{n+1}^{a} (see 5.21), which can be computed by the spectrum of f. Therefore σ_{λ}(f + z_{n+1}^{a}) (λ ≠ 1) is independent on the block structure of the monodromy of f. (In particular, for any a, and λ ≠ 1 one has σ_{λ}(f_{−1,−1;1,1} + z^{a}) = σ_{λ}(f_{−1,1;−1,1} + z^{a}).)

On the other hand, by (5.4.1), the following relation holds:

σ_{1}(f + z_{n+1}^{a}) = − ∑_{{λ^{a}=1 \atop λ≠1}} Σpp_{λ,−}(f).

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IMAR, Bucharest, Romania. Current address:

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e-mail: nemethi@math.ohio-state.edu