CONVOLUTION THEOREM FOR NON-DEGENERATE

MAPS AND COMPOSITE SINGULARITIES

Tomohide Terasoma

Department of Mathematical Science, University of Tokyo, Komaba, Meguro, Tokyo, 153, Japan Max-Planck-Institut für Mathematik, Gottfried Claren Strasse 26, D-53225, Bonn, Germany.

§0 Introduction

Let n be a natural number greater than 1 and f = f(x) = f(x_{1}, …, x_{n}) be a germ of holomorphic function on C^{n} at 0. Let ϵ be a sufficiently small number such that f is defined on B(x) = {(x_{1}, …, x_{n}) ∈ C^{n} || x_{i} |< ϵ}. If δ′ is sufficiently small, the restriction of f : B(x) → C to f^{−1}(B^{0}(s)), where B^{0}(s) = {s ∈ C | 0 <| s |< δ′} is a fiber bundle over B^{0}(s). Therefore for 0 < δ < δ′, π_{1}(B^{0}(s)) = π_{1}(B^{0}(s), δ) acts on H^{i}(f^{−1}(δ), Q). ¿From now on, we use the notation f : B(x) → B(t) for short in this situation, and the morphism f is called a local morphism. If f has only isolated singularities, then H^{i}(f^{−1}(Q), Q) = 0 if i ≠ 0, n − 1. In the same way if m is a natural number greater than 1 and g is a germ of holomorphic function on C^{m} = {y = (y_{1}, …, y_{m})} at 0, a restriction of g defines a fiber bundle g^{−1}(B^{0}(t)) → B^{0}(t) over B^{0}(t). If f and g has only isolated singularities, u = f + g also has only isolated singularity. Let γ_{f} , γ_{g} and γ_{f+g} be the action of the canonical generator of π_{1}(B^{0}(s)), π_{1}(B^{0}(t)) and π_{1}(B^{0}(u)) respectively. Then relation between H^{n+m−1}((f +g)^{−1}(δ), Q), H^{n−1}(f^{−1}(δ), Q) and H^{m−1}(g^{−1}(δ), Q) is given by Thom-Sebastiani theorem [S-T]: There exists an isomorphism

H^{n+m−1}((f + g)^{−1}(δ), Q) ≃ H^{n−1}(f^{−1}(δ), Q) ⊗ H^{m−1}(g^{−1}(δ), Q).

compatible with the actions of the monodromies, i.e. γ_{f+g} = γ_{f} ⊗ γ_{g}.

The Hodge theoretic refinement of Thom-Sebastiani theorem is conjectured by Steenbrink [St], and proved by Varchenko [V], M.Saito [S] and Denef-Loeser [D-L] in different contexts. Let f be a germ of holomorphic function f : B(x) → B(t) such that the restriction of f to f^{−1}(B^{0}(t)) is smooth. After Steenbrink, the space H^{i}(f^{−1}(δ), Q) is equipped with the mixed Hodge structure for a sufficiently small δ and the action of γ_{f} is quasi-unipotent and preserves mixed Hodge structure. The minimal natural number m for which the action of γ^{m} is unipotent for H^{i}(f^{−1}(δ), Q) (0 ≤ i ≤ 2(n − 1) is called the exponent of f. If m is the exponent of f, then the action of N = \frac{1}{m} log(γ^{m}) satisfies NW_{k} ⊂ W_{k−2}. Therefore the action of γ on Gr_{k}^{W} (H^{i}(f^{−1}(δ), Q)) preserves the Hodge structure and has a finite

Typeset by AMS-TEX

image. Therefore [H^{i}(f^{−1}(δ), Q)] = ∑_{k}[Gr_{k}^{W} (H^{i}(f^{−1}(δ), Q))] defines an element of the Grothendieck group K_{MH}(C, μ_{m}) of mixed Hodge structures with an action of μ_{m}. For an element V in K_{MH}(C, μ_{m}), the invariant part and the complement of invariant part under the action of μ_{m} is denoted by V_{1} and V_{≠1}, respectively. The action of μ_{m} preserves the space H^{p,q}, where

H^{p,q} = F^{p}Gr_{k}^{W} (H^{i}(f^{−1}(δ), C)) ∩ F̄^{q}Gr_{k}^{W} (H^{i}(f^{−1}(δ), C))

and χ-part of H^{p,q} is denoted by H^{p,q}(χ), for a finite character χ of π_{1}(B^{0}(t), δ). The set of natural numbers {h^{p,q}(χ)}, where h^{p,q}(χ) = dim H^{p,q}(χ) is called the spectral pair of f and it is an important invariant for singularities. The Hodge analog of Thom-Sebastiani theorem is stated as follows. Let d_{1} and d_{2} be the exponent of f and g respectively and m be the l.c.m. of d_{1} and d_{2}. On the mixed Hodge structure V (d_{1}, d_{2}) = H^{1}({σ^{d_{1}} + τ^{d_{2}} = δ}, Q) the group μ_{d_{1}} × μ_{d_{2}} × π_{1}(B(u) − {0}, δ) acts and the exponent is equal to m = lcm(d_{1}, d_{2}). Therefore V (d_{1}, d_{2}) defines an element [V (d_{1}, d_{2})] in K_{MH}(C, μ_{d_{1}} × μ_{d_{2}} × μ_{m}).

Theorem 0.1 ([V], [S], [D-L]). Under the above notation, the following equality holds in the Grothendieck group K_{MH}(C, μ_{m}).

[H^{n+m−1}((f + g)^{−1}(δ), Q)]
= − ([V (d_{1}, d_{2})] ⊗ [H^{n−1}(f^{−1}(δ), Q)] ⊗ [H^{m−1}(g^{−1}(δ), Q)])^{μ_{d_{1}} ×μ_{d_{2}}}
+ [H^{n−1}(f^{−1}(δ), Q)(χ_{1})]_{≠1} ⊗ [H^{m−1}(g^{−1}(δ), Q)(χ_{2})]_{≠1}
− [H^{n−1}(f^{−1}(δ), Q)(χ_{1})] ⊗ [H^{m−1}(g^{−1}(δ), Q)(χ_{2})].

This theorem is generalized by Nemethi-Steenbrink [N-S] as follows. Let h(s, t) be a function of two variables with an isolated singularity. The composite h ∘ (f, g) of (f, g) : B(x) × B(y) → B(s) × B(t) and h : B(s) × B(t) → B(u) has singularities and it is called a composite singularity of two variables. The spectral pair of the composite singularity of two variables is computed by those of f, g and h in [N-S]. Note that if h(s, t) = s + t, then their theorem is nothing but Theorem 0.1. In this paper, we generalize this theorem for composite singularities of several variables. We explain the situation of our result.

Let n be a natural number greater than 1 and g_{i}(y_{i}) = g_{i}(y_{i1}, …, y_{im_{i}}) be germs of holomorphic functions of m_{i} variables for 1 ≤ i ≤ n. Then x_{i} = g_{i}(y_{i}) defines a local morphism g_{i} : B(y_{i}) → B(x_{i}). Suppose that the restriction of g_{i} to g_{i}^{−1}(B(x_{i}) − {0}) is smooth. Let f be a germ of holomorphic function on B(x) = {(x_{1}, …, x_{n}) || x_{i} |< δ(1 ≤ i ≤ n)} non-degenerate with respect to the Newton boundary. (See Section 1. for the definition of the non-degeneracy condition.) Roughly speaking, our main theorem (Theorem 3.6.1) gives the relation between the spectral pair of the composite singularity f(g_{1}(y_{1}), …, g_{n}(y_{n})) and those of g_{1}, …, g_{n} and f. This theorem is called the convolution theorem for composite singularities. Section 3 is devoted to prove this main theorem.

In the paper of [D-L] they proved the motivic version of Thom-Sebastiani theorem under the assumption of tameness and the existence of the resolution of singularities. Their theorem, Motivic Thom-Sebastiani theorem implies the l-adic analog of Thom-Sebastiani theorem. We prove the l-adic analog of the convolution theorem for composite singularities under the assumption of tameness condition in Section 2. Although the l-adic analog is not a deep theorem, it is worth giving a proof in this paper, because from this result, one can easily expect the formula for the spectral pairs for composite singularities.

§1 Non degenerate maps and reducing multiplicities

§1.1 Toric geometry

Let k be a finite field F_{q} of q elements or the complex number field C and n ≥ 1. Let f = f(x_{1}, …, x_{n}) ∈ k[[x_{1}, …, x_{n}]] be a formal power series on x_{1}, …, x_{n} over k such that f(0) = 0. We assume that for all i = 1, …, n, the coefficient of x_{i}^{m_{i}} is not zero for some m_{i}. The Newton polygon Δ = Δ(f) of f = ∑_{w∈N^{n}} a_{w}x^{w} is

defined by the convex hull of {w ∈ N^{n} | a_{w} ≠ 0} and R_{≥N}^{(i)} = {(0, …, r^{i}, …, 0) | r ≥ N} for a sufficiently large number N. First we recall the notion of nondegeneracy with respect to the Newton boundary. Let σ be a compact face of Δ, L_{σ} be the affine linear hull of σ in R^{n} and L_{σ,0} be the linear subspace of R^{n} (containing 0) parallel to L_{σ}. We define f_{σ} by f_{σ} = ∑_{w∈σ} a_{w}x^{w}. If we choose w_{0} ∈ σ, then f_{σ,0} = f_{σ}/x^{w_{0}} ∈ k[L_{σ,0} ∩ Z^{n}]. If σ is non-compact face, there exists a non-empty subset J of [1, n] such that σ is equal to ∑_{j∈J} R_{+}e_{j} outside of a compact subset. The scheme Spec(k[[x_{j}]]_{j∈J}[x_{j}^{−1}]_{j∈J}) is denoted by B_{σ}^{0}. The function f is said to be non-degenerate with respect to a compact face σ (resp. a non-compact face σ) if {f_{σ,0} = 0} is a smooth variety on Z_{σ} = Spec(k[L_{σ,0} ∩ Z^{n}]) (resp. B_{σ}^{0}). The series f is said to be non-degenerate with respect to the Newton boundary if it is non-degenerate with respect to all the faces σ of Δ. The closed subschemes {x_{i} = 0} and {f = 0} in B(x) = Spec(k[[x_{1}, …, x_{n}]]) are denoted by Z(x_{i}) and Z(f), respectively. If f is non-degenerate with respect to the Newton boundary, we can construct a partial resolution of the singularity of the hypersurface Z(f) ∪ ∪_{i=1}^{n}Z(x_{i}), using toric geometry. Let us define the dual fan F (Δ) of Δ as the set {σ^{∗}(x) | x ∈ Δ}, where σ^{∗}(x) = {l ∈ (R^{n})^{∗} | l(∑_{y∈Δ} R_{+}(y − x)) ≥ 0}. As in [Od], we can associate a toric variety X_{F} for a fan F. Note that the toric variety X_{F (Δ)} associated to F (Δ) is isomorphic to P_{Δ} defined in [Dan]. The dual coordinate of (R^{n})^{∗} is written by w^{∗} = (w_{1}^{∗}, …, w_{n}^{∗}). Let F be a simplicial refinement of F (Δ) such that

(1.1.1)  the restriction of F to the hyperplane {w_{i}^{∗} = 0} consists of

∑_{j∈J} R_{+}e_{j} with i ∉ J ⊂ [1, n].

The natural morphism from X_{F} to the affine space A^{n}(x) = Spec(k[x_{1}, …, x_{n}]) is denoted by π : X_{F} → A^{n}(x). The base change of π by the morphism B(x) = Spec(k[[x_{1}, …, x_{n}]]) → A^{n}(x) is denoted by b(x) : B̂(x) → B(x).

Now we recall the definitions of quasi-smooth varieties and quasi-normal crossing divisors. (See [St].) An integral variety V is called quasi-smooth at a point p ∈ V if there exists a local parameter x_{1}, …, x_{n} at p with the inclusion O_{V,p} ⊂ O_{V,p}[ξ_{1}, …, ξ_{n}], where ξ_{1}^{d_{i}} = x_{1},…,ξ_{1}^{d_{i}} = x_{1} such that (1) O_{V,p}[ξ_{1}, …, ξ_{n}] is smooth with the regular parameters ξ_{1}, …, ξ_{n}, and (2) O_{V,p} is identified with the invariant ring of O_{V,p}[ξ_{1}, …, ξ_{n}] under the action of a subgroup G of μ_{d_{1}} ×⋯×μ_{d_{n}}, where the action of μ_{d_{1}} × ⋯ × μ_{d_{n}} is given by the multiplication of coordinate (ξ_{1}, …, ξ_{n}). If V is quasi-smooth at all p ∈ V, V is called quasi-smooth variety. A divisor D in V is called a quasi-normal crossing divisor at p ∈ D if it is the image of ξ_{1} ⋯ ξ_{s} 0 < s ≤ n under the projection Spec(O_{V,p}[ξ_{1}, …, ξ_{n}]) → Spec(O_{V,p}). The divisor is called quasi-normal corssing if it is quasi-normal crossing at all p ∈ D.

Then it is known that X_{F} and B̂(x) are quasi-smooth, and the reduced part of (b(x))^{−1}(Z(f) ∪ ∪_{i=1}^{n}Z(x_{i})) is a quasi-normal crossing divisor.

Now we consider a covering B(ξ) of B(x). Let d_{1}, …, d_{n} be natural numbers such that μ_{d_{i}} ⊂ k^{×}. We introduce a system of coordinates ξ_{1}, …, ξ_{n} with ξ_{1}^{d_{1}} = x_{1}, …, ξ_{n}^{d_{n}} = x_{n}. Then B(ξ) = Spec(k[[ξ_{1}, …, ξ_{n}]]) is a finite Galois covering of B(x) with Galois group G = μ_{d_{1}} × ⋯ × μ_{d_{n}}. By the base chage of B̂(x) → B(x) by B(ξ) → B(x), we get a morphism b(ξ) : B̂(ξ) → B(ξ). The power series f defines a closed subscheme Z(f(ξ)) of B(ξ). Then B̂(ξ) is quasi-smooth and the reduced part of (b(ξ))^{−1}(Z(f(ξ)) ∪ ∪_{i=1}^{n}Z(ξ_{i})) is a quasi-normal crossing divisor:

((b(ξ))^{−1}(Z(f(ξ)) ∪ ∪_{i=1}^{n}Z(ξ_{i})))_{red} = ∪_{i=0}^{s+n}D_{i},

where D_{0} and D_{s+1}, …, D_{s+n} are proper transforms of Z(f(ξ)) and Z(ξ_{1}), …, Z(ξ_{n}) respectively and D_{1}, …, D_{s} correspond to 1-dimensional cones r_{1}, … r_{s} of F different from R_{+}e_{i}^{∗} (i = 1, …, n). Let B(t) = Spec(k[[t]]). The morphism B(x) → B(t) (resp. B(ξ) → B(t)) defined by t = f(x_{1}, …, x_{n}) (resp. t = f(ξ_{1}^{d_{1}}, …, ξ_{n}^{d_{n}})) is denoted by f(x) (resp. f(ξ)).

