The semi–ring structure and the spectral pairs of sesqui–linear forms

András Némethi^{∗}

October 11, 1997

(Algebra Colloq. 1:1 (1994), 85-95)

1 Introduction

The basic motivation of this paper is related to the theory of complex hypersurface singularities.

Let S_{n} (n ≥ 1) be the set of analytic germs f : (C^{n}, 0) → (C, 0) with isolated singularity at the origin. Let N[S_{n}] be the free semi–group generated by S_{n} and let the graded semi–group N[S] = ⊕_{n≥1}N[S_{n}] be their direct sum. N[S] has a graded semi–ring structure generated by the following multiplicative law: if f ∈ S_{n} and g ∈ S_{m}, then f ⊕ g ∈ S_{n+m} is defined by (f ⊕ g)(x, y) = f(x) + g(y), where x ∈ (C^{n}, 0) and y ∈ (C^{m}, 0).

An important analytic invariant of f ∈ S_{n} is the collection Spp(f) of its spectral pairs. Spp(f) lies in the free semi–group (N[Q×N], +) generated by Q × N [8, 9, 7]. On (N[Q × N], +) there is a compatible semi–ring structure defined by the (multiplicative) law: (α, w) ∗ (β, ω) = (α + β + 1, w + ω + 1).

By [7], the map Spp : (N[S], +, ⊕) → (N[Q × N], +, ∗) is a semi–ring morphism, in particular Spp(f ⊕ g) = Spp(f) ∗ Spp(g).

The most important topological invariant of f ∈ S_{n} is its Seifert form V_{f}. It is well–known that V_{f} is non–degenerate and it is equivalent to the variation map of f.

∗Partially supported by NSF grant No. DMS-9203482

Let (F, +, ⊗_{s}) be the semi–ring of the non–degenerate, graded sesqui– linear forms (i.e. the collection of forms G = (G_{n})_{n≥1}, where each G_{n} is non– degenerate). The additive law is (G+H)_{n} = G_{n} ⊕H_{n}, and the multiplicative one is (G ⊗_{s} H)_{k} = ⊕_{n+m=k}(−1)^{nm}G_{n} ⊗ H_{m}. By a result of Sakamoto [6] (or Deligne, see [1]), if f ∈ S_{n} and g ∈ S_{m}, then V_{f⊕g} = (−1)^{nm}V_{f} ⊗ V_{g}. In particular f ↦ V_{f}⊗1_{C} induces a semi–ring morphism V : N[S] → (F, +, ⊗_{s}).

Consider the natural projection pr_{mod−2} : (N[Q×N], +, ∗) → (N[(Q/2Z)× N], +, ∗) induced by the natural map Q → Q/2Z. In [4] it is proved that the information contained in the complex (equivalently, in the real) Seifert form is equivalent to the information contained in pr_{mod−2}(Spp(f)). In fact, a map Spp_{mod−2} : F → N[(Q/2Z) × N] is constructed, such that the following diagram is commutative:

N[S] ⟶^{Spp}  N[Q × N]
↓ V  ↓ pr_{mod−2}
F ⟶^{Spp_{mod−2}} N[(Q/2Z) × N]

In this note, we study the semi–ring structure of F and we prove that Spp_{mod−2} is a morphism of semi–rings. This seems to be a purely algebraic problem, and it can be formulated independently from the singularity theory.

In the study of isolated singularities it is convenient to study the variation map (Seifert form) V_{f} together with the intersection form b_{f} and the monodromy operator h_{f} (even if b_{f} and h_{f} are determined by V_{f}). Similarly, in the pure algebraic study of the sesqui–linear forms V, we preffer to work with a triplet (V, b, h) (which satisfies some axioms, inspired from the properties of (V_{f}, b_{f}, h_{f})). We call these triplets variation structures. In [4] there is proved that any variation structure (with non–degenerate V) is a direct sum of some indecomposable structures, and the indecomposable structures are listed. Using this, in this paper, we determine the (topological) semi–ring structure of F.

(For more information about the above diagram, and its relation with other invariants of the singularities, see [4].)

2 ε-hermitian variation structures

If U is a finite dimensional vector space then U^{∗} is its dual Hom(U, C). We have the natural isomorphism θ : U → U^{∗∗} given by θ(u)(φ) = φ(u). It is convenient to write ε = ±1 in the form ε = (−1)^{n}. 2.1. Definition. An ε–hermitian variation structure (abbreviated by HVS) over C is a system (U; b, h, V), where

a.) U is a finite dimensional C–vector space,

b.) b : U → U^{∗} is an ε–symmetric C–linear endomorphism, i.e. \overline{b^{∗}∘θ} = εb. c.) h is b–orthogonal automorphism of U, i.e. \overline{h^{∗}}∘b∘h = b. d.) V : U^{∗} → U is a C–linear endomorphism, with

i) \overline{θ^{−1}∘V^{∗}} = −εV ∘\overline{h}^{∗},

ii) V ∘b = h − I.

