THE MIXED HODGE STRUCTURE OF AN ICIS

by ANDRÁS NÉMETHI ^{1}

Department of Mathematics, The Ohio State University 231 West, 18th Avenue, Columbus, OH 43210, U.S.A.

CHAPTER I. INTRODUCTION AND PRELIMINARY RESULTS

§1. Introduction.

The basic object of the paper is a deformation g : (Y, y) → (B, 0) of an isolated complete intersection singularity (icis) (Y_{0}, y) = (g^{−1}(0), y) such that its discriminant locus Δ is a divisor with normal crossings.

In the case when B is one–dimensional, J. Steenbrink constructed a mixed Hodge structure (in the sequel MHS) on the middle cohomology groups H^{k}(Y) and H_{c}^{k}(Y) of the Milnor fiber Y of g [24, 25]. Steenbrink’s construction has two aspects. First, it is local, i.e. the whole construction depends only on the germ g. On the other hand, in [23] he provided a geometric construction of the limit MHS for any projective family f (over (B, 0)), and he proved that if f is a projective extension of g then the local and global MHS’s are compatible. Actually, he also proved that his geometric MHS coincides with Schmid’s limit MHS [22]. All these results are highly generalized in the theory of mixed Hodge modules of M. Saito [18, 19].

In all these constructions and theories, one of the main problems (which is not easy at all) is the description, characterization (as nicely as possible) of the corresponding weight filtrations (see, papers of Steenbrink [25, 28], Steenbrink-Zucker [27], M. Saito [17, 18, 19], Kashiwara [9, 10, 11], etc.). The difficulty in the local case of germs of singularities (presented in a naiv way) is the following: the weight filtration is not the monodromy weight filtration of the infinitezimal monodromy operator. Moreover, in general, the intersection form is degenerated. Hence, the monodromy is not enough to describe the weight filtration, and the intersection form is not enough to give a polarization. The needed new operator is the variation map.

In this paper, we give a construction of the whole polarized MHS’s of the local germ as a quotient of the polarized limit MHS of a projective family suitably chosen (technique already used in [21, 13, 28]). This will provide several characterizations and properties of the weight filtrations (some of them already listed in the literature, some of them new) (see chapter I). Moreover, because of the existence of the limit MHS in the case dim B ≥ 1 given by Schmid [22], it can be generalized for the case of higher dimensional base space (see chapter III).

1Partially supported by NSF Grant No. DMS-9622724 and OSU Seed Grant

Even if in the case of the one-dimensional base space some of the properties presented here (chapter II) are not completely new, or can be derived from the results of Steenbrink or M. Saito with some (sometimes maybe difficult) work, we present them here for the following reasons. Their form and proof, in our approach, is easy to follow, they do not need the machinary of Saito (mixed Hodge modules), of Steenbrink (cohomological mixed Hodge complexes associated with the resolution), or of Kashiwara. On the other hand, we need all these facts about the one–dimensional case for a proper presentation of the case dim B ≥ 1. We emphasize, that even in the one–dimensional case we give strong new relations and properties (see, e.g. (9.7)).

If dim B ≥ 1, using the construction mentioned above, we prove the existence of a MHS on the cohomology group of the Milnor fiber. (Notice that this part cannot be derived from Saito’s or Steenbrink’s work, where the basic objects are the vanishing cycles associated with one–dimensional smoothings.) One of the main results of chapter III is:

Theorem. Let g : (Y, y) → (B, 0) be a deformation of an icis (g^{−1}(0), y) such that its discriminant Δ is a divisor with normal crossings. Then the (middle) cohomology group of its Milnor fiber carries a mixed Hodge structure (H^{k}(Y_{s}), W, F) with the following properties:

a) For any reduced analytic one–dimensional space–germ (C, 0) ⊂ (B, 0) (with (C, 0) ⊄ (Δ, 0)), the MHS on the vanishing cohomology of the deformation g|g^{−1}(C) : (g^{−1}(C) ∩ Y, y) → (C, 0) coincides with (H^{k}(Y_{s}), W, F) (modulo the action of the local complex monodromy cone).

b) For any projective extension f : (X , y) → (B, 0) of g such that f has no critical points outside Y one has an exact sequence of MHS’s:

0 → H^{k}(X_{0}) → H^{k}(X_{s}) → H^{k}(Y_{s}) → H^{k+1}(X_{0}) → H^{k+1}(X_{s}) → 0,

H^{∗}(X_{0}) carries Deligne’s MHS, H^{∗}(X_{s}) Schmid’s limit MHS and H^{k}(Y_{s}) the MHS defined in this paper.

c) The MHS (H^{k}(Y_{s}), W, F) is “compatible” with the action of π_{1}(B^{∗}, ∗), in the sense which will be clarified in section 13.

The (polarized) MHS of the local germ is induced by the limit MHS of Schmid associated with a good projective extension. Moreover, in section 14, we give the local analogs of a theorem of Cattani-Kaplan [2], about the weight filtrations associted with the faces of the monodromy(+variation) cone. For this we need the local version of the relative monodromy weight filtrations. This algebraic facts are separated in section 5.

Our algebraic description of the (relative) weight filtrations in section 5 (which corresponds to our local case) are in the spirit of Deligne presentation [6] (which corresponds to the global situation). We emphasize, that our filtrations are the same as those described by M. Saito (e.g. in [19], Corollary 1.9) but presented in a different way.

Moreover, in the case of one dimensional base space, we prove (see section 9, and 9.6) a decomposition theorem for the cohomology group of the Milnor fiber (by a rather elementary way), which in Saito’s theory corresponds to the semisimplicity of polarized mixed Hodge modules, key fact in that theory (or corresponds to the Beilinson–Bernstein–Deligne decomposition theorem of perverse sheafs). For hypersurfaces it is explicitely formulated in [17], see also [28].

We hope that this work will help in the understanding of Saito’s work as well.

During the paper, we give several applications: connections with the MHS of the real links, signature computation, etc. In section 2 we review some facts about the degeneration of Hodge structures, which is needed in the paper. Using these facts, we can derive all the properties of the local MHS from the properties of Schmid’s limit MHS. In section 3 we prove a decomposition theorem for polarized MHS’s, which will provide our decomposition theorem of the MHS on H^{k}(Y) mentioned above. Section 4 is topological, it gives the existence of a good projective extension (for the one dimensional case, see also [21, 28]).

The preliminary version of the paper [14] was written in September-October 1994. Some of the results were announced in [15]. In the same time when the manuscript was written, J. Steenbrink obtained (independently) the results published in [28] about the MHS of the vanishing cohomology associated with one–dimensional smmoothings. The reader can find some similarities between our chapter II and that paper.