B̂(ξ) ⟶ B̂(x)
b(ξ)↓  ↓^{b(x)}
B(ξ) ⟶ B(x)
f(ξ)↓  ↓^{f(x)}
B(t) ⟶^{=} B(t)

Let f̂(ξ) = f(ξ) ∘ b(ξ). The multiplicities of D_{1}, …, D_{s} in b(ξ)^{−1}(Z(f(ξ))) = (f̂(ξ))^{−1}(0) are computed as follows. Let L(x) = Z^{n} and L(ξ) be the lattice defined by \frac{1}{d_{1}} Z ⊕ ⋯ ⊕ \frac{1}{d_{n}} Z. The primitive generator of r_{i} with respect to the dual lattice L(ξ)^{∗} is denoted by l_{i}. The multiplicity m_{i} of D_{i} in (b(ξ))^{−1}(Z(f(ξ))) is equal to

m_{i} = min{l_{i}(x) | x ∈ Δ}.

Definition 1.1.1. Let p be the characteristic of k and d_{1}, …, d_{n} be natural numbers prime to p. If there exists a simplicial refinement F of F (Δ) with the property (1.1.1) such that m = lcm(m_{i}) is prime to p, the n-tuple of natural numbers (d_{1}, …, d_{n}) is said to be tame with respect to f. The number m is called the exponent of (d_{1}, …, d_{n}) with respect to f. (We fix the simplicial refinement F of F (Δ) once and for all.)

§1.2 Reduction of multiplicity

We reduce the multiplicity of the special fiber of f(ξ) by the base change by the morphism B(τ) → B(t). We use the same notations of §1.1. Let m be lcm(d_{i}) and B(τ) = Spec(k[[τ]]), where τ^{m} = t. We assume (m, p) = 1, i.e. (d_{1}, …, d_{n}) is tame with respect to f. We construct a variety B̃(ξ) over B(τ) with the following commutative diagram:

B̃(ξ) ⟶^{π} B̂(ξ)
f̃↓  ↓f̂
B(τ) ⟶ B(t)

such that (1) f̃^{−1}(0) is reduced and quasi normal crossing and (2) π is a finite Galois covering with a Galois group μ_{m}.

Let F be a simplicial refinement of F (Δ). First we construct the suspension F̃ of F, which is a fan in (R^{n+1})^{∗}. The polar dual Δ^{∗} of the Newton polyhedron Δ is defined by

Δ^{∗} = {x^{∗} ∈ (R^{n})^{∗} | x^{∗}(Δ) ≥ 1}

Since F is a refinement of F (Δ), F defines a decomposition Δ^{∗} = ∐_{σ^{∗}∈F} σ^{∗} ∩ Δ^{∗} of Δ^{∗}. We define lower cone L(σ^{∗}), upper cone U(σ^{∗}) of σ^{∗} and boundary cone B(σ^{∗}) as

L(σ^{∗}) = R_{+}((σ^{∗} ∩ Δ^{∗}) × [0, \frac{1}{m}]), U(σ^{∗}) = (σ^{∗} × [0, ∞] − L(σ^{∗}))^{−},

B(σ^{∗}) = R_{+}((∂Δ^{∗} ∩ σ^{∗}), \frac{1}{m})

The base cone B is defined by B = (R^{n})^{∗} −∪_{σ^{∗}∈F} U(σ^{∗}). We define the suspension F̃ of F as

F̃ = {U(σ^{∗})}_{σ^{∗}∈F} ∪ {B(σ^{∗})}_{σ^{∗}∈F} ∪ {B}.

It is easy to see that if σ^{∗} is a simplicial cone, U(σ^{∗}) and B(σ^{∗}) are also simplicial cones. Even though if F is a simplicial fan, B may not be simplicial cone. Since the suspension F̃(Δ) of F (Δ) is the dual fan F (Δ̃) of the suspension Δ̃, where Δ̃ is the convex hull of 0×R_{≥m} and Δ×0. Therefore, in general, F̃ is a refinement of F (Δ̃). Now we consider an element f(ξ) − τ^{m} = f(ξ_{1}^{d_{1}}, …, ξ_{n}^{d_{n}}) − τ^{m} in k[[ξ_{1}, …, ξ_{n}, τ]]. Let B(ξ, τ) = Spec(k[[ξ_{1}, …, ξ_{n}, τ]]). The fiber product of XF̃ → A(ξ, τ) and B(ξ, τ) → A(ξ, τ) is denoted by B̄(ξ). Since Δ̃ is the Newton polygon of f(ξ)−τ^{m}, f(ξ) − τ^{m} defines a hypersurface B̃(ξ) = Z(f(ξ) − τ^{m}) in B̃(ξ). The composite of B̃(ξ) → B̄(ξ), B̄(ξ) → B(ξ, τ) and B(ξ, τ) → B(τ) is denoted by f̃(ξ).

Proposition 1.2.1. The variety Z(f(ξ) − u^{m}) does not contain the point corresponding to B in B̄(ξ) and quasi-smooth. The ( total ) fiber (f̃(ξ))^{−1}(0) of f̃ at 0 is reduced and quasi-normal crossing.

§2 Tame l-adic sheaves and convolution theorem.

§2.1 Grothendieck groups of etale sheaves.

In this section, we assume that k is a finite field F_{q} and d be an integer such that d | q − 1. The Grothendieck group of etale Q_{l}-sheaves on B(x) = Spec(k[[x]]), B^{0}(x) = Spec(k((x))) and Spec(k) is denoted by K(B(x)), K(B^{0}(x)) and K(k), respectively. The geometric generic point of B^{0}(x) is denoted by x̄. A sheaf F on B^{0}(x) corresponds to the continuous representation (ρ_{F} , Fx̄) of Gal(\overline{k((x))}/k((x))). By Grothendieck’s theorem, there exists an open subgroup H in the inertia group I of Gal(\overline{k((x))}/k((x))) which acts on Fx̄ unipotently. If there exists a subgroup H such that the index [I : H] is prime to p, it is called tame. The Grothendieck group of tame sheaves on B^{0}(x) is denoted by K(B^{0}(x))^{t}. For a tame sheaf F, min{[I : H] | ([I : H], p) = 1, and ρ_{F} |_{H} is unipotent } is called the exponent of F. The Grothendieck group of tame sheaves on B^{0}(x) whose exponents divide d is denoted by K(B^{0}(x))^{t,d}. Note that K(B^{0}(x))^{t,1} is nothing but the Grothendieck group of etale sheaves on B^{0}(x) whose inertia action is unipotent. Let us fix a generator γ of the tame quotient Ẑ(1)′ = ∏_{(p,l)=1} Z̄_{l}(1) of I. Then the logarithm N = log ρ_{F} (γ) of ρ_{F} (γ) acts on Fx̄ nilpotently. The action of N on Fx̄ defines a monodromy filtration W_{k}(Fx̄) on Fx̄ satisfying NW_{k}(Fx̄) ⊂ W_{k−2}(Fx̄). It is characterized by the property:

The k-th iteration N^{k} : Gr_{k}^{W} (Fx̄) → Gr_{−k}^{W}(Fx̄) of the homomorphism induced by N is an isomorphism.

If we define the primitive part P_{k} as the kernel of the homomorphism N^{k+1} : Gr_{k}^{W} (Fx̄) → Gr_{−k−2}^{W}(Fx̄), then F is equal to ∑_{k≥0} ∑_{0≤i≤k} N^{i}P_{k} in K(B^{0}(x))^{t,1}. Since N_{i}P_{k} ≃ P_{k} ⊗ Q_{l}(i) (0 ≤ i ≤ k) corresponds to an unramified representation of Gal(\overline{k((x))}/k((x))), we have the following Lemma.

Lemma 2.1.1. The group K(B^{0}(x))^{t,1} is isomorphic to the group K(k)

Let Π_{x}^{t} = Gal((∪_{(d,p)=1}k((x^{\frac{1}{d}})))⊗k̄/k((x))) be the tame quotient of the absolute Galois group of k((x)). Then we have the exact sequence:

1 → Ẑ(1)′ → Π_{x}^{t} → Ẑ → 1.

The Kummer extension k((τ)), where τ^{d} = x, defines a character Π_{x}^{t} → μ_{d} of Π_{x}^{t}. The kernel of Π_{x}^{t} → μ_{d} × Ẑ can be identified with dẐ(1)′. We fix a generator γ of dẐ(1)′. For a tame etale sheaf F, on B^{0}(x), whose exponent is divisible by d, the action of dẐ(1)′ is unipotent. Let N be the logarithm of ρ_{F} (γ). For any element h̄ ∈ μ_{d} × Ẑ take a lifting of h ∈ Π_{x}^{t}. Then we have hγh^{−1} = γ^{c(h)}, c(h) ∈ (Ẑ′)^{×} and as a consequence, we have hN = c(h)Nh. If we define a filtration W′_{i} = hW_{i}, then W′_{i} is stable under the action of N and the following diagram is commutative.

W_{i}/W_{i−1} ⟶^{N} W_{i−2}/W_{i−3}
c(h)·h↓  ↓^{h}
W′_{i}/W′_{i−1} ⟶^{N} W′_{i−2}/W′_{i−3}

By the characterization of W_{i}, we have W′_{i} = W_{i}. Therefore h induces a homomorphism on Gr_{i}^{W} Fx̄. Since N acts trivially on Gr_{i}^{W} Fx̄, the action of Π_{x}^{t} factors through μ_{d} × Ẑ. As a consequence, K(B^{0}(x))^{t,d} is equal to the Grothendieck group of continuous representations of μ_{d} × Ẑ. Let K(k, μ_{d}) be the Grothendieck group of etale sheaf on Spec(k) with a μ_{d} action. Therefore we have the following lemma.

Lemma 2.1.2. The group K(B^{0}(x))^{t,d} is isomorphic to the group K(k, μ_{d}).

We introduce a notion of equivariant sheaf. Let f : X → S be a scheme over S and G be a finite group acting on Y over S. The action of G on Y is denoted by σ : G → Aut(X/S). Let F be a sheaf on X. A descent data for F is a set of sheaf homomorphisms ϕ_{g} : g^{∗}F → F indexed by elements of G satisfying the 1-cocycle condition ϕ_{gh} = h^{∗}ϕ_{g} ∘ ϕ_{h}. The descent data is called effective if for any x ∈ X and g ∈ G such that g(x) = x, the fiber of the descent data ϕ_{g} |_{x}: (g^{∗}F)_{x} = F_{g(x)} → F_{x} is the identity map. (Note that this terminology is different from usual one.) A sheaf on X with an effective descent data is called (G, σ)-sheaf on X. The Grothendieck group of (G, σ)-sheaf on X is denoted by K(X/S, (G, σ)). Let F be a (X, σ)-sheaf on X. Then the higher direct image sheaves R^{i}f_{∗}F is a sheaf on S with the action of G. Suppose that there exists a quotient scheme Y = X/G of X under the action of G. The structure morphism Y → S is denoted by g and the natural map X → Y is denoted by π. For an etale sheaf G, the pull back π^{∗}G of G by the morphism π has a natural effective descent data. It is known that (R^{i}f_{∗}(π^{∗}G))^{G} ≃ R^{i}g_{∗}G.

Lemma 2.1.3. The group K(B^{0}(x))^{t,d} is isomorphic to the Grothendieck group K(B^{0}(τ), (μ_{d}, σ)) of (μ_{d}, σ)-sheaf on B^{0}(τ).

Remark 2.1.4. This isomorphism depends on the choice of the uniformizer x of k((x)) which is always fixed in our context.

Let F be an etale sheaf on B(x). The sheaf F is said to be tame if the restriction to B^{0}(x) is tame. For a tame sheaf F, the exponent of F is defined by that of the restriction to B^{0}(x). The Grothendieck group of tame sheaf and tame sheaf whose exponent divides d is denoted by K(B(x))^{t} and K(B(x))^{t,d}, respectively. For an etale sheaf F on B(x), the generic geometric fiber and special geometric fiber are denoted by Fx̄ and F¯0 respectively. The specialization map is a Gal(k̄/k)- equivariant map

(1)  sp_{F} : F¯0 → F I ; x̄,

where I is the inertia group of Gal(\overline{k((x))}/k((x))). To give an etale sheaf on B(x) is equivalent to give a triple (Fx̄, F¯0, sp_{F}), where Fx̄ and F¯0 are Gal(\overline{k((x))}/k((x)))- module and Gal(k̄/k)-module respectively and sp_{F} is a Gal(k̄/k)-equivariant homomorphism (1). By considering exact sequence,

0 → j_{!}j^{∗}F → F → i_{∗}i^{∗}F → 0,

we have [F] = [j_{!}j^{∗}F] + [i_{∗}i^{∗}F] in K(B(x)). Since (j_{!}j^{∗}F)¯0 = 0 and (i_{∗}i^{∗}F)x̄ = 0, we have

K(B(x)) = K(B^{0}(x)) ⊕ K(k), K(B(x))^{t} = K(B^{0}(x))^{t} ⊕ K(k),
K(B(x))^{t,d} = K(B^{0}(x))^{t,d} ⊕ K(k).

Now we consider several Grothendieck groups of etale sheaves on B(x) = Spec(k[[x_{1}, …, x_{n}]]). For a subset I of [1, n], we define a closed subscheme B_{I}(x) = Spec(k[[x_{1}, …, x_{n}]]/(x_{i})_{i∈I}) and locally closed subscheme B_{I}^{0}(x) = B_{I}(x)− ∪_{J⊋I}B_{J}(x). A sheaf is said to be stratified by coordinate if the restriction to B_{I}^{0} is locally constant for all I. An etale sheaf F on B(x) stratified by coordinate is called tame if the monodromy of the restriction of F to B_{I}^{0} is tame. The Grothendieck group of tame etale sheaves on B(x) stratified by coordinate is denoted by K_{c}(B(x))^{t}. The tame fundamental group of B_{I}^{0}(x) is denoted by Π_{I}^{t}. Then we have the following exact sequence.

1 → (Ẑ(1)′)^{I^{c}} → Π_{I}^{t} → Ẑ → 1

Let I ⊂ [1, n] and i ∉ I. We set J = I ∪ i and fix a generator e_{i} of geometric monodromy along the divisor x_{i} = 0. Then we have 1 → Ẑ(1)′e_{i} → Π_{I}^{t} → Π_{J}^{t} → 1.