2.2. We have two immediate properties: b∘V = \overline{h}^{∗,−1} − I, and h∘V ∘\overline{h}^{∗} = V. 2.3. Definition. The HVS (U; b, h, V) is called nondegenerate (resp. simple) if b (resp. V) is an isomorphism. 2.4. Remarks.

a.) If b is an isomorphism then V = (h−I)b^{−1} and the HVS (U; b, h, V) is completely determined by the isometric structure (U; b, h) (i.e. triplets with axioms a-b-c and with non–degenerate form b). For their classification, see the papers of Milnor [3] and Neumann [5].

b.) If V is an isomorphism, then h = −εV \overline{V}^{∗,−1} and b = −V^{−1}−ε\overline{V}^{∗,−1}. In particular, the classification of simple HVS-s is equivalent to the classification of C–linear isomorphisms V : U^{∗} → U or to the classification of sesqui–linear forms on finite dimensional vector spaces.

If we would like to emphasize ε then the ε–HVS determined by V is denoted by _{ε}V.

c.) If two real nondegenerate bilinear forms are isomorphic over C, then they are isomorphic over R. In particular, the study of real simple variation structures is equivalent to the study of the complex ones. 2.5. Examples.

1. If V_{i} = (U_{i}; b_{i}, h_{i}, V_{i}) (i = 1, 2) are variation structures, then V_{1} ⊕ V_{2} = (U_{1} ⊕U_{2}; b_{1} ⊕b_{2}, h_{1} ⊕h_{2}, V_{1} ⊕V_{2}) is their direct sum in the category of HVS-s. If V = (U; b, h, V) then −V denotes (U; −b, h, −V) with the same ε.

Assume that ε_{1}, ε_{2} and ε are fixed. If V_{i}, (i = 1, 2) are simple ε_{i}–hermitian variation structures, then the tensor product V_{1} ⊗ V_{2} defines a new simple ε– structure. The corresponding automorphisms are related by^{⊗}h = −εε_{1}ε_{2}h_{1}⊗ h_{2}. In this paper always εε_{1}ε_{2} = −1, i.e. h = h_{1} ⊗ h_{2}.

The conjugate of V = (U; b, h, V) is \overline{V} = (U; \overline{b}, \overline{h}, \overline{V}).

2. In the next examples J_{k} denotes the k × k–Jordan block with eigenvalue = 1.

Consider λ ∈ C^{∗} − S^{1}. (S^{1} denotes the unit circle.) The ε–HVS V^{2k}(λ) is defined by:

V_{λ}^{2k} = (C^{2k}; ( \begin{matrix}0 & I \\ εI & 0 \end{matrix} ) , ( \begin{matrix}λJ_{k} & 0 \\ 0 & \frac{1}{λ̄}J_{k}^{∗,−1} \end{matrix} ) , ( \begin{matrix}0 & ε(λJ_{k} − I) \\ \frac{1}{λ̄}J_{k}^{∗,−1} − I & 0 \end{matrix} )).

Note that V_{λ}^{2k} ≈ V_{1/λ̄}^{2k} ≈ −V_{λ}^{2k}.

3. We are looking for nondegenerate forms b such that b̄^{∗} = εb and J_{k}^{∗}bJ_{k} = b. It is immediate that b_{ij} = 0 if i + j ≤ k and b_{k+1−i,i} = (−1)^{i+1}b_{k,1}. By [3] b is determined by b_{k,1}. Since b is nondegenerate b_{k,1} ≠ 0, so we can assume that b_{k,1} = ω ∈ S^{1}. By the symmetry of b one has \overline{ω} = ε(−1)^{k−1}ω. This equation has two solutions. In conclusion, there are exactly two nondegenerate forms b = b_{±}^{k} (up to isomorphism) with \overline{b}^{∗} = εb and J_{k}^{∗}bJ_{k} = b. Their representatives are chosen so that (b_{±}^{k})_{k,1} = ±i^{−n^{2}−k+1}.

Let λ ∈ S^{1}. If h = λJ_{k}, then by the above argument, there are exactly two nondegenerate ε–HVS-s (up to isomorphism):

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

where ω = (b_{±}^{k})_{k,1} = ±i^{−n^{2}−k+1}.

If λ ≠ 1, then by (d-ii) any HVS with h = λJ_{k} is nondegenerate. If h = J_{k}, then there are some degenerate structures, too.