The table of contents of the paper is the following:

Chapter I. Introduction and preliminary results.

§ 1. Introduction. § 2. Review on degeneration of Hodge structures. § 3. Addendum on polarized MHS’s. § 4. A “homologically good” projective extension of an icis. § 5. Algebraic preliminaries: filtrations and relative filtrations.

Chapter II. The case of one–dimensional base space.

§ 6. Notations and preliminaries. § 7. The mixed Hodge structure on H^{k}(Y). The weight filtration. § 8. The mixed Hodge structure on H_{c}^{k}(Y). The weight filtration. § 9. The decomposition of H^{k}(Y)_{1}˜ and H_{c}^{k}(Y)_{1}˜ . §10. Applications. Connection with the MHS of the links. §11. The Hodge numbers and the signature.

Chapter III. The general case.

§12. The mixed Hodge structure on H^{k}(Y) and H_{c}^{k}(Y). §13. The action of π_{1}(B^{∗}). §14. The weight filtrations associated with the faces of the cone.

§2. Review on degeneration of Hodge structures.

2.1. A Hodge structure of weight i consists of a finitely generated free abelian group H_{Z}, and a finite decreasing filtration F^{•} of subspaces of H = H_{Z} ⊗_{Z} C satisfying H = F^{p} ⊕ F̄^{i−p+1} for any p ∈ Z.

Setting H^{pq} = F^{p} ∩ F̄^{q}, one obtains the associated Hodge decomposition:

H = ⨁_{p+q=i} H^{pq}; H^{pq} = H̄^{pq} with F^{p} = ⨁_{r≥p} H^{r,i−r}.

The Weil operator C : H → H of the Hodge structure (H_{Z}; F) is defined by C(v) = i^{p−q}v for v ∈ H^{pq}.

A bilinear form S on H polarizes the Hodge structure (H_{Z}; F) of weight i if it is defined over Z and:

a) S(u, v) = (−1)^{i}S(v, u) for u, v ∈ H;

(2.2) b) S(F^{p}, F^{i−p+1}) = 0 for any p ∈ Z; and

c) the hermetian form S(C·,¯·) is positive definite.

Let X be an n-dimensional non-singular projective complex variety of dimension k. Assume that an embedding X ⊂ CP^{N} is fixed. Let w ∈ H^{2}(X, Z) be the pull back of the fundamental class ω_{N} ∈ H^{2}(CP^{N}, Z). Our basic example of polarized Hodge structure will be the primitive k-th cohomology P^{k}(X, Z) := free part of {α ∈ H^{k}(X, Z) : α ∪ ω^{n−k+1} = 0} (k ≤ n), which carries a canonical Hodge structure of weight k polarized by the Hodge bilinear form S_{X}(α, β) = (−1)^{k(k−1)/2} · (α ∪ β ∪ ω^{n−k}, [X]). 2.3. If instead of a fixed X one considers a polarized family of non-singular projective varieties, then the local system of primitive cohomology along the fibers underlines a variation of polarized Hodge structure [8].

More precisely, set a projective flat morphism f : X → B where B is a product Δ^{n} of discs and X can be embedded in a projective variety. Assume that f over the product B^{∗} := (Δ^{∗})^{n} of punctured discs is smooth. (Notice that X can be singular over B \ B^{∗}.) Let s_{0} ∈ B^{∗} be a fixed base point and for any s ∈ B take X_{s} := f^{−1}(s). Fix an embedding X ⊂ CP^{N} and define for each X_{s} a fundamental class as above. Then the primitive cohomology groups {P^{k}(X_{s}, Z)}_{s∈B^{∗}} determine a flat bundle P^{k} over B^{∗}, associated with the monodromy representation:

ρ : G := π_{1}(B^{∗}, s_{0}) → Aut(P^{k}(X_{s_{0}}, Z), S)

in the isometries of S = S_{X_{s_{0}}}.

The bundle of Hodge spaces {H^{pq}(X_{s})}_{s} is not flat, it fails to be even holomorphic, but the Hodge filtration F^{p} := {F^{p}(X_{s})}_{s∈B^{∗}} determines a holomorphic bundle. Let P^{k} be Deligne’s extension of P^{k} to B. Then by the Nilpotent Orbit Theorem [22], F^{p} extends to a sub-bundle F^{p} of P^{k}. Then F_{∞} := (F^{p})_{0} is the Hodge filtration of the “limit mixed Hodge structure” associated with the family f. 2.4. Let s = (s_{1}, ..., s_{n}) be local coordinates in B such that B^{∗} = {s_{1}·. . .·s_{n} ≠ 0}. Let γ_{j} ∈ G be represented by a loop in B^{∗} which winds once around {s_{j} = 0} in clockwise direction (j = 1, …, n). Let T_{j} := ρ(γ_{j}) ∈ Aut(H_{Z}, S). By the monodromy theorem T_{j} is quasi-unipotent with Jordan blocks at most of size k + 1. Let m_{j} be the order of its semisimple part and set N_{j} = 1/m_{j} · log T_{j}^{m_{j}}. Obviously, N_{j} is nilpotent and it is an infinitesimal isometry of S, i.e: S(N_{j}x, y) + S(x, N_{j}y) = 0. 2.5. If N is a nilpotent endomorphism of H_{Q} = H_{Z} ⊗ Q and k ∈ Z, then its weight filtration W (H_{Q}; N, k) centered at k, is the unique increasing filtration W of H_{Q} such that

a) NW_{l} ⊂ W_{l−2} for any l ∈ Z, and

b) N^{l} : Gr_{k+l}^{W} → Gr_{k−l}^{W} is an isomorphism for l ≥ 0. Notice that T_{j} is defined over Z, hence N_{j} is defined over Q determining a weight filtration of H_{Q}. 2.6. There exists a relative version of the above definition. Suppose that H_{Q} already carries an increasing filtration W and N is a nilpotent endomorphism of H_{Q} such that NW_{l} ⊂ W_{l} for all l. Then, by Deligne [6], there exists at most one increasing filtration M(H_{Q}; N, W) of H_{Q} such that NM_{l} ⊂ M_{l−2} and the filtration induced by M on Gr_{k}^{W} H_{Q} is precisely W (Gr_{k}^{W} H_{Q}; Gr_{k}^{W} N, k). M is called the monodromy weight filtration of N relative to W. 2.7. For any subset J ⊂ {1, 2, ..., n} define

σ_{J} = {N = ∑_{j∈J} λ_{j}N_{j}; λ_{j} > 0}, and σ_{{1,...,n}} = σ.