The Grothendieck group of tame etale Q_{l}-sheaves stratified by coordinate whose exponent for e_{i} divides d_{i} is denoted by K_{c}(B(x))^{t,d_{1},...,d_{n}}. The generic geometric point of B_{I}^{0}(x) is denoted by x̄_{I}. The specialization map with respect to I and J is a Π_{J}^{t}-equivariant map sp_{I,J} : Fx̄_{J} → F e_{i} ; x̄_{I}. Let K(Π_{I}^{t}) be the Grothendieck group of continuous representations of Π_{I}^{t}. By the same argument as in the 1-variable case, we have

K_{c}(B(x))^{t} = ⊕_{I⊂[1,n]}K(Π_{I}^{t}).

If F_{1}, …, F_{n} are tame sheaves on B(x_{1}), …, B(x_{n}), then the exterior product F_{1} ⊠ ⋯ ⊠ F_{n} = pr_{1}^{∗}F_{1} ⊗ ⋯ ⊗ pr_{n}^{∗}F_{n} is a tame sheaf on B(x) stratified by coordinate.

Lemma 2.1.5. By attaching [F_{1}] ⊗ ⋯ ⊗ [F_{n}] to F_{1} ⊠ ⋯ ⊠ F_{n}, we get a homomorphism

K(B(x_{1}))^{t} ⊗ ⋯ ⊗ K(B(x_{n}))^{t} → K_{c}(B(x))^{t}
K(B(x_{1}))^{t,d_{1}} ⊗ ⋯ ⊗ K(B(x_{n}))^{t,d_{n}} → K_{c}(B(x))^{t,d_{1},...,d_{n}}.

Proof. It is enough to prove that if 0 → F′_{1} → F_{1} → F′′_{1} → 0 is an exact sequence of etale sheaves on B(x_{1}),

0 → F′_{1} ⊠ F_{2} ⊠ ⋯ ⊠ F_{n} → F_{1} ⊠ F_{2} ⊠ ⋯ ⊠ F_{n} → F′′_{1} ⊠ F_{2} ⊠ ⋯ ⊠ F_{n} → 0

is an exact sequence of tame etale sheaves on B(x). It is clear by the description of tame etale sheaves on B(x) stratified by coordinate.

Proposition 2.1.6.

(1) K_{c}(B(ξ), (G, σ_{ξ}))^{t,1} ≃ ⊕_{I⊂[1,n]}K(B_{I}^{0}(ξ), (G_{I}, σ_{I}))^{t,1}, where G_{I} is the image of G in Aut(B_{I}^{0}(ξ)) and σ_{I} be the action of G_{I} on B_{I}^{0}(ξ), i.e. G_{I} = ∏_{i∉I} μ_{d_{i}}.

(2) K_{c}(B_{I}^{0}(ξ), (G_{I}, σ_{I}))^{t,1} ≃ K(k, G_{I}).

§2.2 Vanishing cycle functors and convolution theorem

Let I be a subset of [1, n]. Then the closed subscheme B_{I}(x) of B(x) is defined by B_{I}(x) = {x ∈ B(x) | x_{i} = 0 (i ∈ I)}. Note that B_{∅}(x) = B(x). Let [F_{1}] ⊗ ⋯ ⊗ [F_{n}] be elements of K(B(x_{1}))^{t} ⊗ ⋯ ⊗ K(B(x_{n}))^{t}, respectively. We define a homomorphism ϕ_{I} : K(B(x_{1}))^{t,d_{1}} ⊗ ⋯ ⊗ K(B(x_{n}))^{t,d_{n}} → K(B^{0}(t)) as

ϕ_{I}([F_{1}] ⊗ ⋯ ⊗ [F_{n}]) = [j_{t}^{∗}(Rf(x)_{∗}i_{I}_{∗}i_{I}^{∗}(F_{1} ⊠ ⋯ ⊠ F_{n}))],

where j_{t} : B^{0}(t) → B(t) and i_{I} : B_{I}(x) → B(x) be the natural inclusions. Let B(ξ_{i}) → B(x_{i}) be the μ_{d_{i}}-covering defined in §1.1 and G be the group μ_{d_{1}} × ⋯ μ_{d_{n}} which acts on B(ξ). The action is denoted be σ_{ξ}. We consider the equivariant version

ϕ̃_{I} : K(B(ξ_{1}), (μ_{d_{1}}, σ_{1}))^{t} ⊗ ⋯ ⊗ K(B(ξ_{n}), (μ_{d_{n}}, σ_{n}))^{t} → K(B^{0}(t), G),

of ϕ_{I} by

ϕ̃_{I}([F̃_{1}] ⊗ ⋯ ⊗ [F̃_{n}]) = [j_{t}^{∗}Rf(ξ)_{∗}i_{I∗}i_{I}^{∗}(F̃_{1} ⊠ ⋯ ⊠ F̃_{n})],

for an element [F̃_{1}] ⊗ ⋯ ⊗ [F̃_{n}] ∈ K(B(ξ_{1}), (μ_{d_{1}}, σ_{1}))^{t} ⊗ ⋯ ⊗ K(B(ξ_{n}), (μ_{d_{n}}, σ_{n}))^{t}. Then we have the following commutative diagram.

(2.2.1)

K(B(x_{1}))^{t,d_{1}} ⊗ ⋯ ⊗ K(B(x_{n}))^{t,d_{n}}  ⟶^{ϕ_{I}} K(B^{0}(t))
↓  ↑ ^{G-invariant}
K(B(ξ_{1}), (μ_{d_{1}}, σ_{1}))^{t,1} ⊗ ⋯ ⊗ K(B(ξ_{n}), (μ_{d_{n}}, σ_{n}))^{t,1} ⟶_{ϕ̃_{I}} K(B^{0}(t), G)

First we prove the following proposition.

Proposition 2.2.1. The image of ϕ_{I} is contained in K(B^{0}(t))^{t,m}, where m is the exponent of (d_{1}, …, d_{n}) with respect to f.

Proof. We consider the following commutative diagram:

B̃(ξ) ⟶^{ν} B̂(ξ) ⟶^{b} B(ξ)
f̃(ξ)↓  ↓^{f̂(ξ)}
B(τ) ⟶_{ν′} B(t).

For tame etale sheaves F_{1}, …, F_{n} on B(x_{1}), …, B(x_{n}) whose exponent divides d_{1}, …, d_{n}, the sheaf π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n} are (G, σ_{ξ})-sheaf on B(ξ) with unipotent monodromy. Therefore there exists a filtration F^{i} of (G, σ_{ξ})-sheaves on π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n} such that F^{i}/F^{i+1} has trivial monodromy. Therefore the subquotient (b ∘ ν)^{∗}F^{i}/(b ∘ ν)^{∗}F^{i+1} of (b ∘ ν)^{∗}(π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n}) has also trivial monodromy. Using weight spectral sequence for f̃, j_{τ}^{∗}R^{i}f̃(ξ)_{∗}((b ∘ ν)^{∗}(π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n})) has unipotent monodromy. Since

j_{τ}^{∗}R^{i}f̃(ξ)_{∗}((b ∘ ν)^{∗}(π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n})) ≃ (ν′)^{∗}j_{t}^{∗}R^{i}f(ξ)_{∗}b^{∗}(π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n}),

and the commutative diagram (2.2.1), we have

[j_{t}^{∗}R^{i}f(x)_{∗}(F_{1} ⊠ ⋯ ⊠ F_{n})] = [j_{t}^{∗}R^{i}f(ξ)_{∗}b^{∗}(π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n})]^{G} ∈ K(B^{0}(t))^{t,m}

Definition 2.2.2. Put F = [F_{1}] ⊗ ⋯ ⊗ [F_{n}].

(1)

Φ_{f}(F) = ϕ_{∅}(F) − [(F_{1} ⊠ ⋯ ⊠ F_{n})¯0] ∈ K(k, μ_{m}).

Here we use the identification, ϕ_{∅}(F) ∈ K(B(t)^{0})^{t,m} ≃ K(k, μ_{m}) and [(F_{1} ⊠

⋯ ⊠ F_{n})¯0] ∈ K(k) = K(k, 1) ⊂ K(k, μ_{m}).

(2)

Φ([F_{i}]) = [j_{x_{i}}^{∗}F_{i}] − [F_{i,¯0}] ∈ K(k, μ_{d_{i}}) and
χ̃_{I}(F) = ⊗_{i∉I}[F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}Φ([F_{i}]) ∈ K(k, G),
χ_{I}(F) = ⊗_{i∉I}[F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}Φ([F_{i}])^{μ_{d_{i}}} ∈ K(k, G_{I}),

(3)

Φ̃_{I}(F) =ϕ̃_{I}([π_{1}^{∗}F_{1}] ⊗ ⋯ ⊗ [π_{n}^{∗}F_{n}]) − [(F_{1} ⊠ ⋯ ⊠ F_{n})¯0]
∈ K(B^{0}(t), (G_{I}, σ_{I}))^{t,m} ≃ K(k, G_{I} × μ_{m}),
and Φ̃_{I} = Φ̃_{I}([Q_{l}] ⊗ ⋯ ⊗ [Q_{l}]).

Theorem 2.2.3 (l-adic convolution theorem). Let F = [F_{1}] ⊗ ⋯ ⊗ [F_{n}] in K(B(x_{1}))^{t,d_{1}} ⊗ ⋯ ⊗ K(B(x_{n}))^{t,d_{n}}. Under the notation defined as above,

Φ_{f}(F) = ∑_{I⊂[1,n]}(−1)^{#I}(Φ̃_{I} ⊗ χ_{I}(F))^{G_{I}} ∈ K(μ_{m}, k).

Proof. Since the action of G on Φ̃_{I} factors through G_{I}, it is enough to show that

Φ̃_{f}(F) = ∑_{I⊂[1,n]}(−1)^{#I}(Φ̃_{I} ⊗ χ̃_{I}(F)) ∈ K(G × μ_{m}, k),

by the commutative diagram (2.2.1). Let F_{B} = [π_{1}^{∗}F_{1} ⊠ ⋯ ⊠ π_{n}^{∗}F_{n}]. Using decomposition F_{B} = ∑_{I⊂[1,n]} j_{I}_{!}(F_{B} |_{B_{I}^{0}(ξ)}), where j_{I} : B_{I}^{0}(ξ) → B(ξ) is the natural inclusion. Since F_{B} |_{B_{I}^{0}(ξ)}∈ K(B_{I}^{0}(ξ), (G_{I}, σ_{I}))^{t,1}, we have

j_{I}_{!}(F_{B} |_{B_{I}^{0}(ξ)}) = [j_{I}_{!}Q_{l}] ⊗ ⊗_{i∉I}[F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}[F_{i,¯0}] ∈ K_{c}(B(ξ), (G, σ_{ξ}))^{t,1}.

Here the (G, σ_{ξ})-structure on the right hand side is given by the diagonal action of (G_{I}, σ_{I}) on ⊗_{i∉I}[F_{i,x̄_{i}}] and [j_{I}_{!}Q_{l}]. Using equality [j_{I}_{!}Q_{l}] = ∑_{I⊂K}(−1)^{#(K−I)}[i_{K}_{!}Q_{l}], where i_{K} : B_{K}(ξ) → B(ξ) is the natural inclusion, we have

F_{B} = ∑_{K⊃I}(−1)^{#(K−I)}[i_{K}_{!}Q_{l}] ⊗ ⊗_{i∉I}[F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}[F_{i,¯0}]
= ∑_{K⊂[1,n]}(−1)^{#K}[i_{K}_{!}Q_{l}] ⊗ ( ∑_{K⊃I}(−1)^{#I} ⊗_{i∉I} [F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}[F_{i,¯0}]).

Since

∑_{K⊃I}(−1)^{#I} ⊗_{i∉I} [F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}[F_{i,¯0}]

= ⊗_{i∉K} [F_{i,x̄_{i}}] ⊗ ( ∑_{K⊃I}(−1)^{#I} ⊗_{i∉K−I} [F_{i,x̄_{i}}] ⊗ ⊗_{i∈I}[F_{i,¯0})]
= ⊗_{i∉K} [F_{i,x̄_{i}}] ⊗ ⊗_{i∈K}Φ([F_{i}])
=χ̃_{K}(F),

we have

Φ̃_{f}(F) = ∑_{K⊂[1,n]}(−1)^{#K} Φ̃_{f}([i_{K}_{!}Q_{l}]) ⊗ χ̃_{K}(F)

= ∑_{K⊂[1,n]}(−1)^{#K} Φ̃_{K} ⊗ χ̃_{K}(F).

This completes the proof of the theorem.

We apply Theorem 2.2.3 to composite singularities. Let m_{1}, …, m_{n} be positive integers, y_{i} = (y_{i1}, …, y_{im_{i}}) sets of coordinates. We define B(y_{i}) = Spec(k[[y_{i1}, …, y_{im_{i}}]]). Let g_{i} = g_{i}(y_{i}) be a formal power series of {y_{ij}}_{j} with no constant term. Then g_{i} defines a homomorphism B(y_{i}) → B(x_{i}) by x_{i} = g_{i}(y_{i}). We assume that g_{i} is smooth on B(y_{i}) − g_{i}^{−1}(0) and the exponent d_{i} of g_{i} divides #k^{×}. The fiber product of g_{i} defines a morphism from B(y) = ∏_{i=1}^{n} B(y_{i}) to B(x) = ∏_{i=1}^{n} B(x_{i}) and it is denoted by g. We assume that the exponent m of (d_{1}, …, d_{n}) with respect to f divides #k^{×}.

Lemma 2.2.4. The composite f ∘ g : B(y) → B(x) → B(t) is smooth over B^{0}(t).

Let χ_{g,K}(Q_{l}) = χ_{K}([Rg_{1∗}Q_{l}] ⊗ ⋯ ⊗ [Rg_{n∗}Q_{l}]) and

Φ_{f∘g}(Q_{l}) = Φ_{f}([Rg_{1∗}Q_{l}] ⊗ ⋯ ⊗ [Rg_{n∗}Q_{l}]) = [j_{t}^{∗}R(f ∘ g)_{∗}Q_{l}] − [Q_{l}].

Then the following theorem is a direct consequence of Theorem 2.2.3.

Theorem 2.2.5 (l-adic convolution theorem for composite singularities).

Φ_{f∘g}(Q_{l}) = ∑_{I⊂[1,n]}(−1)^{#I}(Φ̃_{I} ⊗ χ_{g,I}(Q_{l}))^{G_{I}} .

Remark 2.2.6. If we apply this theorem for the function f(x_{1}, x_{2}) = x_{1} + x_{2}, it is nothing but the l-adic version of Thom-Sebastiani theorem. (See also Corollary 3.6.2.) It is proved in [L-D] assuming the existence of resolution of singularities. Here we only assume the tameness of the singularity for g_{1} = 0 and g_{2} = 0.

The rest of this paper is devoted to prove the Hodge analog of Theorem 2.2.5.