4. Suppose that k ≥ 2 and h = J_{k} but b is degenerate. Then kerb = ker(h−I) by (d-ii). Similarly as above, any degenerated form b with kerb = ker(J_{k}−I) and \overline{b}^{∗} = εb and \overline{h}^{∗}bh = b has the properties b_{i,j} = 0 if i + j ≤ k + 1, and b_{k+2−i,i} = (−1)^{i}b_{k,2}. Therefore b_{k,2} ≠ 0 and we can assume that b_{k,2} = ω ∈ S^{1}. By symmetry, \overline{ω} = (−1)^{n+k}ω. Similarly as in the Milnor argument, b is completely determined by b_{k,2}. So, we have exactly two solutions b̃_{±}^{k} with (b̃_{±}^{k})_{k,2} = ±(−1)^{n+1}i^{−(n+1)^{2}−k+1}. Moreover, V is completely determined by h and b (up to isomorphism). In particular, there are exactly two degenerate structures with h = J_{k} and k ≥ 2:

Ṽ_{1}^{k}(±1) = (C^{k}; b̃_{±}^{k}, J_{k}, Ṽ_{±}^{k}),

where (b̃_{±}^{k})_{k,2} = ±i^{−n^{2}−k+2}. In fact:

b = b̃_{±}^{k} = ( \begin{matrix}0 & 0 \\ 0 & b_{±}^{k−1} \end{matrix} ) .

Note that the structure can also be recognized from ((Ṽ_{±}^{k})^{−1})_{k,1} = ±i^{−n^{2}−k+2}; (use the identity b = V^{−1}(h − I)).

By computation we get that Ṽ_{±}^{k} is an isomorphism. In particular, the variation structures V_{λ}^{k}(±1), where λ ∈ S^{1} − {1} resp. k ≥ 1 and Ṽ_{1}^{k}(±1) where k ≥ 2, are simple, and they are determined by the corresponding isometric structures (C^{k}; b, h).

5. Suppose that U = C and h = 1_{C}. Then there are exactly five HVS-s:

V_{1}^{1}(±1) = (C; ±i^{−n^{2}}, 1_{C}, 0);
Ṽ_{1}^{1}(±1) = (C; 0, 1_{C}, ±i^{n^{2}−1});

and

T = (C; 0, 1_{C}, 0).

Note that in Ṽ_{1}^{1}(±1) the variation structure is not determined by its underlying (degenerate) isometric structure.

6. In order to unify the notations of simple structures, we introduce: W_{λ}^{k}(±1) = V_{λ}^{k}(±1) if λ ∈ S^{1} − {1}, and = Ṽ_{1}^{k}(±1) if λ = 1. Set s = 1 if λ = 1 and = 0 otherwise. Then: \overline{W}_{λ}^{k}(±1) = Wk ; \overline{λ}(±(−1)^{−n^{2}−k+1+s}). 2.6. In [4] it is proved the following Theorem. A simple ε–hermitian variation structure is uniquely expressible as a sum of indecomposable ones up to order of summands and isomorphism. The indecomposable ones are:

W_{λ}^{k}(±1) where k ≥ 1; λ ∈ S^{1}; and
V_{λ}^{2k} where k ≥ 1; 0 < |λ| < 1.

This theorem gives a classification of comlex sesquilinear forms (with respect to complex conjugation) over finite dimensional C–vector spaces (cf. 2.6.b). 2.7. If we do not want to relate this presentation and classification to the singularity and Hodge theory, then the sign–convention can be simplified. For the sign–motivation, see [4].

3 The multiplicative structure

3.1. If we want to study only the sesqui-linear forms, then the sign of ε is irrelevant. But in some cases, we want to see the invariants of the associated ε–symmetric form b, too. From this reason, it is convenient to work in a Z_{2}–graded theory; the gradation given by ε. On the other hand, we have a substructure given by the automorphism h. The nilpotent element log J_{k} gives a filtration of the ambient space of W_{λ}^{k}(±1). In fact, the action of log J_{k} can be extended to an irreducible representation of sl_{2}(C). In the classical theory of the representations of sl_{2}(C)–modules this weight filtration is centered at zero. Inspired again by the Hodge theory, we want to work with arbitrary centers. This gives an N–graded theory. Since in the theory of singularities, the symmetry of the bilinear form and the center of the weight filtration are related and are coordinated by the dimension, we will use (only) an N–gradation. (Our results can be extended to a Z–graded version in a trivial way.) 3.2. Definition. A graded hermitian variation structure over C (abbreviated by GHV S) is a system of N–graded objects V = ⊕_{n≥0}V_{n+1} (finite direct sum), such that V_{n+1} is a (−1)^{n}–hermition variation structure. V is called simple if for any n ≥ 0 V_{n+1} is either simple or zero.

The direct sum and the tenzor product (twisted with a sign) are defined by:

(V ⊕ W)_{n+1} = V_{n+1} ⊕ W_{n+1};

(V ⊗_{s} W)_{k+1} = ⨁_{n+m+1=k}(−1)^{(n+1)(m+1)}V_{n+1} ⊗ W_{m+1}.