2.8. Theorem. ([2], see also [27])

a) For any fixed J, any N ∈ σ_{J} has the same weight filtration W (H; N, k) (denoted by W (σ_{J})).

b) If N ∈ σ_{J}, N′ ∈ σ′_{J}, and N′′ ∈ σ_{J∪J′}, then

M(H_{Q}; N; W (N′, k)) = W (H_{Q}; N′′, k).

Now, the constructed objects S, F_{∞} and W (σ) form an interesting structure. This motivates the following definition: 2.9. Definition. [2] Fix a real vector space H_{R}, a real (−1)^{k}-symmetric bilinear form S on it and denote by g_{R} the real infinitesimal isometries of S. An increasing filtration W (defined over R), and a decreasing filtration F of H_{R} ⊗ C, and a nilpotent element N ∈ g_{R} (with N^{k+1} = 0) form a (real) polarized mixed Hodge structure (H; S, W, F, N, k) if:

a) W is the weight filtration of N centered at k;

b) (W, F) is a mixed Hodge structure, i.e for any l the filtration, induced by F on Gr_{l}^{W}, is a Hodge structure of weight l; c) NF^{p} ⊂ F^{p−1} for any p ∈ Z; and

d) for any l ≥ 0, the Hodge structure induced by F on the primitive subspace P_{k+l} := ker(N^{l+1} : Gr_{k+l}^{W}H → Gr_{k−l−2}^{W}H) ⊂ Gr_{k+l}^{W} is polarized by the bilinear form S(·, N^{l}·).

Now, an important consequence of Schmid’s Nilpotent Orbit theorem is:

2.10. Theorem. [22]

If F_{∞} is the limit Hodge filtration of the variation (P^{k}, ∇, F^{k}, S), and W (σ) is the weight filtration of the monodromy cone σ, then (S, W (σ), F_{∞}, k) is a mixed Hodge structure polarized by any N ∈ σ.

An important property of the polarized mixed Hodge structures is the following: 2.11. Descent Lemma. [4]

Let (H; S, W, F, σ, k) be a mixed Hodge structure which is polarized by any N ∈ σ. Fix T ∈ σ̄ and define on T H the filtrations (F_{T})^{p} := T H ∩ F^{p}; (W_{T})_{l} = T H_{R}∩W_{l}. Set S_{T} (T u, T v) := S(u, T v). Then (T H; S_{T} , W_{T} , F_{T} , σ, k− 1) is a mixed Hodge structure which is polarized by any element of the cone σ. (Here σ acts by taking the restriction.)

The dual result is the following:

Let (H; S, W, F, σ, k) be a polarized mixed Hodge structure as above. Fix T ∈ σ, and on the quotient space H/ ker T define the filtrations qW and qF where q is the projection H → H/ ker T. Define the (−1)^{k+1}–symmetric form on H/ ker T by S_{T} (qx, qy) := S(x, T y). Then (qH; S_{T} , qW, qF, σ, k + 1) is a mixed Hodge structure polarized by σ.

2.12. The morphisms of type (p, q) of the mixed Hodge structure (W, F) satisfy T W_{l} ⊂ W_{l+p+q} and T F^{r} ⊂ F^{r+p}. In particular any T ∈ σ̄(= the closure of the monodromy cone) is a morphism of type (−1, −1).

A mixed Hodge structure always admits a splitting compatible with all its morphisms [5]. Set

(2.13) I^{pq} = F^{p} ∩ W_{p+q} ∩ (F̄^{q} ∩ W_{p+q} + \overline{∑_{j≥0} F^{q−1−j} ∩ W_{p+q−2−j})}.

Then W_{l} = ⊕_{p+q≤l}I^{pq}, F^{p} = ⊕_{r≥p}I^{rs} and I^{pq} = Ī^{qp}(mod ⊕_{{r<p \atop s<q}} I^{rs}) [3].

The mixed Hodge structure (W, F) splits over R if its decomposition (2.13) satisfies I^{pq} = \overline{I^{qp}}. In that case I^{pq} = F^{p} ∩ F̄^{q} ∩ W_{p+q}.

§3. Addendum on polarized MHS’s.

3.1. To any mixed Hodge structure (H; W, F) (where H = H_{R} ⊗C, and W is defined over R), there are various ways of associating another one that splits over R (cf. 2.12). One such way, introduced by Deligne (see [3], (2.20)), is functorial in the following sense.

We first recall the construction. Consider the generalized Hodge decomposition {I^{pq}}_{pq} of (H; W, F) (cf. 2.13) and let T be the semisimple automorphism of H defined by T |I^{pq} = (p + q − k) · id_{I^{pq}}. Then (H; W, F) splits over R if and only if T = T̄. But even if T ≠ T̄, there exists a unique real endomorphism δ of H such that:

i) T̄ = Ad(e^{−2iδ}) · T, and

ii) δ(I^{pq}) ⊂ ⊕_{{r<p \atop s<q}} I^{rs} for any p and q.

Then for F̃ := e^{−iδ} · F the mixed Hodge structure (H; W, F̃) splits over R, and Gr^{W} F = Gr^{W} F̃.

The construction is compatible with the polarization. If σ is a monodromy cone, and the mixed Hodge structure (H; S, W, F, k) is polarized by any N ∈ σ, then δ ∈ g_{R}, and(H; S, W, F̃, k) is polarized by any N ∈ σ.

We will need the following result. 3.2. Proposition. The functor (H; W, F) → (H; W, F̃) is exact on the category of mixed Hodge structures. Proof. It is enough to verify the assertion for short exact sequences. Let (K; W ∩ K, F ∩ K) be a sub-mixed Hodge structure of (H; W, F). Then T_{K} = T_{H}|K. Consider the nilpotent Lie algebra

L_{K}^{−1,−1} = \{X ∈ gl(H); X(I^{pq}) ⊂ ⊕_{{r<p \atop s<q}} I^{rs}, X(K) ⊂ K\}.

Then T̄ ∈ T + L_{K}^{−1,−1}. Since L_{K}^{−1,−1} is ad T –invariant, proposition (2.2) of [3] assures the existence of δ′ ∈ L_{K}^{−1,−1} ∩ gl(H_{R}) such that T̄ = Ad(e^{−2iδ′}) · T.

The unicity property assures that δ′ = δ (δ constructed above), i.e δ(K) ⊂ K. Again, by unicity, one has δ|K = δ_{K}. □ 3.3. Notation. We write H˜ instead of (H; W, F̃), when there is no danger of confusion.