§3 Mixed Hodge structure of composite singularities

§3.1 Local space

Let N be an analytic space and X and Y be closed subspaces such that X is a projective algebraic variety. The triple (N, X, Y) is called a local space if

(1) Any irreducible component of Y is not contained in X.

(2) N − (X ∪ Y) is a smooth variety

(3) There exists a proper bimeromorphic map b : Ñ → N such that b is an isomorphism on b^{−1}(N − (X ∪ Y)) and b^{−1}(X ∪ Y) is a normal crossing divisor of Ñ.

The irreducible decomposition of the proper transform Ỹ of Y is written by Ỹ = ∪_{i∈I} Ỹ_{i}. For a subset J of I, we define Ỹ_{J} = ∩_{j∈J} Ỹ_{j}.

(4) For any subset J of I, Ỹ_{J} retracts to Ỹ_{J} ∩ X̃, where X̃ = b^{−1}(X). (In

particular, Ñ retracts to the subvariety X̃.) Note that for a triple with the properties (1)-(3), there exists a sufficiently small neighborhood N′ of X such that (N′, X, Y ∩ N′′) is a local space. We introduce an equivalence relation on the set of local spaces by (N, X, Y) ∼ (N′, X, Y ∩ N′), where N′ is a sufficiently small neighborhood of X. ¿From now on, equivalent local spaces are always identified.

Example 3.1.1. Let B(x) = {(x_{1}, …, x_{n}) || x_{i} |< ϵ} and f(x_{1}, …, x_{n}) be a holomorphic function on B(x) to B(t) = {t || t |< ϵ}. Then for a sufficiently small ϵ, (B(x), 0, f^{−1}(0)) is a local space.

In this section, we introduce a mixed Hodge structure on the cohomology H^{i}(N − (X ∪ Y), Q). To introduce a Hodge structure, we fix a resolution of singularities Ñ → N of (3). Let Ỹ = ∪_{i∈I} Ỹ_{i} be the irreducible decomposition of the proper transform Ỹ of Y and X̃ = ∪X̃_{i} be the irreducible decomposition of b^{−1}(X). Put Ỹ = ∪_{i}Ỹ_{i}. Then (Ñ, X̃, Ỹ) is also a local space. First we construct a mixed Hodge structure on H^{i}(Ñ − (X̃ ∪ Ỹ), Q) = H^{i}(N − (X ∪ Y), Q). Later we prove that it is independent of the choice of the resolution of singularities. The local space (N, X, Y) is called normal crossing if N is smooth and X ∪ Y is a normal crossing divisor in N. We assume that (N, X, Y) is a normal crossing local space and X = ∪_{i=1}^{q}X_{i} and Y = ∪_{i=1}^{p}Y_{i} are the irreducible decomposition. We can define the logarithmic de Rham complex Ω_{N}^{•}(log D) as in [Del]. By the result of [Del], there exist quasi-isomorphisms Rj_{∗}C → j_{∗}Ω_{N−D}^{•} and Ω_{N}(log D) → j_{∗}Ω_{N−D}^{•}, where j : N − D → N is the natural inclusion. Let i_{I} : X_{I} → X be the natural inclusion. If I ⊂ J, then there exists a natural morphism (i_{I})_{∗}Rj_{∗}Q → (i_{J})_{∗}Rj_{∗}Q.

Let N be an analytic manifold and D = ∪_{i}D_{i} be a simple normal crossing divisor. For a subset J of I, we put D_{J} = ∩_{j∈J}D_{j} and the ideal sheaf of D_{J} is denoted by I_{J}. The sheaf of logarithmic differential forms Ω_{N}^{i} (log D) is a locally free sheaf on N and I_{D_{J}} ⊗ Ω_{N}^{i} (log D) is identified with a subsheaf of Ω_{N}^{i} (log D). If z_{i} is a local equation of D_{j} and ω ∈ Ω_{N}^{i} (log D), then we have d(z_{ω}) = z_{i}(\frac{dz_{j}}{z_{j}} ∧ ω) + z_{j}dω, therefore I_{D_{J}} ⊗ Ω_{N}^{•} (log D) is a sub-complex of Ω_{N}^{•} (log D). Therefore the quotient sheaves Ω_{N}^{i} (log D) ⊗ O_{D_{J}} forms a complex of sheaves Ω_{N}^{•} (log D) ⊗ O_{D_{J}}.

Lemma 3.1.2. Let j : N − D → N be the natural inclusion. Then there exists a quasi-isomorphism

Rj_{∗}C |_{X_{J}} ≃ Ω_{N}^{•} (log D) ⊗ O_{X_{J}} .

Proof. First consider the following diagram where all the morphisms are filtered quasi isomorphisms;

(3.1.1) (Rj_{∗}C, τ_{•}) → (Rj_{∗}Ω_{N −D}, τ_{•}) ⟵ (Ω_{N} (log D), τ_{•}) → (Ω_{N} (log D), W_{•})

Let W_{k}(Ω_{N} (log D) ⊗ O_{D_{J}}) = Im(W_{k}(Ω_{N} (log D)) → Ω_{N} (log D) ⊗ O_{D_{J}}). Then we have the following lemma.

Lemma 3.1.3.

(1) The Poincare residue induces the isomorphism

Gr_{k}^{W} (Ω_{N}^{i} (log D) ⊗ O_{D_{J}}) ≃ ⊕_{#K=k}Ω_{D_{J∪K}}^{i−k}

(2) The natural morphism π : (Ω_{N}^{•} (log D) |_{D_{J}} , W_{•}) → (Ω_{N}^{•} (log D) ⊗ O_{D_{J}} , W_{•}) is a filtered quasi isomorphism

Proof. (1) is well known and easy to check.

(2) The associated graded morphism of π is equal to

⊕_{#K=k}Ω_{D_{K}}^{•} |_{D_{J∪K}} [−k] → ⊕_{#K=k}Ω_{D_{J∪K}}^{•} [−k]

and they are quasi-isomorphic to C_{D_{J∪K}}. This proves the lemma 3.1.3.

By (3.1.1) and Lemma 3.1.3 (2), we have Lemma 3.1.2.

Now we return to the situation of a normal crossing local space (N, X, Y). We define the following complexes K_{Q} = K_{Q}(N, X, Y) and K_{C} = K_{C}(N, X, Y) on X.

K_{Q} : ⊕_{#I=1}(i_{I})_{∗}Rj_{∗}Q |_{X_{I}} → ⊕_{#I=2}(i_{I})_{∗}Rj_{∗}Q |_{X_{I}} → ⋯

and

K_{C} : ⊕_{#I=1}(i_{I})_{∗}Ω_{N}^{•} (log(D)) ⊗ O_{X_{I}} → ⊕_{#I=2}(i_{I})_{∗}Ω_{N}^{•} (log(D)) ⊗ O_{X_{I}} → ⋯

The following lemma is a direct consequence of Lemma 3.1.2.

Lemma 3.1.4. There exists a quasi isomorphism of complexes of sheaves:

Rj_{∗}Q |_{X}→ K_{Q}, K_{Q} → K_{C}.

Definition 3.1.5. We introduce a weight filtration W on K_{Q}, K_{C}, and a Hodge filtration F on K_{C} as follows.

W_{k}K_{Q} : ⊕_{#I=1} τ_{k}Rj_{∗}Q |_{X_{I}} → ⊕_{#I=2}τ_{k+1}Rj_{∗}Q |_{X_{I}}
→ ⋯ → ⊕_{#I=j}τ_{k+j−1}Rj_{I}_{∗}Q → ⋯

W_{k}K_{C} : ⊕_{#I=1} W_{k}Ω_{N}^{•}(log(D)) ⊗ O_{X_{I}} → ⊕_{#I=2}W_{k+1}Ω_{N}^{•}(log D) ⊗ O_{X_{I}}
→ ⋯ → ⊕_{#I=j}W_{k+j−1}Ω_{N}^{•}(log D) ⊗ O_{X_{I}} → ⋯

F^{p}K_{C} : ⊕_{#I=1} σ^{p}Ω_{D}^{•}(log(D)) ⊗ O_{X_{I}} → ⊕_{#I=2}σ^{p}Ω_{N}^{•}(log D) ⊗ O_{X_{I}}
→ ⋯ → ⊕_{#I=j}σ^{p}Ω_{N}^{•}(log D) ⊗ O_{X_{I}} → ⋯,

where τ_{•} and σ^{•} are the canonical and the stupid filtration, respectively.

These filtration on the complex of sheaves induces the weight filtration W on RΓ(X, K_{Q}) and RΓ(K, K_{C}) and the Hodge filtration F on RΓ(X, K_{C}) up to filtered quasi-isomorphism.

Definition 3.1.6. The pair ((L_{Q}, W), (L_{C}, W, F)) of filtered complex of sheaves and bi-filtered complex of sheaves on X is a cohomological mixed Hodge complex (=CMHC for short), if

(1) (L_{Q}, W) ⊗ C and (L_{C}, W) are filtered quasi-isomorphism.

(2) The spectral sequence for the filtered complex of sheaves (RΓ(X, Gr_{k}^{W} (L_{C})), Gr_{k}^{W} (F)) on X degenerates at E_{1}-term.

(3) The filtration Gr_{k}^{W} (F^{p}) defines a Hodge structure of weight m + k on H^{m}(X, Gr_{k}^{W} (L_{C})).

Theorem 3.1.7. The pair ((K_{Q}, W), (K_{C}, W, F)) defines a CMHC on X.

It is sufficient to prove that Gr_{k}^{W} (K_{C}) is a cohomological Hodge complex of weight k, where Gr_{k}^{W} (K_{C}) is given by the double complex:

⊕_{#I=k,#J=1}Ω_{Y_{I}∩X_{I}}^{•} [−k](−k) →^{0} ⊕_{#I=k,#J=2}Ω_{Y_{I}∩X_{I}}^{•} [−k − 1](−k − 1) →^{0}

Moreover, the associated graded module Gr_{F}^{p} Gr_{k}^{W} (K_{C}) for the induced filtration Gr_{k}^{W} (F^{p}) is equal to

⊕_{#I=k,#J=1}Ω_{Y_{I}∩X_{I}}^{p−k} [−p](−k) →^{0} ⊕_{#I=k,#J=2}Ω_{Y_{I}∩X_{I}}^{p−k−1}[−p](−k − 1) →^{0}

Therefore the m-th hypercohomology of Gr_{F}^{p} Gr_{k}^{W} (K_{C}) is equal to

⊕_{j≥1} ⊕_{#I=k+j−1,#J=j} H^{m−p−j+1}(Ω_{Y_{I}∩X_{J}}^{p−k−j+1})(−k − j + 1).

Therefore by the classical Hodge theory the following spectral sequence degenerates at the E_{1}-term.

E_{1}^{p,q} = H^{p+q}(X, Gr_{F}^{p} Gr_{k}^{W} (K_{C})) → E_{∞}^{p+q} = H^{p+q}(X, Gr_{k}^{W} (K_{C})),

and the filtration defined by this spectral sequence defines a pure Hodge structure of weight m + k on H^{m}(X, Gr_{k}^{W} (K_{C})).

We prove the independence of the choice of the resolution of singularity. We use the same argument as [Del]. Let (N_{1}, X_{1}, Y_{1}) and (N_{2}, X_{2}, Y_{2}) be normal crossing local spaces, (N, X, Y) a local space and f_{i} : N_{i} → N be proper morphisms for i = 1, 2 such that f^{−1}(X ∪ Y) = X_{i} ∪ Y_{i} and f_{i} |_{N_{i}−(X_{i}∪Y_{i})}: N_{i} − (X_{i} ∪ Y_{i}) → N − (X ∪ Y) are isomorphic for i = 1, 2. Then by taking a fiber product of f_{1}, f_{2} and resolving it, there exists the third normal crossing local space (N_{3}, X_{3}, Y_{3}) and proper morphism g_{i} : (N_{3}, X_{3}, Y_{3}) → (N_{i}, X_{i}, Y_{i}) such that g_{i}^{−1}(X_{i} ∪ Y_{i}) = X_{3} ∪ Y_{3} and the restrictions of g_{i} to N_{3} − (X_{3} ∪ Y_{3}) are isomorphisms.

(N_{3}, X_{3}, Y_{3}) ⟶ (N_{1}, X_{1}, Y_{1})
↓  ↓
(N_{2}, X_{2}, Y_{2}) ⟶ (N, X, Y)

The morphism g_{i} induces a homomorphism from the mixed Hodge structure of (N_{3}, X_{3}, Y_{3}) to that of (N_{i}, X_{i}, Y_{i}) which is bijective. Since the category of mixed Hodge structures is an abelian category, this gives rise to the isomorphism of mixed Hodge structures. In the same way, we can prove the following functoriality of the mixed Hodge structures of local spaces.

Proposition 3.1.8. Let (N_{1}, X_{1}, Y_{1}) and (N_{2}, X_{2}, Y_{2}) be local spaces and f : N_{1} → N_{2} be a morphism such that f(N_{1} − (X_{1} ∪ Y_{1})) ⊂ N_{2} − (X_{2} ∪ Y_{2}) and f |_{N_{1}−(X_{1}∪Y_{1})} is an immersion. Then f induces a homomorphism of mixed Hodge structures of local spaces.

Proof. We reduce the proof to the case where both (N_{1}, X_{1}, Y_{1}) and (N_{2}, X_{2}, Y_{2}) are normal crossing local spaces. In this case, the proof is exactly the same as [Del].

Remark 3.1.9. Let (N, X, Y) be a normal crossing local space and X_{1} be a closed subvariety of X, N_{1} be a neighborhood of X_{1}, and Y_{1} be the closure of (X ∪ Y) ∩ N_{1} − X_{1}. Then (N_{1}, X_{1}, Y_{1}) is a local space and the open immersion N_{1} ⊂ N satisfies the condition (1).

Corollary 3.1.10. Let (N, X, Y) be a local space and W be a subvariety of X. Then H_{c}^{i}(W, Rj_{∗}Q) has a natural mixed Hodge structure.

Proof. Let Z_{1} be the closure of W in X and put Z_{2} = Z_{1} −W. The open immersion from W to Z_{1} is denoted by k : W → Z_{1}. Since

H_{c}^{i}(W, Rj_{∗}Q) ≃ H^{i}(Z_{1}, k_{!}(Rj_{∗}Q |_{W}))

and

k_{!}(Rj_{∗}Q |_{W}) → Rj_{∗} |_{Z_{2}} →^{rest} Rj_{∗} |_{Z_{1}} →^{+1}

is the distinguished triangle, it is enough to prove that Rj_{∗}Q |_{Z_{1}} and Rj_{∗}Q |_{Z_{2}} are cohomological mixed Hodge complex and the morphism rest is a morphism of mixed Hodge complex. For the first statement, we may assume that W is the closed subvariety of X. By modifying, there exists a modification of local spaces π : (Ñ, X̃, Ỹ) → (N, X, Y) such that (Ñ, X̃, Ỹ) is normal crossing and W̃ = π^{−1}(W) is a divisor of Ñ contained in X̃. By using the construction of remark, we can see that Rj̃_{∗}Q |W̃ is a cohomological mixed Hodge complex on W̃. Therefore,

RΓ(W, Rj_{∗}Q |_{W}) ≃ RΓ(W, Rπ_{∗}(Rj̃_{∗}Q |W̃)) ≃ RΓ(W̃ , Rj̃_{∗}Q |W̃)

is a mixed Hodge complex. We can prove that the morphism rest is a morphism of CMHC’s in the same way.