(Here, by the above definition, V_{n+1} ⊗ W_{m+1} is (−1)^{n+m+1}–symmetric.) 3.3. The additive structure of GHVS-s is completely determined by theorem 2.6. As a byproduct of the multiplication structure, we must reobtain the behaviour of the tensor product J_{k}⊗J_{l}. In terms of sl_{2}(C)–representations, this problem is equivalent to the splitting of the tensor product of two irreducible representation in irreducible ones, i.e. in the Clebsch–Gordan series (see for example [2]). Now, each h = λJ_{k} (λ ∈ S^{1}) belongs to exactly two HVS, but, as we will see, the parameter space of the eigenvalues and this Z_{2}–sign fit together in a moduli space which is a Z_{2}–covering of the parameter space of the eigenvalues. Therefore, the multiplicative structure is coordinated by two objects: the Clebsch–Gordan series and this moduli space.

3.4. A computation (or an argument using the signature) gives for ξ ∈ S^{1}: lim_{λ→ξ;λ∉S^{1}} V_{λ}^{2k} = V_{ξ}^{k}(+1) ⊕ V_{ξ}^{k}(−1); (cf. 3.8). This, and also the next discussions, motivate the following definition: for ξ ∈ S^{1} set V_{ξ}^{2k} = V_{ξ}^{k}(+1) ⊕ V_{ξ}^{k}(−1). 3.5. Theorem. The multiplication table, given on homogeneous elements, is:

a)

V_{λ}^{2k} ⊗_{s} V_{η}^{2l} = ⨁_{t=1}^{min(k,l)} V_{λη}^{2(k+l+1−2t)} ⊕ ⨁_{t=1}^{min(k,l)} V_{λ/η̄}^{2(k+l+1−2t)}

where λ ∉ S^{1} and η ∉ S^{1}; b)

V_{λ}^{2k} ⊗_{s} W_{ξ}^{l}(±1) = ⨁_{t=1}^{min(k,l)} V_{λξ}^{2(k+l+1−2t)}

where λ ∉ S^{1} and ξ ∈ S^{1}; c)

(W_{λ}^{k}(u))_{n+1} ⊗_{s} (W_{η}^{l}(v))_{m+1} = ⨁_{t=1}^{min(k,l)} W_{λη}^{k+l+1−2t}(uv(−1)^{t+1}s(λ, η))

where λ = e^{−2πiα}; η = e^{−2πiβ}; u = ±1; v = ±1 and s(λ, η) = (−1)^{[α+β]−[α]−[β]}. Proof. The first two parts follow from the classification theorem (2.6) and the Clebsch–Gordan theorem. (As a remark, note that V_{λ}^{2k} ⊗ · is not one-to-one.)

We prove c) in several steps. For the left hand side of 3.5.c we use the notation V^{⊗} = (b^{⊗}, h^{⊗}, V^{⊗}). Step 1: The case λ = 1, η ≠ 1 and l = 1. a.) If k = 1, then V^{⊗} = (−1)^{(n+1)(m+1)}ui^{n^{2}−1}·v(η−1)i^{m^{2}} = uv(η−1)i^{(n+m+1)^{2}}. This corresponds to V_{η}^{1}(uv)_{n+m+1}. b.) If k ≥ 2, then V^{⊗} = (−1)^{(n+1)(m+1)}(η − 1)vi^{m^{2}} Ṽ_{u}^{k}. Therefore, b^{⊗} = (V^{⊗})^{−1}(ηJ_{k}−I) = (V^{⊗})^{−1}[η(J_{k}−I)+(η−1)I] = (−1)^{(n+1)(m+1)}vi^{−m^{2}}[(Ṽ_{u}^{k})^{−1}+ \frac{η}{η−1}b̃_{u}^{k}]. Hence (b^{⊗})_{k,1} = (−1)^{(n+1)(m+1)}vi^{−m^{2}}(Ṽ_{u}^{k})_{k,1}^{−1} = (−1)^{(n+1)(m+1)}vi^{−m^{2}}ui^{−n^{2}−k+2} = uvi^{−(n+m+1)^{2}−k+1}. Step 2: The case λ = η = 1.

a) The case l = 1 follows from ((V^{⊗})^{−1})_{k,1} = (−1)^{(n+1)(m+1)}(Ṽ_{u}^{k})_{k,1}^{−1} ·(Ṽ_{v}^{1})_{1,1}^{−1} = (−1)^{(n+1)(m+1)}ui^{−n^{2}−k+2}vi^{−m^{2}−1+2} = uvi^{−(n+m+1)^{2}−k+2}.