The main result of this section is the following: 3.4. Theorem. Let (H; S, W, F, N, k) be a polarized MHS which splits over R. Let (K; W ∩K, F ∩K) be a sub-MHS of (H; W, F) such that K ⊂ ker N. Then, there exists a canonical direct sum decomposition

(H; S, W, F, N, k) = ⊕_{i=1,2}(H_{i}; S_{i}, W_{i}, F_{i}, N_{i}, k)

in polarized MHS’s, such that:

a) (H_{1})_{R} ⊥ (H_{2})_{R} orthogonality with respect to S_{R},

b) ker(N_{1})_{R} = K_{R}. Moreover, there exists a Lie–algebra morphism ρ : sl_{2}(R) → g_{R} such that

c) ρ ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) = N,

d) ρ ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) = Y provides a self–dual grading ⊕I_{l} of H, i.e.: H =

⊕_{l}I_{l}, S(I_{i}, I_{j}) = 0 if i + j ≠ 2k, and Y |I_{l} = (l − k) · id_{I_{l}};

e) (H_{i})_{R}(i = 1, 2) are sl_{2}(R)–invariant subspaces of H_{R}. Proof. Let {I^{pq}}_{pq} be the generalized Hodge decomposition of (H; W, F). Then K = ⊕_{p,q}I^{pq} ∩ K, i.e. {I_{K}^{pq} := K ∩ I^{pq}}_{pq} is the Hodge decomposition of K. Define Y by Y |I^{pq} = (p + q − k) · id_{I^{pq}}. Then Y ∈ g_{R} and Y (K) ⊂ K. Moreover, {I_{l} = ⊕_{p+q=l}I^{pq}}_{l} gives an S–dual grading of (H, W) (as in d)). By (2.8) [3], there exists a Lie–algebra homomorphism ρ : sl_{2}(R) → End_{R}(H) which satisfies c) and d). Set

N^{+} = ρ ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) .

Then N^{+} ∈ g_{R} and it is of type (1,1) (cf. [3], 3.10). Let (H_{1})_{R} be the sl_{2}(R)–invariant subspace of H_{R} generated by K_{R}. Since for p + q > k one has I_{K}^{pq} = 0 (because of K ⊂ ker N), the Lefschetz decomposition of N^{+}|H_{1} reads as:

(3.5) H_{1} = ⊕_{p,q} ⊕_{0≤t≤k−p−q} (N^{+})^{t}I_{K}^{pq}.

Let I_{0}^{pq} = I^{pq} ∩ ker N^{p+q−k} be the set of primitive subspaces of H. Define I_{H_{1},0}^{pq} = H_{1} ∩ I_{0}^{pq}, then:

(3.6) I^{pq} ∩ H_{1} = ⊕_{t≥0}N^{t}(I_{H_{1},0}^{p+t,q+t}).

Now, by the general theory:

(3.7) I^{pq} = ⊕_{t≥0}N^{t}(I_{0}^{p+t,q+t})

is an orthogonal decomposition with respect to S, and

(3.8) \{ \begin{matrix}S(I_{0}^{pq}, N^{l}I_{0}^{rs}) = 0 for r + s = k + l = p + q; (p, q) ≠ (s, r), \\ i^{p−q}S(u, N^{l}ū) > 0 for u ∈ I_{0}^{pq}, u ≠ 0. \end{matrix}

Therefore, the decomposition (3.6) is orthogonal with respect to S|H_{1} and S induces on H_{1} a non–degenerate form which has the polarization properties. Define F_{1}^{p} = ⊕_{r≥p}I^{rs} ∩ H_{1}, W_{1,l} = ⊕_{r+s≤l}I^{rs} ∩ H_{1}. Then

(H_{1}; S_{R}|(H_{1})_{R}, W_{1}, F_{1}, N|H_{1}, k)

is a polarized MHS, which splits over R and ker(N|H_{1}) = K.

Let (H_{2})_{R} be the orthogonal subspace of (H_{1})_{R} in H_{R}, with respect to S_{R}. Then (H_{2})_{R} is sl_{2}(R)–invariant and S|H_{2} is nondegenerate.

Let (p, q) be such that p + q = k + l for some l ≥ 0. Define I_{H_{2},0}^{pq} = I_{0}^{pq} ∩ H_{2} = {x ∈ I_{0}^{pq}; S(x, N^{l}I_{H_{1},0}^{qp}) = 0}. Then by (3.8) one has dim I_{H_{2},0}^{pq} = dim I_{0}^{pq} − dim I_{H_{1},0}^{pq}, hence I_{0}^{pq} = I_{H_{1},0}^{pq} ⊕ H_{H_{2},0}^{pq}. Moreover H_{2} = ⊕_{p,q} ⊕_{t≥0} N^{t}I_{H_{2},0}^{pq}, I^{pq} ∩ H_{2} = ⊕_{t≥0}N^{t}I_{H_{2},0}^{p+t,q+t}, and the filtrations F_{2}^{p} = ⊕_{r≥p}I^{pq} ∩ H_{2} and W_{2,l} = ⊕_{p+q≤l}I^{pq} ∩ H_{2} defines a polarized MHS which splits over R. □ 3.9. Another method to prove (3.4) is based on the following fact. Any polarized MHS which splits over R can be constructed by the operations of direct sum, duality, tensor and symmetric products from three basic types of R–split polarized MHS (see, for example, [3] p. 478). 3.10. Example. Theorem (3.4) is not true if the mixed Hodge structure does not split over R. As a counter–example, consider a non–real complex number λ, k = 1, and

H = Cv_{00} ⊕ Cv′_{00} ⊕ Cv_{11} ⊕ Cv′_{11},

Nv_{11} = v_{00}, Nv′_{11} = v′_{00},

S(v_{11}, v_{00}) = 1, S(v′_{11}, v′_{00}) = 1,

I^{00} = Cv_{00} ⊕ Cv′_{00}, I^{11} = Cv_{11} ⊕ C(v′_{11} + λv_{00}); and K_{R} = Rv′_{00}.

The proof of the next proposition is similar to the proof of (3.4), and it is left to the reader.