§3.2 Mixed Hodge structures associated to hypersurface singularities

In this section, we investigate mixed Hodge structures of germs of hypersurface singularities.

Let B(y) = {(y_{1}, …, y_{n}) || y_{i} |< ϵ} and f = f(y_{1}, …, y_{n}) be a holomorphic function on B(x) such that f(0) = 0. The function f defines a holomorphic map from B(y) to B(t) = {t ∈ C || t |< ϵ} and put Z(f) = {y ∈ B(y) | f(y) = 0}. We assume that f is smooth outside of Z(f). Let m be a positive integer. The variety B_{m}(y) is defined by the following fiber product:

B_{m}(y) ⟶^{π} B(y)
f′↓  ↓^{f}
B(τ) ⟶ B(t),

where B(τ) = {τ ∈ C || τ |< ϵ} and τ^{m} = t. Then (B_{m}(y), π^{−1}(0), π^{−1}(Z(f))) is a local space and H^{i}(B_{m}(y) − π^{−1}(Z(f)), Q) has a mixed Hodge structure defined in §2.1. The covering transformation group μ_{m} = Gal(B(τ)/B(t)) acts on the mixed Hodge structure on H^{i}(B_{m}(y) − π^{−1}(Z(f)), Q) by the functoriality. Let K_{MH}(C, μ_{m}) be the Grothendieck group of Q-mixed Hodge structure with a μ_{m}- action. Then K_{MH}(C, μ_{m}) has a natural ring structure arising from tensor product and graded by weight. Since [Q − Q(1)] = [Q] − [Q(1)] is a non zero divisor on K_{MH}(C, μ_{m}), the natural homomorphism from K_{MH}(C, μ_{m}) to the localization K_{MH}(C, μ_{m})_{loc} of K_{MH}(C, μ_{m}) with respect to (Q − Q(1)) is injective. (See [D-L].)

The element Ψ_{f,m}(Q) = [H^{∗}(B_{m}(y)−π^{−1}(Z(f)), Q)] in K_{MH}(C, μ_{m}) is defined by ∑_{i=0}^{2 dim B(x)}(−1)^{i}[H^{i}(B_{m}(y) − π^{−1}(Z(f)), Q)]. We define

(3.2.1)  ϕ_{f,m}(Q) = \frac{1}{[Q − Q(−1)]}Ψ_{f,m}(Q) ∈ K_{MH}(C, μ_{m})_{loc}, and
Φ_{f,m}(Q) = ϕ_{f,m}(Q) − [Q] ∈ K_{MH}(C, μ_{m})_{loc}.

Note that ϕ_{f,m}(Q) corresponds to the cohomology of the Milnor fiber if the exponent of monodromy divides m. Using the weight spectral sequence, one can show ϕ_{f,m} is actually an element in K_{MH}(C, μ_{m}), but we do not use this fact in the rest of this paper.

Let y_{i} = (y_{i1}, …, y_{im_{i}}) be sets of coordinates as in the end of §2.2. We define B(y_{i}) = {(y_{i1}, …, y_{i,m_{i}}) ∈ C^{m_{i}} || y_{ij} |< ϵ (1 ≤ j ≤ m_{i})}. Let g_{i} = g_{i}(y_{i}) be a holomorphic function on B(y_{i}) with g_{i}(0) = 0. Then g_{i} defines a homomorphism B(y_{i}) → B(x_{i}) by x_{i} = g_{i}(y_{i}). We assume that g_{i} is smooth on B(y_{i})−g_{i}^{−1}(0). The fiber product g of g_{i} from B(y) = ∏_{i=1}^{n} B(y_{i}) to B(x) = ∏_{i=1}^{n} B(x_{i}) is defined as in §2.2. The sequence of positive numbers (d_{1}, …, d_{n}) is denoted by d. We define B_{d}(x), B̃_{d}(x), B_{d}^{0}(x) and B̃_{d}^{0}(x) by the following cartesian squares.

B_{d}(y) ⟶ B(y)  B_{d}^{0}(y) ⟶ B^{0}(y)
↓  ↓  ↓  ↓
B(ξ) ⟶ B(x),  B^{0}(ξ) ⟶ B^{0}(x)
B̃_{d}(y) ⟶ B_{d}(y)  B̃_{d}^{0}(y) ⟶ B_{d}^{0}(y)
↓  ↓  ↓  ↓
B(τ) ⟶ B(t),  B(τ) ⟶ B(t).

We introduce the equivariant version of the definition (3.2.1). The natural map B̃_{d}(y) → B(τ) is denoted by f̃(y). We define ϕ̃_{f∘g,m}(Q) by

ϕ̃_{f∘g,m}(Q) =\frac{1}{[Q − Q(−1)]}[H^{∗}(B̃_{d}(y) − (f̃(y))^{−1}(0), Q)] − [Q]
∈ K_{MH}(C, G × μ_{m})_{loc}.

Then it is easy to see that Φ_{f∘g,m}(Q) = Φ̃_{f∘g,m}(Q)^{G}. For a subset I of [1, n], we define B_{I}(ξ) by ∏_{i∉I} B(ξ_{i}). The fiber product of morphism B_{I}(ξ) → B(ξ) → B(t) and B(τ) → B(t) is denoted by B_{I,m}(ξ). Put G_{I} = ∏_{i∉I} μ_{d_{i}}. Then G_{I} × μ_{m} acts on B_{I,m}(ξ). The natural morphism from B_{I,m}(ξ) → B(τ) is denoted by f_{I}(ξ). We define ϕ̃_{f_{I},m}(Q) by

ϕ̃_{f_{I},m}(Q) =\frac{1}{[Q − Q(−1)]}[H^{∗}(B_{I,m}(ξ) − (f_{I}(ξ))^{−1}(0), Q)] − [Q]
∈ K_{MH}(C, G_{I} × μ_{m})_{loc}.

Let B̃_{I,d}^{0} = B̃_{I,d} − ∪_{J⊋I} B̃_{J,d} and the k_{I} be the natural inclusion

k_{I} : B̃_{I,d}^{0} − (f̃(y))^{−1}(0) → B̃_{I,d} − (f̃(y))^{−1}(0).

Then we use the following equality.

(3.2.2) [H^{∗}(B̃_{d}(y) − (f̃(y))^{−1}(0), Q)] = ∑_{I⊂[1,n]}[H^{∗}(B̃_{I,d}(y) − (f̃(y))^{−1}(0), k_{I!}Q)]

Now we introduce several notation related to the resolution of singularities. Let b(y_{i}) : B̂(y_{i}) → B(y_{i}) be a resolution of singularity for g_{i}^{−1}(0) ⊂ B(y_{i}), i.e, b(y_{i}) is a birational projective morphism and b(y_{i})^{−1}(g^{−1}(0))_{red} is a simple normal crossing divisor. Let d_{i} be the least common multiple of the multiplicities of the components in b(y_{i})^{−1}(g^{−1}(0)). Let B(ξ_{i}) → B(x_{i}) be the morphism defined by ξ_{i}^{d_{i}} = x_{i}. The normalization of B̂(y_{i}) ×_{B(x_{i})} B(ξ_{i}) is denoted by B̃(y_{i}). The second projection B̃(y_{i}) → B(ξ_{i}) is denoted by g̃_{i}. Then the morphism g̃_{i} is μ_{d_{i}}-equivariant and B̃(y_{i}) is a quasi-smooth variety with reduced quasi-normal crossing divisor g̃_{i}(0).

B̃(y_{i}) ⟶ B̂(y_{i})
g̃_{i}↓  ↓ĝ_{i}
B(ξ_{i}) ⟶ B(x_{i})

Let V = B̃(y) = ∏_{i=1}^{n} B̃(y_{i}) be the product of B̃(y_{i}) and the product V → B(ξ) = ∏_{1=1}^{n} B(y_{i}) of g_{i} is denoted by h. The base change of h by the morphism B̂(ξ) → B(ξ), B̃(ξ) → B(ξ) and B̄(ξ) → B(ξ) are denoted by ĥ : V̂ → B̂(ξ), h̃ : Ṽ → B̃(ξ) and h̄ : V̄ → B̄(ξ). Then we have the following diagram.

V̂ ⟶ V
ĥ↓  ↓^{h}
B̂(ξ) ⟶^{b(ξ)} B(ξ) ⟶^{f(ξ)} B(t)
↑  ↑
B̃(ξ) ⟶ B̄(ξ) ⟶ B(τ)
h̃↑  ↑h̄
Ṽ ⟶ V̄

The morphism h : V → B(ξ) is G-equivariant and G×μ_{m} acts on Ṽ as B(τ)-variety. Let B^{0}(ξ) = ∏_{i=1}^{n} B^{0}(ξ_{i}), D_{B} = B(ξ) − B^{0}(ξ) and D_{F} = (f(ξ))^{−1}. The proper transform of D_{B} and D_{F} in B̂(ξ) is denoted by D̂_{B} and D̂_{F} and the exceptional divisor for b(ξ) is denoted by D̂_{E}. The pullbacks of D̂_{B}, D̂_{F} and D̂_{E} are denoted by D̃_{B}, D̃_{F} and D̃_{E}, respectively. The group G × μ_{m} acts on D̃_{E}. We define divisors on V, V̂ , Ṽ and V̄ as follows:

h^{−1}(D_{B}) = V_{B}, h^{−1}(D_{F}) = V_{F}
ĥ^{−1}(D̂_{B}) = V̂_{B}, ĥ^{−1}(D̂_{F}) = V̂_{F} , ĥ^{−1}(D̂_{E}) = V̂_{E}
h̃^{−1}(D̃_{B}) = Ṽ_{B}, h̃^{−1}(D̃_{F}) = Ṽ_{F} , h̃^{−1}(D̃_{E}) = Ṽ_{E}

Let j̃_{V} : Ṽ − (Ṽ_{F} ∪ Ṽ_{E}) → Ṽ be the natural immersion. Then the mixed Hodge structure of H^{∗}(Ṽ − (Ṽ_{F} ∪ Ṽ_{E}), Q) is given by H^{∗}(Ṽ , Rj̃_{V ∗}Q |Ṽ_{E}). The inverse image of (0, …, 0) under the morphism V → ∏_{i=1}^{n} B(y_{i}) is denoted by V_{c}. The action of G on V induces that of V_{c}. The inverse image of V_{c} under the morphism Ṽ → V is denoted by Ṽ_{c}. Then we have Ṽ_{c} = V_{c} × D̃_{E} and the action of G × μ_{m} is equal to the diagonal action. The restriction of h̃ to Ṽ is denoted by h̃_{c} : Ṽ_{c} → D̃_{E}. It is easy to see that V_{c} → V and Ṽ_{c} → Ṽ are retractions. The group G × μ_{m} acts on the mixed Hodge structure H^{i}(Ṽ , Rj̃_{V ∗}Q) ≃ H^{i}(Ṽ_{c}, Rj̃_{V ∗}Q |Ṽ_{E}). Therefore we have

[H^{∗}(Ṽ , Rj̃_{V ∗}Q)] = [H^{∗}(Ṽ − (Ṽ_{E} ∪ Ṽ_{F}), Q)] ∈ K_{MH}(C, G × μ_{m}).

Let D̃_{E} = ∐_{σ∈F′} Z̃_{σ}^{0} be the stratification indexed by F′ and Z̃_{σ} be the closure of Z̃_{σ}^{0} in D̃_{E}. Note that this stratification is stable under the action of G × μ_{m}. Let U_{σ} be a tubular neighborhood of Z̃_{σ}, j_{σ} : U_{σ} − (D̃_{E} ∪ D̃_{F}) → U_{σ} and j̃_{σ} : h̃^{−1}(U_{σ} − (D̃_{E} ∪ D̃_{F})) → h̃^{−1}(U_{σ}) be the natural inclusion.

h̃^{−1}(U_{σ} − (D̃_{E} ∪ D̃_{F})) ⟶^{j̃_{σ}} h̃^{−1}(U_{σ}) ⟵^{⊃} h̃_{c}(Z̃_{σ}^{0})
↓  ↓  ↓
U_{σ} − (D̃_{E} ∪ D̃_{F}) ⟶^{j_{σ}} U_{σ}  ⟵_{⊃} Z̃_{σ}^{0}

Let G_{σ} be the stabilizer of Z̃_{σ}^{0} in G × μ_{m}. Then H_{c}^{i}(h̃_{c}^{−1}(Z̃_{σ}^{0}), Rj̃_{V ∗}Q) is a G_{σ}- module and we have

[H^{i}(Ṽ_{c}, Rj̃_{V ∗}Q)] = ∑_{[σ]∈F′/(G×μ_{m})} Ind_{G_{σ}}^{G}[H_{c}^{i}(h̃_{c}^{−1}(Z̃_{σ}^{0}), Rj̃_{V ∗}Q)]

= ∑_{[σ]∈F′/(G×μ_{m})} Ind_{G_{σ}}^{G}[H_{c}^{i}(h̃_{c}^{−1}(Z̃_{σ}^{0}), Rj̃_{σ∗}Q)].

Put I(σ) = {i | e_{i} ∈ σ} and Ψ_{V,m,σ}(Q) = [H_{c}^{i}(h̃_{c}^{−1}(Z̃_{σ}^{0}), Rj̃_{σ∗}Q)]. Then we have

[H^{∗}(B̃_{d}(y) − (f ∘ g)^{−1}(0), k_{∅!}Q)] = ∑_{{σ∈F′,I(σ)=∅}/(G×μ_{m})} Ind_{G_{σ}}^{G}(Ψ_{V,m,σ}(Q))

in K_{MH}(C, G×μ_{m}). In the next section, we compute R^{i}j̃_{σ∗}Q using toric geometry.

§3.3 The structure of R^{i}j̃_{σ∗}Q.