b) If l = 2, then by Clebsch–Gordan theorem, we know that V^{⊗} = Ṽ_{1}^{k+1}(x) ⊕ Ṽ_{1}^{k−1}(y). If {e_{1}, …, e_{k}} resp. {f_{1}, f_{2}} are the standard bases such that log J_{k}e_{i} = e_{i−1} (i = 2, …, k) and log J_{2}f_{2} = f_{1}, then Ṽ_{1}^{k+1}(x) is generated by e_{k} ⊗ f_{2} over C[h^{⊗}], where h^{⊗} = J_{k} ⊗ J_{2}. Therefore xi^{−(n+m+1)^{2}−(k+1)+2} = sign(V^{⊗})^{−1}(e_{k} ⊗f_{2})((log J_{k} ⊗J_{2})^{k}(e_{k} ⊗f_{2})) = sign(V^{⊗})^{−1}(e_{k} ⊗f_{2})(e_{1} ⊗f_{1}) = (−1)^{(n+1)(m+1)}(Ṽ_{u}^{k})_{k,1}^{−1} · (Ṽ_{v}^{2})_{2,1}^{−1}. Therefore x = uv.

Now, by b = V^{−1}(h − I), we deduce that (Ṽ_{x}^{k+1})_{i,j}^{−1} = 0 if i + j ≤ k + 1. Therefore, for any z ∈ Ṽ_{1}^{k+1}(x) with the property (log J_{k+1})^{k−1}z = 0, we have (Ṽ_{x}^{k+1})^{−1}(z, (log J_{k+1})^{k−2}z) = 0. On the other hand, if z = ∑_{i=1}^{k−1} a_{i}e_{i} ∈ Ṽ_{1}^{k−1}(y), then (Ṽ_{y}^{k−1})^{−1}(z, (log J_{k−1})^{k−2}z) = a_{k−1}^{2}(Ṽ_{y}^{k−1})_{k,1}^{−1} = a_{k−1}^{2}yi^{−(n+m+1)^{2}−(k−1)+2}. Therefore, we can recover y, if we find a z ∈ ker(log h^{⊗})^{k−1} so that z^{⊗} = (V^{⊗})^{−1}(z, (log J_{k} ⊗ J_{2})^{k−2}z) ≠ 0. Take z = e_{k−1} ⊗ f_{2} − (k − 1)e_{k} ⊗ f_{1}. Then (log J_{k} ⊗ J_{2})^{k−2}z = e_{1} ⊗ f_{2} − e_{2} ⊗ f_{1}; thus (−1)^{(n+1)(m+1)}z^{⊗} = (Ṽ_{u}^{k})_{k−1,1}^{−1} · (Ṽ_{v}^{2})_{2,2}^{−1} − (Ṽ_{u}^{k})_{k−1,2}^{−1} · (Ṽ_{v}^{2})_{2,1}^{−1} − (k − 1)(Ṽ_{u}^{k})_{k,1}^{−1} · (Ṽ_{v}^{2})_{1,2}^{−1} + (k − 1)(Ṽ_{u}^{k})_{k,2}^{−1} · (Ṽ_{v}^{2})_{1,1}^{−1}. Since (Ṽ_{±}^{k})_{i,j}^{−1} = 0 if i + j ≤ k, the first and the last term is zero. Moreover, (Ṽ_{±}^{k})_{k−1,2}^{−1} = −(Ṽ_{±}^{k})_{k,1}^{−1}, therefore z^{⊗}/k = yi^{−(n+m+1)^{2}−k+3} = (−1)^{(n+1)(m+1)}uvi^{−n^{2}−k+2−m^{2}}, i.e. y = −uv.

c) For the general case we can assume that 3 ≤ l ≤ k. By the case l = 2 we have: (Ṽ_{1}^{l−1}(v))_{m}⊗_{s}(Ṽ_{1}^{2}(+1))_{1} = (Ṽ_{1}^{l}(v))_{m+1}⊕(Ṽ_{1}^{l−2}(−v))_{m+1}. Now, multiply this by (Ṽ_{1}^{k}(u))_{n+1} and use the obtained identity as an inductive step. Step 3: The case λ ≠ 1, η = 1. By Step 1: (Ṽ_{λ}^{k}(u))_{n+1} = (V_{λ}^{1}(+1))_{1} ⊗_{s} (Ṽ_{1}^{k}(u))_{n}. Now use Step 2 and Step 1 again. Step 4: (V_{λ}^{1}(+1))_{1} ⊗_{s} (V_{η}^{1}(+1))_{1} = (W_{λη}^{1}(s(λ, η))_{2}; where λ ≠ 1 and η ≠ 1.

a) The case λη = 1. Recall : V_{λ}^{1}(+1) = (C; 1, λ, λ − 1). Therefore V^{⊗} = −(λ − 1)(η − 1) = 2(Re(λ) − 1) < 0. But s(λ, η) = −1, so we are finished.