3.11. Proposition.

Let H_{i} = (H_{i}; S_{i}, W_{i}, F_{i}, N_{i}, k) (i = 1, 2) be two polarized MHS’s which split over R. Assume that the induced MHS’s (ker N_{i}; ker N_{i} ∩ W_{i}, ker N_{i} ∩ F_{i}) (i = 1, 2) are isomorphic as MHS’s. Then H_{1} is isomorphic to H_{2} as a polarized MHS. 3.12. Example. Proposition (3.11) is not true for arbitrary MHS’s. For example take k = 1, and the structures: H_{1} = H_{2} = Cv_{00} ⊕ Cv_{11}, such that S_{1}(v_{11}, v_{00}) = S_{2}(v_{11}, v_{00}) = 1, N_{1}v_{11} = N_{2}v_{11} = v_{00} and I_{1}^{00} = I_{2}^{00} = Cv_{00}; but I_{1}^{11} = Cv_{11} and I_{2}^{11} = C(v_{11} + λv_{00}), where λ ∈ C \ R. Then ker N_{1} = ker N_{2} (as MHS’s), but H_{1} ≠ H_{2}.

§4. A “homologically good” projective extension of an icis.

4.1. Let g : (Y, y) → (B, 0) be a deformation of an icis (Y_{0}, y) := (g^{−1}(0), y). We will assume that its generic fiber is smooth. It is well-known (see, for example, [26]), that there exists a projective extension f of g. In this section we will prove the existence of a projective extension with some additional homological properties. For the one–dimensional case, see [21]. (Actually, in the one–dimensional case, even more general extension theorems exist [13, 28].)

This extension creates the algebraic context for our study of the local situation. 4.2. Let f : X → B be a projective morphism as in (2.3), and x ∈ X. Assume that X \ {x} is non–singular, f(x) = 0, and (f^{−1}(0), x) is an icis. Let τ be a real analytic function with non–negative values defined in a neighbourhood of x such that τ^{−1}({0}) = {x}. Then for sufficiently small 0 < ϵ << 1, and after possibly shrinking B, f : Y := f^{−1}(B) ∩ τ^{−1}([0, ϵ]) → B is a ”good representative” [12] of the icis f : (X , x) → (B, 0). Let k := dim f^{−1}(0).

Let Y_{s} be the Milnor fiber f^{−1}(s) ∩ Y (s ∈ B^{∗} := B \ Δ and Δ := the discriminant locus of g). It is a smooth manifold of real dimension 2k with boundary ∂Y_{s} = f^{−1}(s) ∩ τ^{−1}(ϵ) and has the homotopy type of a bouquet of spheres ⋁_{μ} S^{k}, where μ is the Milnor number of the icis (f^{−1}(0), x).

Fix an embedding X ⊂ CP^{N} and let ω be the fundamental class in H^{2}(X_{s}, Z) as in section 2. For any s ∈ B^{∗}, define P^{k}(X_{s}, Z) = {x ∈ H^{k}(X_{s}, Z)_{fr} : x ∪ ω = 0}, and P_{k}(X_{s}Z) = {x ∈ H_{k}(X_{s}, Z)_{fr} : x ∩ ω = 0}. Here ∪ω is the cup product, ∩ω is the cap product; and for a Z–modul V we denote its free part by V_{fr}.

4.3. Theorem.

Let g : (Y, y) → (B, 0) be a deformation of an icis as in (4.1). After possibly shrinking Y and B, there exist X ⊂ CP^{N} and a flat projective map f : X → B, and an open embedding of Y in X such that:

a) g is analytically equivalent to f|_{Y} and f has no critical points outside Y;

b) i_{∗} : H_{k}(Y_{s}, Z) → H_{k}(X_{s}, Z) is one–to–one for any s ∈ B, or equivalently:

i^{∗} : H^{k}(X_{s}, Z) → H^{k}(Y_{s}, Z) is onto for any s ∈ B.

c) for any embedding X ⊂ CP^{N} and s ∈ B^{∗}: H_{k}(Y_{s}, Z) ⊂ P_{k}(X_{s}, Z), and the composition: P^{k}(X_{s}, Z) ↪ H^{k}(X_{s}, Z) → H^{k}(Y_{s}, Z) is onto. Proof.

It is enough to prove the result for the versal deformation of (g^{−1}(0), x). Therefore, we can assume that g : (C^{l+k}, 0) → (C^{l}, 0) is a flat, finitely determined polynomial mapping. Choose an integer r such that g is r– determined, and polynomials h = (h_{1}, ..., h_{l}) such that:

a) all monomials of h_{i} (i = 1, …, l) have degree bigger than r;

b) h : (C^{l+k}, 0) → (C^{l}, 0) defines an icis, such that the boundary of its Milnor fiber is an integer homology sphere. Take, for example, Brieskorn polynomials h_{i} = ∑_{j} a_{ij}x_{j}^{n_{ij}}, where (a_{ij}) is a generic coefficient matrix, n_{ij} are big different prime numbers.

Consider now the icis α : (C × C^{l+k}, 0) → (C × C^{l}, 0), given by α(t, x) = (t, g_{t}(x)), where g_{t}(x) = tg(x) + h(x). Let r′ > r be an integer such that α is (r′−1)–determined, and the degree of the polynomials g_{i} and h_{i} is strictly less than r′. By Sard’s theorem, there exist homogenious polynomials (G_{1}, ..., G_{l}) of degree r′ such that the closure of (g_{t} + G)^{−1}(0) in CP^{l+k} has 0 as its only singular point. Set:

F (t, s, x_{0}, ..., x_{l+k}) = x_{0}^{r′} · g_{t}(x_{1}/x_{0}, ..., x_{l+k}/x_{0}) − G(x_{1}, ..., x_{l+k}) − sx_{0}^{r′}, X′ = {(t, s, [x]) ∈ D × B × CP^{l+k} : F = 0}, and f′ : X′ → D × B, given by (t, s, [x]) → (t, s). Let x_{t} = (t, 0, [1 : 0... : 0]) be the unique singular point of (f′)^{−1}(t, 0).

Notice that the germ α̃ : (C × C^{l+k}, 0) → (C × C^{l}, 0), given by (t, x) → (t, g_{t}(x) − G(x)) is an icis and x_{t} is a point on its critical locus. Therefore there is a (non–canonical) embedding of the Milnor fiber Y of the icis f′ : (X′, x_{t}) → (D × B, (t, 0)) in the Milnor fiber Ỹ of α̃. On the other hand, the Milnor fiber Ỹ embeds in the projective fiber X of f′. Consider the composition:

H_{k}(Y, Z) →^{i_{1}} H_{k}(Ỹ , Z) →^{i_{2}} H_{k}(X, Z),

where the maps i_{1} and i_{2} are induced by the above embeddings. The map i_{1} is injective by the general local theory of an icis [12]. On the other hand, since ∂Ỹ is a homological sphere, by Mayer–Victoris theorem, H_{k}(Ỹ , Z) is a direct summand in H_{k}(X, Z).