First we describe a stratification of D̃_{E} in terms of toric geometry. The space B^{0}(ξ) × B^{0}(τ) can be considered as an open set of B̄(ξ). The complement D̄ = B̄(ξ) − B^{0}(ξ) × B^{0}(τ) is a quasi normal crossing divisor whose irreducible components are indexed by the set of 1-dimensional cones r in F̃. Let {r_{0}, r_{1}, …, r_{s}, r_{s+1}, …, r_{s+n}} be the set of 1-dimensional cones of F̃ such that r_{0}, r_{s+1}, …, r_{s+n} correspond to the proper transform of {τ = 0}, {ξ_{1} = 0}, …, {ξ_{n} = 0}, respectively. The corresponding divisors are written as D̄_{0}, D̄_{1}, …, D̄_{s}, D̄_{s+1}, …, D̄_{s+n}. The intersection D̄_{i_{1}} ∩ ⋯ ∩ D̄_{i_{k}} is non-empty and as a consequence defines a stratum of D̄ if and only r_{i_{1}}, …, r_{i_{k}} generate a cone σ of the fan F̃. We denote Z̄_{σ} = D̄_{i_{1}} ∩ ⋯ ∩ D̄_{i_{k}}. Since D̄_{0} ∩ B̃(ξ) = D̃_{F}, (∪_{i=1}^{n}D̄_{s+i}) ∩ B̃(ξ) = D̃_{B} and (∪_{i=1}^{s}D̄_{i})∩B̃(ξ) = D̃_{E}, Z̃_{σ} = Z̄_{σ} ∩B̃(ξ), where σ is a simplicial cone of F̃, defines a stratification of D̃_{F} ∪ D̃_{E} ∪ D̃_{B}. It is easy to see that Z̃_{σ} is not empty if and only if dim Z̄_{σ} ≥ 1 i.e. dim σ < n + 1. We put Z̄_{σ}^{0} = Z̄_{σ} − ∪_{τ>σ}Z̄_{τ}. Then Z̄_{σ}^{0} is isomorphic to a torus and Z̃_{σ}^{0} is a non-singular hypersurface in Z̄_{σ}^{0}. As a consequence, we have a stratification

D̃_{B} ∪ D̃_{E} ∪ D̃_{F} = ∪_{σ: simplicial ,dim σ<n+1}Z̃_{σ}^{0}.

Under this stratification, D̃_{E} corresponds to σ’s whose generator contains at least one of r_{1}, …, r_{s}. Let A_{σ} = C[σ̂ ∩ L(ξ, τ)] and Spec(A_{σ}) → XF̃ be the natural map. Let U_{σ} be the pull back of B(ξ) in Spec(A_{σ}). Let Lσ̂ be the maximal linear subspace contained in σ̂ and Ā_{σ} = C[Lσ̂ ∩ L(ξ, τ)]. The kernel of the natural map A_{σ} → Ā_{σ} is denoted by I_{σ}. Then the corresponding closed subvariety is identified with Z̄_{σ}^{0}.

Z̄_{σ}^{0} ⟶^{⊂} U_{σ}  ⟶ B̄(ξ)
∩↓  ↓^{∩}
Spec(A_{σ}) ⟶ XF̃

By taking the intersections with B̃(ξ), we have the following diagram.

h̃_{c}^{−1}(Z_{σ}^{0}) ⟶^{⊂} h̃^{−1}(Ũ_{σ}) ⟵^{j̃_{σ}} h̃^{−1}(Ũ_{σ}) − D̃
↓  ↓  ↓
Z̃_{σ}^{0}  ⟶_{⊂}  Ũ_{σ}  ⟵_{⊃}  Ũ_{σ} − D̃

We use these varieties to compute H^{∗}(h̃_{c}^{−1}(Z_{σ}^{0}), Rj̃_{σ}Q |_{h_{c}^{−1}(Z̃_{σ}^{0})})

The natural morphism Ṽ → V and Ṽ → B̃(ξ) defines a closed immersion Ṽ → V × B̃(ξ). This morphism defines a morphism of local spaces:

(Ṽ , Ṽ_{c}, D̃_{E} ∪ D̃_{F} ∪ D̃_{B}) →^{α̃} (V × B̃(ξ), V_{c} × D̃_{E}, V × B̃(ξ) − V^{0} × B^{0}(ξ)),
(V̄ , V̄_{c}, D̄_{E} ∪ D̄_{F} ∪ D̄_{B}) →^{ᾱ} (V × B̄(ξ), V_{c} × D̄_{E}, V × B̄(ξ) − V^{0} × b^{0}(ξ, τ)

where V^{0} = h^{−1}(B^{0}(ξ)). Note that V_{c} × D̃_{E} ≃ Ṽ_{c}. The inclusion of divisors D̃_{E} ∪ D̃_{F} → D̃_{E} ∪ D̃_{F} ∪ D̃_{B} defines the following open immersion of local spaces:

(Ṽ , Ṽ_{c}, D̃_{E} ∪ D̃_{F} ∪ D̃_{B}) →^{β̃} (Ṽ , Ṽ_{c}, D̃_{E} ∪ D̃_{F}).
(V̄ , V̄_{c}, D̄_{E} ∪ D̄_{F} ∪ D̄_{B}) →^{β̄} (V̄ , V̄_{c}, D̄_{E} ∪ D̄_{F})

We investigate these inclusions on the open set h̃^{−1}(Ũ_{σ}) ⊂ Ṽ and V × Ũ_{σ}.

Let y be a point in B̃(y) and z_{11}, …, z_{1m_{1}}, …, z_{n1}, …, z_{nm_{n}} be a local coordinate near y such that g_{1}, …, g_{n} can be written as

g_{1} = z_{11}^{m_{11}} ⋯ z_{1l_{1}}^{m_{1l_{1}}} , …, g_{n} = z_{n1}^{m_{n1}} ⋯ z_{nl_{n}}^{m_{nln}} .

Then the local equation of V at a lifting ỹ of y can be written as

ξ_{1}^{d′_{1}} = z_{11}^{m′_{11}} ⋯ z_{1l_{1}}^{m′_{1l_{1}}} , …, ξ_{n}^{d′_{n}} = z_{n1}^{m′_{n1}} ⋯ z_{nl_{n}}^{m′_{nln}} ,

where

d′_{i} = \frac{1}{gcd(d_{i}, m_{i,j})_{j}} d_{i}, m′_{ij} = \frac{1}{gcd(d_{i}, m_{i,j})_{j}} m_{ij}.

Let r_{1}, …, r_{a} be a generator of σ∩L(ξ, τ)⊗Q, such that ∑_{i=1}^{a} Nr_{i} ⊂ (σ∩L(ξ, τ)). Suppose that r_{1}, …, r_{k} correspond to the components of D̄_{B} and r_{k+1}, …, r_{a} correspond to the components of D̄_{E} ∪ D̄_{F}. We take r_{a+1}, …, r_{n+1} such that ∑_{i=1}^{n+1} Zr_{i} is finite index in L(ξ, τ)^{∗}. Then by changing the numbering ξ_{i}, r_{1}, …, r_{a} can be written as

r_{1} = (r_{11}, 0, …, 0), …, r_{k} = (0, …, r_{kk}, 0, …, 0),
r_{k+1} = (r_{k+1,1}, …, r_{k+1,n}, r_{k+1,n+1}), …, r_{a} = (r_{a1}, …, r_{an}, r_{a,n+1})

Then C[σ̂ ∩ L(ξ, τ)] is a subring of C[u_{1}, …, u_{a}, u_{a+1}^{±}, …, u_{n+1}^{±}], where {u_{i}}_{i} is the multiplicative expression of the dual base {r_{i}^{∗}}_{i} of {r_{i}}_{i}. The morphism Spec(C[u_{1}, …, u_{a}, u_{a+1}^{±}, …, u_{n+1}^{±}] → Spec(C[ξ_{1}, …, ξ_{n}]) is given by

ξ_{1} = u_{1}^{r_{11}} ⋯ u_{a}^{r_{a1}} · ( unit ), …, ξ_{n} = u_{1}^{r_{1n}} ⋯ u_{a}^{r_{an}} · ( unit ),
τ = u_{1}^{r_{1,n+1}} ⋯ u_{a}^{r_{a,n+1}} · ( unit )

First we investigate the inclusion ᾱ and β̄. The local equation for V̄ is given by

[u_{1}^{r_{11}} ∏_{p=k+1}^{a} u_{p}^{r_{p1}}]^{d′_{1}} · ( unit ) = z_{11}^{m′_{11}} ⋯ z_{1l_{1}}^{m′_{1l_{1}}}
⋯
[u_{k}^{r_{kk}} ∏_{p=k+1}^{a} u_{p}^{r_{pk}}]^{d′_{k}} · ( unit ) = z_{k1}^{m′_{k1}} ⋯ z_{kl_{k}}^{m′_{kl_{k}}}

[ ∏_{p=k+1}^{a} u_{p}^{r_{p,k+1}}]^{d′_{k+1}} · ( unit ) = z_{k+1,1}^{m′_{k+1,1}} ⋯ z_{k+1,l_{k+1}}^{m′_{k+1,l_{k+1}}}
⋯
[ ∏_{p=k+1}^{a} u_{p}^{r_{pn}}]^{d′_{n}} · ( unit ) = z_{n1}^{m′_{n1}} ⋯ z_{nl_{n}}^{n′_{nln}}

Moreover, the open set corresponding to V̄ − (V̄_{E} ∪ V̄_{F} ∪ V̄_{B}) and V̄ − (V̄_{E} ∪ V̄_{F}) is given by

u_{1} ∈ B(u_{1})^{0}, …, u_{a} ∈ B(u_{a})^{0}, and
u_{k+1} ∈ B(u_{k+1})^{0}, …, u_{a} ∈ B(u_{a})^{0}

respectively. Let j̄_{EF B} : V̄ − (V̄_{E} ∪ V̄_{F} ∪ V̄_{B}) → V̄, j̄_{EF} : V̄ − (V̄_{E} ∪ V̄_{F}) → V̄ and j̄_{M} : V^{0} × B^{0}(ξ, τ) → V × B̄(ξ) be the natural inclusions. Then

(3.3.1)  R^{i}(j̄_{EF B})_{∗}Q = ∧^{i}R^{1}(j̄_{EF B})_{∗}Q,
R^{i}(j̄_{EF})_{∗}Q = ∧^{i}R^{1}(j̄_{EF})_{∗}Q,
R^{i}(j̄_{M})_{∗}Q = ∧^{i}R^{1}(j̄_{M})_{∗}Q,

for i ≥ 1. Using the local expression of V̄ − (V̄_{E} ∪ V̄_{F} ∪ V̄_{B}), V̄ − (V̄_{E} ∪ V̄_{F}) and V^{0} × B(ξ, τ)^{0} → V × B̄(ξ), the stalks if the natural homomorphism

R^{1}(j̄_{M})_{∗}Q →^{ᾱ^{∗}} R^{1}(j̄_{EF B})_{∗}Q ⟵^{β̄^{∗}} R^{1}(j̄_{EF})_{∗}Q

at v̄ are computed as follows.

(R^{1}(j̄_{M})_{∗}Q)v̄ = [Q^{a} ⊕ (⊕_{i=1}^{n}Q^{l_{i}} ⊕ Q)] ⊗ Q(−1)
(R^{1}(j̄_{EF B})_{∗}Q)v̄ = [Coker(Q^{n+1} →^{A_{EF B}} (Q^{a} ⊕ (⊕_{i=1}^{n}Q^{l_{i}} ⊕ Q)))] ⊗ Q(−1)
(R^{1}(j̄_{EF})_{∗}Q)v̄ = [Coker(Q^{n−k+1} →^{A_{EF}} (Q^{a−k} ⊕ (⊕_{i=k+1}^{n}Q^{l_{i}} ⊕ Q)))] ⊗ Q(−1),

where

A_{EF B}(e_{i}) = d′_{i}(r_{1i}, …, r_{ai}) ⊕ (0, …, 0, (m′_{i1}, …, m′_{i,l_{i}}), 0, …, 0), (1 ≤ i ≤ n)

A_{EF B}(e_{n+1}) = d′_{i}(r_{1,n+1}, …, r_{a,n+1}) ⊕ (0, …, 0, 1),

and

A_{EF} (e_{i}) = d′_{i}(r_{k+1i}, …, r_{ai}) ⊕ (0, …, 0, (m′_{i1}, …, m′_{i,l_{i}}), 0, …, 0), (k + 1 ≤ i ≤ n)

A_{EF} (e_{n+1}) = d′_{i}(r_{k+1,n+1}, …, r_{a,n+1}) ⊕ (0, …, 0, 1).

The morphism ᾱ^{∗} is identified with the natural projection. Since the diagram

Q^{n+1} ⟶^{A_{EF B}} Q^{a} ⊕ (⊕_{i=1}^{n}Q^{l_{i}} ⊕ Q)
↑  ↑
Q^{n−k+1} ⟶_{A_{EF}} Q^{a−k} ⊕ (⊕_{i=k+1}^{n}Q^{l_{i}} ⊕ Q)

is commutative, β̄^{∗} is identified with the homomorphism induced by the natural inclusions. By (3.3.1) and the expression of ᾱ^{∗} and β̄^{∗}, we have the following proposition.

Proposition 3.3.1.

(1) The morphism ᾱ^{∗} is injective an β̄^{∗} is surjective.

(2) The morphism R^{i}(j̄_{M})_{∗}Q → R^{i}(j̄_{EF B})_{∗}Q and R^{i}(j̄_{EF})_{∗}Q → R^{i}(j̄_{EF B})_{∗}Q are identified with ∧^{i}ᾱ^{∗} and ∧^{i}β̄^{∗}. Moreover they are surjective and injective respectively.

Since B̃(ξ) meets D̄_{E} ∪ D̄_{F} ∪ D̄_{B} transversally, we get the following proposition.

Proposition 3.3.2. For v̄ ∈ Ṽ, we have

(R^{i}(j̃_{EF B})_{∗}Q)v̄ = (R^{i}(j̄_{EF B})_{∗}Q)v̄,
(R^{i}(j̃_{EF})_{∗}Q)v̄ = (R^{i}(j̄_{EF})_{∗}Q)v̄,
(R^{i}(j̃_{M})_{∗}Q)v̄ = (R^{i}(j̄_{M})_{∗}Q)v̄.

As a consequence, Proposition 3.3.1 holds by replacing j̄_{EF B}, j̄_{EF} and j̄_{M} by j̃_{EF B}, j̃_{EF} and j̃_{M}.