b) If λη ≠ 1, then the left hand side is V_{λη}^{1}(x) for a suitable x. Now, b^{⊗} = −\frac{λη−1}{(λ−1)(η−1)} therefore x = sign(i b^{⊗}) = s(λ, η). Step 5: For the last case λ ≠ 1 and η ≠ 1, use the “decomposition” of V_{λ}^{k} resp. of V_{η}^{l} as in the proof of Step 3, and use Step 4. 3.6. The formula (3.5.c) suggests that the four types W_{ξ}^{k}(±1) (ξ ∈ S^{1}) fit in a single parameter space. In order to see this, fix ξ = e^{−2πiβ} (0 ≤ β < 1) on S^{1}, and consider the result of the multiplication of V_{ξ}^{k}(+1) with V_{e^{−2πiα}}(+1) for α ∈ (0, 1):

V_{e^{−2πiα}}^{1}(+1) ⊗_{s} V_{ξ}^{k}(+1) = \{ \begin{matrix}V_{ξe^{−2πiα}}^{k}(+1) & 0 < α < 1 − β \\ Ṽ_{1}^{k}(−1) & α + β = 1 \\ V_{ξe^{−2πiα}}^{k}(−1) & 1 − β < α < 1. \end{matrix}

So {W_{e^{−2πiα}}^{k}(±1)}_{α,±1} fit in the following parameter space:

[figure: α:; 0; 1; 2; Ṽ_{1}^{k}(±1); V_{e^{−2πiα}}^{k}(±1); Ṽ_{1}^{k}(∓1); V_{e^{−2πiα}}^{k}(∓1); Ṽ_{1}^{k}(±1)]

Define V^{k}(α) = W_{e^{−2πiα}}^{k}((−1)^{[α]}). Then (3.5.c) has the following form: 3.7. Proposition.

(V^{k}(α))_{n+1} ⊗_{s} (V^{l}(β))_{m+1} = ⨁_{t=1}^{min(k,l)} V^{k+l+1−2t}(α + β + t + 1).

3.8. Fix k ≥ 1. Let D^{∗} = {z ∈ C^{∗}, |z| < 1}, the open punctured disc, and \overline{D}^{∗} = \{z ∈ C^{∗}, |z| ≤ 1\} the closed one. By the above discussion, the parameter space of the indecomposable nondegenerated sesquilinear forms of type V_{λ}^{2k}, (λ ∉ S^{1}), and W_{λ}^{k}(±1), (λ ∈ S^{1}), is D^{∗} ∪ R/2Z. In order to see the right topology on this (i.e.”lim_{λ→e^{−2πiα},|λ|<1} = {α, α + 1}”), we make the following construction. We define an equivalence relation on \overline{D}^{∗}, generated by: z_{1} ∼ z_{2} if z_{1}, z_{2} ∈ D^{∗} and z_{1}^{2} = z_{2}^{2}. The factor set M = \overline{D}^{∗}/_{∼} (with the factor topology) is the moduli space of our forms. 3.9. If X is a set, let N[X] be the free abelian semi-group generated by (the base) X: N[X] = {finite sums ∑ n_{i}x_{i}; n_{i} ∈ N, x_{i} ∈ X}. On N[M] we define the following multiplication, given on generators:

[z_{1}] ∗ [z_{2}] = \{ \begin{matrix}[z_{1}z_{2}] & if |z_{1}| = 1 or |z_{2}| = 1 \\ [z_{1}z_{2}] + [z_{1}/\overline{z}_{2}] & if |z_{1}| < |z_{2}| < 1 \\ [z_{1}z_{2}] + [z_{1}/\overline{z}_{2}] + [−z_{1}/z̄_{2}] & if |z_{1}| = |z_{2}| < 1 \end{matrix}

The resulted semi–ring (N[M], +, ∗) is denoted by H.

If R is a semi–ring then the additive structure of R[N^{∗}] = {∑_{i≥1} r_{i}X^{i} : r_{i} ∈ R} can be completed in several ways to a semi–ring structure. One of them is the polynomial structure, denoted by R[X], given by r_{i}X^{i} · r_{j}X^{j} = r_{i}r_{j}X^{i+j}. We define another one as follows. Let A : R → R be an automorphism of R. We define the “twisted Clebsch–Gordan ”multiplication on R by:

r_{k}Y^{k} • r_{l}Y^{l} = ∑_{t=1}^{min(k,l)} A^{t+1}(r_{k}r_{l})Y^{k+l+1−2t}.

In the case of R = H we have a natural automorphism induced by z ↦ −z (z ∈ \overline{D}^{∗}). The twisted Clebsch–Gordan semi-ring structure H[tcg], by definition, is (H[N^{∗}], +, •). Its multiplication is :

[z]Y^{k} • [w]Y^{l} = ∑_{t=1}^{min(k,l)}[(−1)^{t+1}zw]Y^{k+l+1−2t}.