Therefore, for X = X′∩{t = sufficently small fixed number} and f = f′|X the condition a) holds and b) is true for any s ∈ B^{∗}. If s ∈ B \ B^{∗}, then for s′ ∈ B^{∗} sufficiently close to s, there exists a diagram of the following form:

0 → K → H_{k}(Y_{s′}, Z) → H_{k}(Y_{s}, Z) → 0
↓=  ↓ i_{∗}  ↓ i_{∗}
0 → K → H_{k}(X_{s′}, Z) → H_{k}(X_{s}, Z) → 0

Hence b) holds for any s ∈ B.

For part c) consider (X , f) which satisfies a) and b) and an arbitrary embedding X ⊂ CP^{N}. Let L be a generic hyperplane in CP^{N} such that x ∉ L. Consider the representative Y so small that Y ∩ L = ∅. Then Y_{s} ∩ L = ∅, in particular H_{k}(Y_{s}, R) ⊂ ker(· ∩ ω_{R}). Since H_{k}(Y_{s}.Z) is free, the homological part of c) follows. For the cohomological part, consider the surjective map:

H^{k}(X_{s}, Z) = H^{k}(Ỹ_{s}, Z) ⊕ H^{k}(X_{s} \ Ỹ_{s}, Z) → H^{k}(Ỹ_{s}, Z) → H^{k}(Y_{s}, Z).

Since H^{k+2}(Ỹ_{s}, Z) = 0 one has H^{k}(Ỹ_{s}, Z) ↪ P^{k}(X_{s}, Z) . □

§5. Algebraic preliminaries: filtrations and relative filtrations.

5.1. If N_{p} is a nilpotent endomorphism of the vector space H_{p}, and k is an integer, then the monodromy weight filtration W (H_{p}; N_{p}, k) (cf. 2.5) can be provided by the explicit formula:

W_{k+l}(H_{p}; N_{p}, k) = ∑_{t≥max(0,l)} imN_{p}^{−l+t} ∩ ker N_{p}^{t+1}

(cf. [27] p. 499). 5.2. In this section H and H_{c} are finite dimensional Q–vector spaces. Consider two linear maps V : H → H_{c} and j : H_{c} → H such that N := j ∘ V is nilpotent (or, equivalently N_{c} := V ∘ j is nilpotent). Fix an integer k.

We define an increasing filtration W (H; N, V, k) on H by the following ”convolution”:

(5.3) W_{l+k}(H; N, V, k) = ∑_{t≥max(0,l)} imN^{−l+t} ∩ ker V N^{t}.

Notice that it depends only on the nilpotent endomorphism N of H and the subspace ker V ⊂ H. The following lemma is an easy consequence of (5.1): 5.4. Lemma:

a) If V is an isomorphism, then W (H; N, V, k) = W (H; N, k + 1).

b) If j is an isomorphism, then W (H; N, V, k) = W (H; N, k).

On the space H_{c} we define an increasing filtration by W_{k+l}^{c}(H_{c}; N_{c}, V, k) =

(5.5)  \{ \begin{matrix}∑_{t≥max(0,l)}im N_{c}^{−l+t−1}V ∩ ker N_{c}^{t+1} & if l < 0 \\ ker N_{c}^{l+1} + ∑_{t≥max(0,l)}im N_{c}^{−l+t−1}V ∩ ker N_{c}^{t+1} & if l ≥ 0. \end{matrix}

It depends only on the nilpotent endomorphism N_{c} of H_{c} and on the subspace im V ⊂ H_{c}. Again, as above, one has: 5.6. Lemma.

a) If V is an isomorphism, then: W^{c}(H_{c}; N_{c}, V, k) = W (H_{c}; N_{c}, k − 1).

b) If j is an isomorphism, then: W^{c}(H; N_{c}, V, k) = W (H_{c}; N_{c}, k) .

Now, we can replace the system (H, N, V) with (H_{c}, N_{c}, j), and we define the filtration W (H_{c}; N_{c}, j, k) of H_{c} as in (5.3). Similarly, if we replace (H_{c}, N_{c}, V) with (H, N, j) we can define the filtration W^{c}(H; N, j, k) of H. In order to avoid the confusion, we give the first one:

W_{k+l}(H_{c}; N_{c}, j, k) = ∑_{t≥max(0,l)} imN_{c}^{−l+t} ∩ ker jN_{c}^{t}.

Then (5.4) and (5.6) read as: 5.7. Lemma. Assume that either V or j is an isomorphism, then:

W (H; N, V, k) = W^{c}(H; N, j, k + 1), and
W^{c}(H_{c}; N_{c}, V, k) = W (H_{c}; N_{c}, j, k − 1) .

5.8. Remark.

i) In general, the identities of (5.7) do not hold. Take for example H and H_{c} such that dim H = dim H_{c}, and any linear map j which is not an isomorphism; and set V = 0. Then Gr_{l}^{W} H = Gr_{l}^{W^{c}}H_{c} = 0 if l ≠ k. On the other hand, the weight filtrations (given in terms of j) on the right hand side of the identities in (5.7) are not pure.

ii) Lemma 5.7 has the following easy generalization. Its importance consists of the fact, that this will be exactly our geometric situation (cf. 9.6).

Assume that one has the direct sum decomposition

(H; H_{c}, V, j) = ⊕_{i=1,2}(H_{i}; H_{c,i}, V_{i}, j_{i})

such that V_{1} and j_{2} are isomorphisms. Then the identities in (5.7) still hold.

The above decomposition condition is a completely non–trivial restriction for the maps V and j, and in general, it is not easy to verify whether they satisfy it or not. For example, take:

j = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{matrix} ) and V = ( \begin{matrix}0 & −1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

They verify all the “naiv” algebraic conditions of our geometric maps j and V (listed in §6), but not the above decomposition condition. In our situation, the above decomposition will be provided by the presence of a polarized mixed Hodge structure. This type of decomposition is a key fact (and principle) in the theory of mixed Hodge modules (and perverse sheafs). In that general context, it follows from the semisimplicity of the polarized Hodge modules. 5.9. The connection between the monodromy weight filtration of a nilpotent endomorphism and the filtrations (5.3) and (5.5) are clarified in the following theorem. 5.10. First characterization. Set V : H → H_{c} and j : H_{c} → H such that N_{c} = V ∘ j and N = j ∘ V are nilpotent. Fix an integer k.

a) There exists a diagram:

[figure: H_{p}; H_{p}; H_{p}; H; H_{c}; H; V; j; N_{p}; q; i; q; =; ∩]

such that q is onto, i is one–to–one. The diagram assures that N_{p} is nilpotent.