§3.4 Supplement for mixed Hodge structures

Let X be a topological space and K = ((K_{Q}, W), (K_{C}, W, F)) and K′ = ((K′_{Q}, W), (K′_{C}, W, F)) be cohomological mixed Hodge complexes (=CMHC’s for short). A morphism from K to K′ is defined by the pair of morphisms of filtered and bifiltered complexes φ_{Q} : K_{Q} → K′_{Q} and φ_{C} : K_{C} → K′_{C} where (φ_{Q} ⊗ C, W) and (φ_{C}, W) are filtered congruent to each other. After [Dur], we define the cone Cone(φ) of φ by

Cone(φ)_{A}^{p} = K_{A}^{p} ⊕ (K′)_{A}^{p−1},
d : Cone(φ)_{A}^{p} → Cone(φ)_{A}^{p+1}; (x, y) ↦ (dx, φ(x) + dy),
W_{k} Cone(φ)_{A}^{p} = W_{k}K_{A}^{p} ⊕ W_{k+1}(K′)_{A}^{p−1}, for A = Q, C and
F^{q} Cone(φ)_{C}^{p} = F^{q}K_{C}^{p} ⊕ F^{q}(K′)_{C}^{p−1}.

According to [Dur], Cone(φ) give rise to be a CMHC.

Definition 3.4.1. A morphism φ : K → K′ of CMHC’s is called weakly equivalent if the underlying morphism K_{Q} → K′_{Q} is a quasi-isomorphism.

Remark 3.4.2. By weak equivalence, we do not impose that φ_{Q} or φ_{C} are filtered quasi-isomorphism.

The following lemma will be useful later.

Lemma 3.4.3. Let f : K → L and b : L′ → L be morphisms of CMHC’s on X. Suppose that b is weak equivalent. Then there exists a cohomological mixed Hodge complex K′ and morphisms a : K′ → K and f′ : K′ → L′ of CMHC’s on X such that a is a weak equivalence and f ∘ a = b ∘ f′.

Proof. Let K′ = Cone((f, b) : K ⊕ L′ → L) and a : K′ → K and f′ : K′ → L′ be the natural morphisms. Then K′ is equipped with CMHC and a and f are morphism of CMHC’s which satisfy the conditions of the lemma.

We define direct sum decomposition of CMHC’s.

Definition 3.4.4.

(1) Let K, L and M be CMHC’s and p_{1} : K → L and p_{2} : K → M be morphisms of CMHC’s. p_{1} ⊕ p_{2} : K → L ⊕ M is called a direct sum decomposition if it is weakly equivalent.

(2) Two direct sum decompositions f_{1} : K → L_{1} ⊕ M_{1} and f_{2} : K → L_{2} ⊕ M_{2} are said to be equivalent if there exists a third direct sum decomposition p_{3} : K → L_{3} ⊕ M_{3} and weak equivalences g_{i} : L_{3} → L_{i} and h_{i} : M_{3} → M_{i} (i = 1, 2) such that the following diagram commutes

K ⟶^{f_{3}} L_{3} ⊕ M_{3}
=↓  ↓^{g_{i}⊕h_{i}}
K ⟶_{f_{i}} L_{i} ⊕ M_{i}

Lemma 3.4.5. Let f : K → L ⊕ M be a direct sum decomposition and K → K′ be a weak equivalence. Then there exist CMHC’s L′, M′ and weak equivalences a : L → L′, b : M → M′ and f′ : K′ → L′ ⊕ M′ such that the following diagram commutes.

K ⟶^{g} K′
f↓  ↓^{f′}
L ⊕ M ⟶_{(a,b)} L′ ⊕ M′

Proof. Since f is weakly equivalent, the natural morphism f^{∗} : Cone(K → L) ⊕ Cone(K → M) → K is weakly equivalent. Therefore the composite

g ∘ f^{∗} : Cone(K → L) ⊕ Cone(K → M) → K′

is weakly equivalent. Therefore

f′ : K′ → Cone(Cone(K → L) → K′)[1] ⊕ Cone(Cone(K → M) → K′)[1]

is a direct sum decomposition. We put L′ = Cone(Cone(K → L) → K′)[1] and M′ = Cone(Cone(K → M) → K′)[1]. Since the natural morphisms L → Cone(Cone(K → L) → K) and M → Cone(Cone(K → M) → K) are weakly equivalent, the composites

a : L → Cone(Cone(K → L) → K) → Cone(Cone(K → L) → K′) = L′
b : M → Cone(Cone(K → M) → K) → Cone(Cone(K → M) → K′) = M′

are weakly equivalent. Thus we have the required CMHC’s and weak equivalences.

Definition 3.4.6 (Quasi-canonical filtration).

(1) Let K be a CMHC on X. A sequence κ_{i} : K_{i} → K and κ_{i,i+1} : K_{i} → K_{i+1} of morphism of CMHC’s is called quasi-canonical filtration if (1) there exists K′_{i} → K such that the diagram is commutative

K′_{i} ⟶^{a} K_{i}
b↓  ↓
τ_{i}K ⟶ K,

where a and b are quasi-isomorphism. Note that we do not impose that either K′_{i} or τ_{i}K are equipped with structures of CMHC’s, and (2) κ_{i} = κ_{i+1} ∘ κ_{i+1}.

(2) The decomposition Cone(K_{i−1} → K_{i})[1] → L_{i} ⊕ M_{i} of the cone Cone(K_{i−1} → K_{i}) is called a decomposition associated to quasi-canonical filtration {K_{i} → K, K_{i−1} → K_{i}}.

Definition 3.4.7 (Associated decomposition). Let K_{i} → K, K_{i−1} → K_{i} be a quasicanonical filtration and Cone(K_{i−1} → K_{i})[1] → C^{i} ⊕ I^{i} be a decomposition associated to the quasi canonical filtration. Let L → K be a morphism of CMHC’s.

The decomposition is associated to the morphism L → K if there exists complexes A, B and morphisms f : A → B, a : A → H^{i}L, b : B → H^{i}K and b′ : B → Cone(K_{i−1} → K_{i})[1] such that

(1) The morphisms a, b and b′ are quasi-isomorphism.

(2) The following diagram commutes

A ⟶^{f′} B ⟶^{b′} Cone(K_{i−1} → K_{i})[1]
a↓  ↓^{b}
H^{i}L ⟶_{H^{i}(f)} H^{i}K

(3) The composite A → B → Cone(K_{i−1} → K_{i})[1] → I^{i} is a quasi isomorphism.

Remark 3.4.8. We do not impose the complexes of sheaves A, B and H^{i}L are equipped with the structure of CMHC’s.

Now we can prove the following proposition.

Proposition 3.4.9. Let L, K be CMHC’s, K_{i} → K, K_{i−1} → K_{i} be a finite quasi-canonical filtration, f_{i} : Cone(K_{i−1} → K_{i}) → C^{i} ⊕ I^{i} be decompositions of Cone(K_{i−1} → K_{i}) and φ : L → K be a morphism of CMHC’s. Suppose that the morphism φ is associated to the decompositions f_{i}. Then there exists a quasicanonical filtration {L_{i} → L, L_{i−1} → L_{i}} such that Cone(L_{i−1} → L_{i}) is weakly equivalent to I^{i}.

Proof. We construct L_{m}, L_{m} → K_{m} and L_{m−1} → L_{m} by descending induction. For a sufficiently large m, K_{m} → K is a quasi isomorphism and K_{i} → K_{i+1} is isomorphic for i ≥ m. By Lemma 3.4.3, we can take a CMHC L^{m} such that the following diagram of CMHC’s commutes

L_{m} ⟶ K_{m}
i_{m}↓  ↓
L ⟶ K

such that i_{m} is a weak equivalence. Therefore we can take L_{m} → L for a sufficiently large m. Suppose that L_{m}, L_{m} → K_{m} and L_{m} → L_{m+1} (m ≥ k) satisfying the conditions of the proposition are given by induction. Let′I^{k} = Cone(Cone(K_{k−1} → K_{k})[1] → C^{k}). Then the composite ′I^{k} → Cone(K_{k−1} → K_{k})[1] → I^{k} is a weak equivalence and it is denoted by b. The morphism L_{k} → K_{k} → Cone(K_{k−1} → K_{k})[1] → I^{k} is denoted by a. Then we have the following diagram;

′I^{k} ⟶ Cone(K_{k−1} → K_{k})[1]
↓^{b}
L^{k} ⟶^{a} I_{k},

where b is weakly equivalent. Then by Lemma, there exists L′_{k}, a′′ : L′_{k} →′ I^{k} and b′ : L′_{k} → L_{k} such that b′ is weakly equivalent. Let L_{k−1} = Cone(L′_{k} →′ I^{k}). Then there are natural morphisms

L_{k−1} = Cone(L′_{k} →′ I^{k}) → Cone(K_{k} → Cone(K_{k−1} → K_{k})[1]) → K_{k−1} and L_{k−1} = Cone(L′_{k} →′ I^{k}) → L′_{k} → L_{k}. This proceeds the steps of the induction.

Corollary 3.4.10. Under the notations as in Proposition 3.4.9,

[H^{∗}(X, L)] = ∑_{i}(−1)^{i}[H^{∗}(X, I^{i})]

in K_{MH}(C)

Proof. Since L is finite, L_{m} is exact for m << 0 and L_{m} → L is weakly equivalent. for m >> 0. Therefore,

[H^{∗}(X, L)] = ∑_{i}(−1)^{i}[H^{∗}(X, Cone(L_{i−1} → L_{i}))]

= ∑_{i}(−1)^{i}[H^{∗}(X, I^{i})]

§3.5 Direct sum decomposition for normal crossing local spaces

Let (N, X, Y) be a normal crossing local space and ∪_{i∈I}D_{i} be the irreducible decomposition of D = X ∪ Y. Let us define the complex subvariety D_{J} of X as ∩_{j∈J}D_{j}. The complex of sheaves Ω_{N}^{•}(log D) ⊗ O_{D_{j}} is denoted by Ω^{•}(J). We have introduced a structure of cohomological mixed Hodge complex on Rj_{∗}Q |_{D_{I}}, where j : N − (X ∪ Y) → N is the natural inclusion, via the quasi-isomorphism

(Rj_{∗}Q |_{D_{I}}) ⊗ C ≃ Ω^{•}(J)

with the filtrations W_{k} and F^{p} on the right hand side. Let Z_{J} = ∪_{i∉J}D_{J∪i} and D_{J}^{0} = D_{J} − Z_{J} and the natural inclusion from D_{J}^{0} to D_{J} is denoted by j_{J} : D_{J}^{0} → D_{J}. As in §?, the structure of CMHC on (j_{J})_{!}(Rj_{∗}Q |_{D_{J}^{0}} ) is given by the complex

K_{J}^{•} : Ω^{•}(J) → ⊕_{#K=1,K∪J=∅}Ω^{•}(J ∪ K) → ⊕_{#K=2,K∪J=∅}Ω^{•}(J ∪ K) → ⋯

with the filtrations

W_{k}K_{J}^{•} : W_{k}Ω^{•}(J) → ⊕_{#K=1,K∪J=∅}W_{k+1}Ω^{•}(J ∪ K)
→ ⊕_{#K=2,K∪J=∅}W_{k+2}Ω^{•}(J ∪ K) → ⋯, and

F^{p}K_{J}^{•} : F^{p}Ω^{•}(J) → ⊕_{#K=1,K∪J=∅}F^{p}Ω^{•}(J ∪ K)
→ ⊕_{#K=2,K∪J=∅}F^{p}Ω^{•}(J ∪ K) → · · · .

We define sub-complex K_{J,i}^{•}, W_{k}K_{J,i}^{•} and F^{p}K_{J,i}^{•} by

K_{J,i}^{•} : W_{i}Ω^{•}(J) → ⊕_{#K=1,K∪J=∅}W_{i}Ω^{•}(J ∪ K)
→ ⊕_{#K=2,K∪J=∅}W_{i}Ω^{•}(J ∪ K) → ⋯,

W_{k}K_{J,i}^{•} = K_{J,i}^{•} ∩ W_{k}K_{J}^{•} and F^{p}K_{J,i}^{•} = K_{J,i}^{•} ∩ F^{p}K_{J}^{•}.

Proposition 3.5.1.

(1) The sequence of morphism K_{J,i}^{•} → K_{J}^{•}, K_{J,i−1}^{•} → K_{J,i}^{•} are quasi canonical filtration.

(2) The quotient complex K_{J,i}/K_{J,i−1} is weakly equivalent to L_{J}^{•} ⊗ ∧^{i}(Q^{J}) ≃ L_{J}^{•} ⊗ (⊕_{#L=i,L⊂J}Q), where

L_{J}^{•} : Ω_{D_{J}}^{•} [−i] → ⊕_{#K=1}Ω_{D_{J∪K}} [−i] → ⋯

(3) Let Q^{J} = M_{1} ⊕ M_{2} be a direct sum decomposition of Q^{J}. Then the direct sum decomposition a direct sum decomposition ∧^{i}M_{1} ⊕ M_{2} ∧ ∧^{i}(Q^{J}) give reise to a direct sum decomposition L_{J}^{•} ⊗ ∧^{i}M_{1} ⊕ L_{J}^{•} ⊗ (M_{2} ∧ ∧^{i}(Q^{J})) of K_{J,i}/K_{J,i−1}.

Proof. First we show that K_{J,i}^{•} is a CMHC. The sheaf Gr_{k}^{W} K_{J,i} is given by

Gr_{i}^{W} K_{J,i}^{•} =(Gr_{k}^{W} Ω^{•}(J) →^{0} ⊕_{#K=1,K∪J=∅}Gr_{k+1}^{W}Ω^{•}(J ∪ K) →^{0} ⋯ →^{0}

⊕_{#K=i−k,K∪J=∅} Gr_{i}^{W} Ω^{•}(J ∪ K) → 0)

=(⊕_{#L=k}Ω_{D_{L∪J}}^{•} [−k] →^{0} ⊕_{#K=1,#L=k+1,K∪J=∅}Ω_{D_{L∪J∪K}}^{•} [−k − 1] →^{0}

⋯ →^{0} ⊕_{#K=i−k,#L=k+(i−k),K∪J=∅}Ω_{D_{L∪J∪K}}^{•} [−i] → 0)

Therefore

H^{m}(Gr_{F}^{p} Gr_{k}^{W} K_{J,i}) = ⊕_{#L=k} H^{m−p}(Ω_{D_{L∪J}}^{p−k})⊕
⊕_{#K=1,#L=k+1,K∪J=∅} H^{p−m−1}(Ω_{D_{L∪J∪K}}^{p−k−1} )
⊕ ⋯ ⊕ ⊕_{#K=i−k,#L=k+(i−k),K∪J=∅}H^{m−i+k−p}(Ω_{D_{L∪J∪K}}^{p−i}).

and Gr_{k}^{W} (K_{J,i}^{•}) is a cohomological Hodge complex. Since K_{J,i}^{•} for i << 0 and K_{J,i} ≃ K_{J} for i >> 0, it is sufficient to prove that H^{p}(K_{J,i}^{•}/K_{J,i−1}^{•}) = 0 if p ≠ i. Since K_{J,i}^{•} is quasi isomorphic to

τ_{i}Rj_{∗}Q |_{D_{J}} → ⊕_{#K=1,K∪J=∅}τ_{i}Rj_{∗}Q |_{D_{J∪K}} → ⋯,

the quotient complex K_{J,i}/K_{J,i−1} is quasi isomorphic to

R^{i}j_{∗}Q |_{D_{J}} [−i] → ⊕_{#K=1,K∪J=∅}R^{i}j_{∗}Q |_{D_{J∪K}} [−i] → ⋯,

and it is quasi isomorphic to (j_{J})_{!}(Rj_{∗}Q |_{D_{J}^{0}} )[−i]. This proves the proposition. (2) The associated graded complex of sheaves K_{J,i}/K_{J,i−1} is isomorphic to

⊕_{#L=i}Ω_{L∪J}^{•}[−i] → ⊕_{#L=i,#K=1,K∩J=∅}Ω_{L∪J∪K}^{•}[−i] → ⋯

and therefore it is a direct sum of M_{L}^{•}, (#L = i), where

M_{L}^{•} = (Ω_{L∪J}^{•}[−i] → ⊕_{#K=1,K∩J=1}Ω_{L∪J∪K}^{•}[−i] → ⋯).