The inclusion R/2Z ⊂ \overline{D}^{∗}, given by α ↦ e^{−πiα}, induces the corresponding sub-semi-ring structures on N[R/2Z] (with multiplication: [α]∗[β] = [α+β]), and on N[R/2Z][tcg][X] (with automorphism A[α] = [α + 1]).

Denote GHV S(S^{1}) = {V ∈ GHV S : the eigenvalues of h lie on S^{1}}. 3.10. Theorem. We have the following semi–ring isomorphisms:

a)

(H[tcg][X], +, •) ≈ (GHV S, ⊕, ⊗_{s});

b)

(N[R/2Z][tcg][X], +, •) ≈ (GHV S(S^{1}), ⊕, ⊗_{s}).

c)

(H[tcg], +, •) ≈ {simple ε–HVS, ⊕, ⊗} (with fixed ε)

≈ {nondegenerated sesqui-linear forms, ⊕, ⊗};

Proof. Use the correspondence (V_{λ}^{2k})_{n+1} ⟷ [\sqrt{λ}]Y^{k}X^{n+1}, (λ ∈ D^{∗}), and (V^{k}(α))_{n+1} ⟷ [e^{−πiα}]Y^{k}X^{n+1}. 3.11. Remark. The complex conjugation (cf. 2.5.6) has the following effect:

([z]Y^{k}X^{n+1})^{−} = \{ \begin{matrix}[\overline{z}] Y^{k}X^{n+1} & if |z| < 1 \\ [e^{−πiβ}] Y^{k}X^{n+1} & if z = e^{−πiα} and β = 2[α] − α + n + k \end{matrix}

4 Hodge numbers and spectral pairs associated with variation structures

Let V = (U; b, h, V) be a simple variation structure.

In the sequel, we assume that the eigenvalues of our structures are on the unit circle. Recall that s = s(λ) = 0 if λ ≠ 1 and = 1 otherwise.

We want to construct a weight filtration W on U_{λ}^{∗} and a mod–2 decomposition on Gr_{∗}^{W} U_{λ}^{∗}. By the decomposition theorem, it is enough to define them on the indecomposable homogeneous elements. The weight filtration is given by h̄^{∗,−1}, the center of the filtration associated to the homogeneous element (W_{λ})_{n+1}, is, by definition, n + s. In fact, it is the unique filtration with center n + s and with the properties: dimGr_{l}^{W} (W_{λ}^{r+1}(u)) = 1 if l = n + s − r, …, n + s + r, and log \overline{J}_{r+1}^{∗,−1}(W_{l}) ⊂ W_{l−2}.

The Z_{2}–decomposition

Gr_{l}^{W} W_{λ}^{r+1}(u) = F_{+}Gr_{l}^{W} W_{λ}^{r+1}(u) ⊕ F_{−}Gr_{l}^{W} W_{λ}^{r+1}(u)

is given by:

dimF_{v}Gr_{n+s+r−t}^{W}W_{λ}^{r+1}(u) = \{ \begin{matrix}1 & if uv(−1)^{t} = 1 \\ 0 & otherwise \end{matrix}

In other words: Gr_{n+s+r−t}^{W}W_{λ}^{r+1}(u) = F_{(−1)^{t}u}Gr_{n+s+r−t}^{W}W_{λ}^{r+1}(u).

If V_{λ} = ∑_{u=±1,r≥0} p_{λ}^{r+1}(u)W_{λ}^{r+1}(u) then we redefine p_{λ}^{n+s+r,u} = p_{λ}^{r+1}(u), u = ±1, r ≥ 0; (n+s+r is the weight of the “primitive element” of W_{λ}^{r+1}(u)); and define h_{λ}^{w,u} = dimF_{u}Gr_{w}^{W} U_{λ}^{∗}. By these notations, we have the following relations:

p_{λ}^{w,u} = h_{λ}^{w,−u} − h_{λ}^{w+2,u} w ≥ n + s;

h_{λ}^{w,u} = ∑_{l≥0} p_{λ}^{w+2l,(−1)^{l}u} w ≥ n + s;

h_{λ}^{n+s−k,u} = h_{λ}^{n+s+k,(−1)^{k}u}.

In particular, V = ∑ p_{λ}^{r+1}(u)W_{λ}^{r+1}(u) is completely determined by the numbers {p_{λ}^{w,u}; w ≥ n + s} or by {h_{λ}^{w,u}}.

The Z_{2}–spectral pairs lies in the semiring

N[(R/2Z) × N] = { ∑_{(α,w)}(α, w), α ∈ R/2Z, w ∈ N},

where the additive structure is the natural one, but the multiplicative one is given by:

(α, w) ∗ (β, ω) = (α + β + 1, w + ω + 1).