In the sequel, such a diagram will be called an “algebraic extension” of the pair V : H → H_{c}, j : H_{c} → H.

b) The filtration qW (H_{p}; N_{p}, k) of H is independent of the choice of the extension, and it is exactly W (H; N, V, k).

c) The filtration i^{−1}W (H_{p}; N_{p}, k) of H_{c} is independent of the choice of the extension, and it is exactly W^{c}(H_{c}; N_{c}, V, k). Proof. First construct i and q such that j = q ∘ i, then define N_{p} := iV q, and use (5.1). □ 5.11. In practice, it is inconvenient to work with the convolution filtrations (5.3) and (5.5). The following theorem will gives a different characterization of them. First we recall that the monodromy weight filtration W (H_{p}; N_{p}, k) (cf. 2.6 and 5.1) can be constructed inductively as follows (see [6] and [27]).

If W_{k+i}(N_{p}) is defined for |i| > l (l ≥ 0), then:

(5.12) \{ \begin{matrix}W_{k+l}(N_{p}) = ker(N_{p}^{l+1} : H_{p} → H_{p}/W_{k−l−2}(N_{p})) & (l ≥ 0) \\ W_{k−l}(N_{p}) = N_{p}^{l}W_{k+l}(N_{p}) & (l > 0). \end{matrix}

Our inductive construction is the following. 5.13. Second characterization. Let V and j be as in (5.10). The filtrations (5.3) and (5.5) are the unique filtrations W (H) and W (H_{c}) on H and H_{c} respectively with the following properties:

\begin{matrix}a) W_{k+l}(H) = ker(V N^{l} : H → H_{c}/W_{k−l−2}(H_{c})) & (l ≥ 0) \\ b) W_{k+l}(H_{c}) = ker(N_{c}^{l+1} : H_{c} → H_{c}/W_{k−l−2}(H_{c})) & (l ≥ 0) \\ c) W_{k−l}(H) = N^{l}W_{k+l}(H) & (l > 0) \\ d) W_{k−l}(H_{c}) = V N^{l−1}W_{k+l}(H) & (l > 0) . \end{matrix}

Proof. Obviously, the iterative properties (a–d) assure the unicity of the filtrations. Via (5.10), it is enough to verify that (a–d) are satisfied by the filtrations qW (N_{p}) and i^{−1}W (N_{p}) induced by the monodromy weight filtration W (N_{p}) in an “algebraic extension” (5.10.a). We leave to the reader to verify that (5.12) provides exactly the relations (a–d). □

Finally, our last list of properties is the following: 5.14. Third characterization. Let V and j be as in (5.10).

i) Consider the filtrations W (H) = W (H; N, V, k) and W (H_{c}) = W^{c}(H_{c}; N_{c}, V, k). Then the following holds:

a) V W_{l}(H) ⊂ W_{l−2}(H_{c}) and jW_{l}(H_{c}) ⊂ W_{l}(H), in particular NW_{l}(H) ⊂ W_{l−2}(H), and N_{c}(W_{l}(H_{c})) ⊂ W_{l−2}(H_{c}).

b) V N^{l−1} : Gr_{k+l}^{W}H → Gr_{k−l}^{W}(H_{c}) is an isomorphism if l > 0.

c) Assume that (H; H_{c}, V, j) = ⊕_{i=1,2}(H_{i}; H_{c,i}, V_{i}, j_{i}) such that V_{1} and j_{2} are isomorphisms. Then: N^{l}j : Gr_{k+l}^{W}(H_{c}) → Gr_{k−l}^{W}(H) is an isomorphism if l ≥ 0 (cf. 5.8.ii).

ii) Assume that the hypothesis (i.c) holds. Then the conditions (a-b-c) characterize uniquely the filtrations W (H) and W (H_{c}). Proof.

i) Consider an extension as in (5.10.a). Then the relation N_{p}W_{l}(H_{p}) ⊂ W_{l−2}(H_{p}) (cf. 2.5.a) gives a). The second part is exactly (5.13.a) and (5.13.d). For c) concider the two cases when V or j is an isomorphism and apply (2.6.b). ii) From (a-b-c) one has:

\begin{matrix}(a) W_{k+l}(H) = ker(V N^{l} : H → H_{c}/W_{k−l−2}(H_{c})) & (l ≥ 0) \\ (b′) W_{k+l}(H_{c}) = ker(N^{l+1}j : H_{c} → H/W_{k−l−2}(H)) & (l ≥ −1) \\ (c′) W_{k−l}(H) = N^{l}jW_{k+l}(H_{c}) & (l ≥ 0) \\ (d) W_{k−l}(H_{c}) = V N^{l−1}W_{k+l}(H) & (l > 0). \end{matrix}

Indeed, by descent induction over l, it can be verified that (a) and (d) follows from (i.a-b) and (b’) and (c’) from (i.a) and (i.c). Then these properties give an inductive construction of the filtrations, which assures the unicity. □ Notice that under the splitting hypothesis of (ii), (5.13.a–d) is equivalent to (a-b’-c’-d) above. 5.15. Remark. The map N^{l}j from (5.14.i.c) is not an isomorphism in general. For example, consider the situation described in (5.8.i). 5.16. Remark. Since the filtrations W (H; N, V, k) and W^{c}(H_{c}; N_{c}, V, k) depend only on the operators N, N_{c} and V, it is natural to ask about a natural set of properties of the system (N, N_{c}, V) such that the associated filtrations still satisfy the properties listed above. Notice that if N = j ∘ V and N_{c} = V ∘ j as in (5.2), then: (*) N and N_{c} are nilpotent; V N = N_{c}V, and ker V ⊂ ker N, im N_{c} ⊂ im V.