Moreover, the direct sum is compatible with the filtration F^{•} and W_{•} induced by that of K_{J,i}^{•} and it is easy to see that these filtrations defines structure of CMHC.

This morphism defines a morphism Cone(K_{J,i−1} → K_{J,i})[1] → ⊕_{#L=i}M_{L}^{•} of CMHC which is a quasi-isomorphism, i.e. a weak equivalence of CMHC. (Note that in general if 0 → K → L → M → 0 is an exact sequence of CMHC, then the natural map Cone(K → L)[1] → M is a weak equivalence. Therefore H^{i}(X, Cone(K → L)) ≃ H^{i}(X, M) is an isomorphism of mixed Hodge structures. cf. [D], §2.5 for X = a point.) We show that M_{L}^{•} is quasi-isomorphic to 0 if L ⊈ J. Let a ∈ L − J and put J′ = J ∪ {a}. We define complexes ′M_{L}^{•} and ′′M_{L}^{•} as

′M_{L}^{•} = (Ω_{L∪J′} → ⊕_{#K=1,K∩J′=∅}Ω_{L∪J′∪K} → ⋯)
′′M_{L}^{•} = (0 → ⊕_{#K=1,K∩J=∅,a∈K}Ω_{L∪J′∪K} → ⊕_{#K=2,K∩J=∅,a∈K}Ω_{L∪J′∪K} → ⋯)

Then it is easy to see that M_{L}^{p} =′ M_{L}^{p} ⊕′′ M_{L}^{p}. The bijection between {K′ | #K′ = p, K′ ∩ J′ = ∅} and {K | #K = p + 1, K ∩ J∅, a ∈ K} given by K′ = K − {a} gives an isomorphism of complexes ′M_{L}^{•} →′′ M_{L}^{•}. As a complex, it is easy to check that M_{L}^{•} = Cone(′M_{L}^{•} →′′ M_{L}^{•}) and this proves the exactness of M_{L}^{•} if L ⊈ J. Moreover, if L ⊂ J, then M_{L}^{•} ≃ L_{J}^{•}. Therefore we get the statement (2) of the proposition.

The stetement (3) is a direct consequence of (2).

§3.6 Convolution theorem for Hodge theory.

First, we compute Ψ_{V,m,σ}(Q) using the result of §3.3, §3.4 and §3.5. The normal crossing divisor V_{B} of V defines a natural stratification and V_{c} is expressed as a union of this stratification: ∪_{τ∈K}S_{τ}^{0}. The morphism R^{i}(j̃_{M})_{∗}Q → R^{i}(j̃_{EF B})_{∗}Q is surjective and identified with a projection for direct sum decomposition of CMHC’s by the expression of R^{i}(j̃_{M})_{∗}Q |Z̃_{σ}^{0} in Proposition 3.3.2 and the expression of α̃^{∗} in Proposition 3.3.1.(2). Therefore R^{i}(j̃_{EF B})_{∗}Q |_{S_{τ}^{0}×Z̃_{σ}^{0}} has a structure of CMHC which is weakly equivalent to ∧^{i}(Q(−1)^{r̃(σ,τ)}), as a CMHC, where r̃(σ, τ) = dim R^{1}(j̃_{EF B})_{∗}Q |v̄ for v̄ ∈ S_{τ}^{0} ×Z̃_{σ}^{0} using Proposition 3.5.1.(3) and Theorem 2.4.9. We use the same argument again to compute R^{i}(j̃_{EF})_{∗}Q |_{S_{τ}^{0}×Z̃_{σ}^{0}} and we get an isomorphism R^{i}(j_{EF})_{∗}Q ≃ ∧^{i}Q(−1)^{r(σ,τ)} as a CMHC,s, where r(σ, τ) = ∑_{i=k+1}^{n} l_{i} + a − n. Here we used the same notations l_{k+1}, …, l_{n} and a in §3.3. Therefore, we have an isomorphism

H_{c}^{∗}(S_{τ}^{0} × Z̃_{σ}^{0}, R^{i}(j̃_{EF})_{∗}Q) ≃ H_{c}^{∗}(S_{τ}^{0}, Q) ⊗ H_{c}^{∗}(Z̃_{σ}^{0}, Q) ⊗ ∧^{i}(Q(−1)^{r(σ,τ)}),

as G_{σ,τ} -Hodge structure, where G_{σ,τ} is the stabilizer of S_{τ} × Z̃_{σ}^{0} in G × μ_{m}. As a consequence, we have

[H_{c}^{∗}(S_{τ}^{0} × Z̃_{σ}^{0}, R^{i}(j̃_{EF})_{∗}Q)] = [H_{c}^{∗}(S_{τ}^{0}, Q)] ⊗ [H_{c}^{∗}(Z̃_{σ}^{0}, Q)] ⊗ [Q − Q(−1)]^{r(σ,τ)}.

in K_{MH}(C, G_{σ,τ}). Let r′(σ, τ) = ∑_{i=k+1}^{n}(l_{i} − 1) and r′′(σ, τ) = a − k. Then we have the following equality:

∑_{[τ]∈K/G} Ind_{G_{τ}}^{G} [H_{c}^{∗}(S_{τ}^{0}, Q)] ⊗ [Q − Q(−1)]^{r′(σ,τ)} = ⊗_{i∈I}[H_{c}^{∗}(V_{c,i}, Q)] ⊗_{i∉I} ϕ_{g_{i},d_{i}}(Q),

in K_{MH}(C, G), where I = I(σ) = {i | e_{i} ∈ σ}. Therefore, we have

Ψ_{V,m,σ} = ⊗_{i∈I}[H_{c}^{∗}(V_{c,i}, Q)] ⊗ ⊗_{i∉I}ϕ_{g_{i},d_{i}}(Q) ⊗ [H_{c}^{∗}(Z̃_{σ}^{0}, Q)] ⊗ [Q − Q(−1)]^{r′′(σ,τ)} in K_{MH}(C, G × μ_{m}). For any I ⊂ [1, n], we have

∑_{{I(σ)=I}/G} Ind_{G_{σ}}^{G}[H_{c}^{∗}(Z̃_{σ}^{0}, Q)] ⊗ (Q − Q(−1))^{r′′(σ,τ)} = H_{c}^{∗}(Z̃_{I}^{0}, Rj̃_{∗}Q),

where j̃ : B̃(ξ) − (D̃_{E} ∪ D̃_{F}) → B̃(ξ) and Z̃_{I}^{0} = ∪_{I(σ)=I} Z̃_{σ}^{0}. As a consequence, we have

[H^{∗}(B̃_{d}(y) − (f(ξ) ∘ g)^{−1}(0), k_{∅!}Q)]

= ∑_{K×{σ|I(σ)=∅}/G×μ_{m}} Ind_{G_{sigma,τ}}^{G×μ_{m}} [H_{c}^{∗}(S_{τ}^{0}, Q)] ⊗ [H_{c}^{∗}(Z̃_{σ}^{0}, Q)] ⊗ [Q − Q(−1)]^{r(σ,τ)}
= ⊗_{i=1}^{n} ϕ_{g_{i},d_{i}}(Q) ⊗ [H_{c}^{∗}(Z̃_{∅}^{0}, Rj̃_{∗}Q)],

where k_{∅} and B̃_{d}^{0}(y) is defined at the end of §3.2. In the same way, for a subset I of [1, n], we have

[H^{∗}(B̃_{I,d}(y) − (f_{I}(ξ) ∘ g)^{−1}(0), k_{I!}Q)] = ⊗_{i∉I}ϕ_{g_{i},d_{i}}(Q) ⊗ [H_{c}^{∗}(Z̃_{I}^{0}, Rj̃_{∗}Q)].

Using the equality (3.2.2), we have

[H^{∗}(B̃_{d}(y) − (f(ξ) ∘ g)^{−1}(0), Q)]
= ∑_{I⊂[1,n]}[H^{∗}(B̃_{I,d}(y) − (f_{I}(ξ) ∘ g)^{−1}(0), k_{I!}Q)]
= ∑_{I⊂[1,n]} ⊗_{i∉I}ϕ_{g_{i},d_{i}}(Q) ⊗ [H_{c}^{∗}(Z̃_{I}^{0}, Rj̃_{I∗}Q)]
= ∑_{I⊂[1,n]}(−1)^{#I} ⊗_{i∉I} ϕ_{g_{i},d_{i}}(Q) ⊗ ⊗_{i∈I}(ϕ_{g_{i},d_{i}}(Q) − [Q]) ⊗ [H_{c}^{∗}(Z̃_{I}, Rj̃_{I∗}Q)]
= ∑_{I⊂[1,n]}(−1)^{#I} ⊗_{i∉I} ϕ_{g_{i},d_{i}}(Q) ⊗ ⊗_{i∈I}(ϕ_{g_{i},d_{i}}(Q) − [Q])
⊗ ([H_{c}^{∗}(Z̃_{I}, Rj̃_{I∗}Q)] − [Q − Q(−1)]
+ [Q − Q(−1)]

we have

Φ̃_{f∘g,m}(Q) = (\frac{1}{[Q − Q(−1)]}[H_{c}^{∗}(B̃_{d}(y) − (f(ξ) ∘ g)^{−1}(0), Q)]) − Q

= ∑_{I⊂[1,n]}(−1)^{#I} ⊗_{i∉I} ϕ_{g_{i},d_{i}}(Q) ⊗ ⊗_{i∈I}Φ_{g_{i},d_{i}}(Q) ⊗ Φ̃_{f_{I},m}(Q).

By taking G-invariant part of the above equality, we have the following theorem. Note that the action of G on H_{c}^{∗}(Z̃_{I}^{0}, Rj̃_{∗}Q) factors through G_{I}.

Theorem 3.6.1 (Convolution Theorem for Hodge structures).

Φ_{f∘g,m}(Q) = ∑_{I⊂[1,n]}(−1)^{#I}(⊗_{i∉I}ϕ_{g_{i},d_{i}}(Q) ⊗ Φ̃_{f_{I},m}(Q))^{G_{I}} ⊗ ⊗_{i∈I}Φ_{g_{i},d_{i}}(Q)^{μ_{d_{i}}}

By specializing to the case where n = 2 and f = x_{1} + x_{2}, we have the following corollary.

Corollary 3.6.2 (Thom-Sebastiani theorem for Hodge structures). Let g_{1} and g_{2} be germs of holomorphic functions on C^{n_{1}} and C^{n_{2}} at 0 with isolated singularities. Let d_{1} and d_{2} be the exponents of g_{1} and g_{2} respectively and m = lcm(d_{1}, d_{2}). Then we have

Φ_{g_{1}+g_{2},m} =(Φ_{g_{1},d_{1}} ⊗ Φ_{g_{2},d_{2}} ⊗ Φ̃_{ξ_{1}^{d_{1}} +ξ_{2}^{d_{2}}} )^{μ_{d_{1}} ×μ_{d_{2}}}
− Φ_{g_{1},m,≠1} ⊗ Φ_{g_{2},m,≠1} + Φ_{g_{1},m} ⊗ Φ_{g_{2},m},

where Φ_{g_{1}+g_{2},m} = Φ_{g_{1}+g_{2},m}(Q) e.t.c.

Proof. By the Theorem 3.6.1, we have

(3.6.1)  Φ_{g_{1}+g_{2},m} =(ϕ_{g_{1},d_{1}} ⊗ ϕ_{g_{2},d_{2}} ⊗ Φ̃_{ξ_{1}^{d_{1}} +ξ_{2}^{d_{2}} ,m})^{μ_{d_{1}} ×μ_{d_{2}}}
− (ϕ_{g_{1},d_{1}} ⊗ Φ̃_{ξ_{1}^{d_{1}} ,m})^{μ_{d_{1}}} ⊗ Φ_{g_{2},d_{2}}^{μ_{d_{2}}}
− (ϕ_{g_{2},d_{2}} ⊗ Φ̃_{ξ_{2}^{d_{2}} ,m})^{μ_{d_{2}}} ⊗ Φ_{g_{1},d_{1}}^{μ_{d_{1}}}
+ Φ_{g_{1},d_{1}}^{μ_{d_{1}}} ⊗ Φ_{g_{2},d_{2}}^{μ_{d_{2}}}.

On the other hand, we have

(3.6.2)  Φ̃_{ξ_{1}^{d_{1}} ,m} = [Q[μ_{d_{1}}]] − [Q]

in K_{MH}(C, μ_{d_{1}} × μ_{m}), where μ_{m} acts on μ_{d_{1}} through its quotient. Therefore we have

(ϕ_{g_{1},d_{1}} ⊗ Φ̃_{ξ^{d_{1}} ,m})^{μ_{d_{1}}} = ϕ_{g_{1},d_{1},≠1} = Φ_{g_{1},d_{1},≠1}.

If (Φ̃_{ξ_{1}^{d_{1}} +ξ_{2}^{d_{2}} ,m} ⊗ C)(χ_{1}, χ_{2}) ≠ 0 for a charcters χ_{1} and χ_{2} of μ_{d_{1}} and μ_{d_{2}}, then χ_{1} ≠ 1 and χ_{2} ≠ 1. Therefore we have

(3.6.3) (ϕ_{g_{1},d_{1}}⊗ϕ_{g_{2},d_{2}}⊗Φ̃_{ξ_{1}^{d_{1}} +ξ_{2}^{d_{2}} ,m})^{μ_{d_{1}} ×μ_{d_{2}}} = (Φ_{g_{1},d_{1}}⊗Φ_{g_{2},d_{2}}⊗Φ̃_{ξ_{1}^{d_{1}} +ξ_{2}^{d_{2}} ,m})^{μ_{d_{1}} ×μ_{d_{2}}}

By (3.6.1), (3.6.2) and (3.6.3), we have the corollary.

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