The system of equations:

\{ \begin{matrix}e^{−2πiα} = λ \\ (−1)^{w−n−[−α]} = u \end{matrix}

has exactly one solution α = α_{λ,w,u} ∈ R/2Z. We associate with the space F_{u}Gr_{w}^{W} U_{λ}^{∗} the spectral pairs (α, w−s(λ)) with multiplicity h_{λ}^{w,u} = dimF_{u}Gr_{w}^{W} U_{λ}^{∗}. The collection of the Z_{2}–spectral pairs of V is:

Spp_{mod−2}(V) = ∑_{λ,w,u} h_{λ}^{w,u}(α_{λ,w,u}, w − s(λ)).

It is clear that passing to the spectral pairs we do not lose any information: we can recuperate V from its spectral invariants. Moreover, Spp_{mod−2}(V_{1}⊕V_{2}) = Spp_{mod−2}(V_{1}) + Spp_{mod−2}(V_{2}).

The symmetry of the weight filtration gives the invariance of Spp_{mod−2}(f) with respect to the transformation (α, n + k) ⟷ (α − k, n − k). If the structure comes from a real one, the stability with respect to the complex conjugation gives an addition invariance with respect (α, n + k) ⟷ (n − 1 − α, n − k). 4.1. Theorem.

h_{λ}^{w,u}(V ⊗_{s} W) = ∑ h_{ξ}^{p,v}(V)h_{η}^{q,z}(W)

where the sum is over the following combinations:

λ = ξη, w = p + q + 1 − s(ξ) − s(η) + s(ξη), u = vz · s(ξ, η). Proof. It is enough to verify the theorem only on the indecomposable homogeneous elements. Instead of a direct verification, we give the following argument. The relation λ = ξη is clear. The weights (eigenvalues of the diagonal element) of the tensor product of two irreducible sl_{2}(C)–representations are the corresponding sums of the factors’ weights. Since our centers are shifted with n + s(ξ), m + s(η) resp. n + m + 1 + s(ξη) the second relation follows. For the third one, it is enough to verify the sign behaviour only for the top weights (thanks to the compatibility of the twisted Clebsch–Gordan multiplication). 4.2. This theorem can be considered as the topologic version of (8.11–8.12) in [7]. That result works in analytic context and its proof is based on the theory of D–modules. In fact, our basic motivation to prove theorem (4.1) was the decision to give a topological support to that result. 4.3. Apparently, introducing different centers and separating the cases λ ≠ 1 and λ = 1, we mixed up the multiplication structure comletely. A not very careful analysis of theorem (4.1) suggests the same fact. But, hopefully, an appropiate organisation of the informations will give a very nice multiplication law. 4.4. Theorem.

Spp_{mod−2}(V_{1} ⊗_{s} V_{2}) = Spp_{mod−2}(V_{1}) ∗ Spp_{mod−2}(V_{2}).

Proof. Again, it is enough to verify the relation only on the indecomposable homogeneous elements. For these, Spp((W_{λ}^{r+1}(u))_{n+1}) = ∑_{k=0}^{r}(α − k, n + r − 2k) where λ = e^{−2πiα} and (−1)^{s(λ)+r−[−α]} = u. Now, use the identity s(λ) − [−α] = [α] + 1 and ∪_{k=0}^{r} ∪_{j=0}^{l} {k + j} = ∪_{t=0}^{min(r,l)} ∪_{i=0}^{l+r−2t} {t + i}. 4.5. Corollary.

Spp_{mod−2} : (GHV S(S^{1}), ⊕, ⊗_{s}) → (N[(R/2Z) × N], +, ∗)

is a monomorphism of semi-rings.

References

[1] Demazure, M.: Classification des germes a point critique isolé et a nombres de modules 0 ou 1 (d’apres Arnold). Sém. Bourbaki 26^{e} année, 1973/74, 443, (1974).

[2] Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Spriger–Verlag, 1972.

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

[4] Némethi, A.: The real Seifert form and the spectral pairs of isolated hypersurface singularities, submitted.

[5] Neumann, W.D.: Invariants of plane curve singularities. Monographie N^{∘}31 de L’Enseignement Math.(1983)

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

[7] Scherk,J. and Steenbrink, J.H.M.: On the Mixed Hodge Structure on the Cohomology of the Milnor Fibre. Math. Ann. 271, 641-665(1985)

[8] Steenbrink, J.H.M.: Mixed Hodge Structures on the Vanishing Cohomology. Nordic Summer School/NAVF, Symposium in Math. Oslo(1976)

[9] Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. Proc. of Symp. in Pure Math. 40, Part 2(1983)

IMAR, Bucharest, Romania. Current address:

The Ohio State University

Department of Mathematics

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Columbus, OH 43210-1174

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