We leave to the reader to verify that if the maps (N, N_{c}, V) satisfy (*) then there exists j such that N = j ∘ V and N_{c} = V ∘ j. 5.17. In the sequel we study the relative version of the above definitions and results. For the classical case see [6] and [27], (cf. 2.8). In order to prove our theorem (the uniqness of the relative filtrations), we need a technical result, a generalization of the relations (5.13 a–b),which can be interesting on its own. 5.18. Proposition. Let V and j be as in (5.10) and W (H) = W (H; N, V, k) and W (H_{c}) = W^{c}(H_{c}; N_{c}, V, k). Then for any j ≥ k and l ≥ 0 one has:

(α) W_{j+l}(H) = ker(V N^{l} : H → H_{c}/W_{j−l−2}(H_{c})

(β) W_{j+l}(H_{c}) = ker(N_{c}^{l+1} : H_{c} → H_{c}/W_{j−l−2}(H_{c})). Proof. Consider an extension as in (5.10.a), then (for j ≥ k and l ≥ 0) use the identity:

W_{j+l}(H_{p}; N_{p}, k) = ker(N_{p}^{l+1} : H_{p} → H_{p}/W_{j−l−2}(H_{p})). □

5.19. Now, consider the maps V : H → H_{c} and j : H_{c} → H such that N = j ∘ V and N_{c} = V ∘ j are nilpotent. Assume that there exist two filtrations W (H) and W (H_{c}) of H and H_{c} respectively, such that

V W_{l}(H) ⊂ W_{l}(H_{c}) and jW_{l}(H_{c}) ⊂ W_{l}(H) .

5.20. Theorem. There exists at most one pair of filtrations M(H) and M(H_{c}) of H and H_{c} respectively such that

i) V (M_{l}(H)) ⊂ M_{l−2}(H_{c}) and jM_{l}(H_{c}) ⊂ M_{l}(H);

ii) M(H) induces on Gr_{i}^{W} H the filtration W (Gr_{i}^{W} H; Gr_{i}^{W} N, Gr_{i}^{W} V, i); iii) M(H_{c}) induces on Gr_{i}^{W} H_{c} the filtration W^{c}(Gr_{i}^{W} H_{c}; Gr_{i}^{W} N_{c}, Gr_{i}^{W} V, i) . If the filtrations M(H) and M(H_{c}) exist, then they satisfy the following

identities too:

Let k := min{l : Gr_{i}^{W} H = Gr_{i}^{W} H_{c} = 0 for any i > l}. Then, for any j ≥ k and l ≥ 0 one has

(α) M_{j+l}(H) = ker(V N^{l} : H → H_{c}/M_{j−l−2}(H_{c}))

(β) M_{j+l}(H_{c}) = ker(N_{c}^{l+1} : H_{c} → H_{c}/M_{j−l−2}(H_{c})).

The pair of filtrations M(H) and M(H_{c}) is called the filtration associated with V and j, relative to the pair (W (H), W (H_{c})). Proof. Assume that M(H) and M(H_{c}) exist, i.e. they satisfy (i-ii-iii). We have to prove their uniqness and that (α) and (β) hold. We will use induction over dim H + dim H_{c}. Since all the properties are compatible with an index shift, we can assume that k = 0. Consider the exact sequences:

0 → W_{−1}(H) → H →^{p} Gr_{0}^{W} H → 0

0 → W_{−1}(H_{c}) → H_{c} →^{p} Gr_{0}^{W} H_{c} → 0 .

Now, the induced filtrations {W_{−1}(H) ∩ M_{l}(H)}_{l} and {W_{−1}(H_{c}) ∩ M_{l}(H_{c})}_{l} satisfy the above properties applied for the spaces W_{−1}(H), W_{−1}(H_{c}) and for the induced maps V |W_{−1}(H), j|W_{−1}(H_{c}) and induced filtrations W (H)|W_{−1}(H) and W (H_{c})|W_{−1}(H), therefore the relative filtrations M(W_{−1}(H)) and M(W_{−1}(H_{c})) exist and

(∗) \{ \begin{matrix}M_{l}(W_{−1}(H)) = M_{l}(H) ∩ W_{−1}(H) \\ M_{l}(W_{−1}(H_{c})) = M_{l}(H_{c}) ∩ W_{−1}(H_{c}) . \end{matrix}

Moreover, by the inductive step, these filtrations satisfy the (appropriate) relations (α) and (β). Notice that k(W_{−1}) = min{l : Gr_{i}^{W} W_{−1}H = Gr_{i}^{W} W_{−1}H_{c} = 0 for i > l} < 0.

Now, we prove the relations (α) and (β) for M(H) and M(H_{c}). The relation (α) The inclusion ⊂ follows from i). For the invers inclusion fix j ≥ 0 and l ≥ 0. Take x ∈ H such that V N^{l}x ∈ M_{j−l−2}(H_{c}). Let x̂ := px its class in Gr_{0}^{W} H. Then (Gr_{0}^{W} V N^{l})x̂ ∈ pM_{j−l−2}(H_{c}) = W_{j−l−2}(Gr_{0}^{W} H_{c}; Gr_{0}^{W} N_{c}, Gr_{0}^{W} V, 0). Then, by (5.18.α) x̂ ∈ W_{j+l}Gr_{0}^{W} H = pM_{j+l}(H). Therefore, there exists y ∈ M_{j+l}(H) such that z := x − y ∈ W_{−1}(H). Consider now V N^{l}z. Since V N^{l}x ∈ M_{j−l−2}(H_{c}) and V N^{l}y ∈ M_{j−l−2}(H_{c}) (by (i)) one has that V N^{l}z ∈ W_{−1}(H_{c})∩M_{j−l−2}(H_{c}) = M_{j−l−2}(W_{−1}(H_{c})). Since j ≥ 0 > k(W_{−1}), and M(W_{−1}) satisfy the condition (α), one has z ∈ M_{j+l}(W_{−1}(H)). Therefore x = y + z ∈ M_{j+l}(H) (by (*)). The relation (β) is proved similarly (use (5.18.β),(*), and the inductive step).

Now, we prove the uniqness of M. Notice that (α) and (β) for j = 0 read as:

(a”) M_{l}(H) = ker(V N^{l} : H → H_{c}/M_{−l−2}(H_{c})) for l ≥ 0,

(b”) M_{l}(H_{c}) = ker(N_{c}^{l+1} : H_{c} → H_{c}/M_{−l−2}(H_{c})) for l ≥ 0. Consider also the identities:

(c”) M_{−l}(H) = N^{l}M_{l}(H) + M_{−l}(W_{−1}(H)) for l > 0,

(d”) M_{−l}(H_{c}) = V N^{l−1}M_{l}(H) + M_{−l}(W_{−1}(H_{c})) for l > 0. The proof of (c”) and (d”) is the following. The inclusions ⊃ follow by (i) and (*). For the invers inclusion of (c”) take x ∈ M_{−l}(H). Then x̂ ∈ pM_{−l}(H) = W_{−l}(Gr_{0}^{W} H; Gr_{0}^{W} N, Gr_{0}^{W} V, 0). Now, by (5.13.c) x̂ ∈ (Gr_{0}^{W} N)^{l}pM_{l}(H), hence there exists y ∈ M_{l}(H) such that x − N^{l}y ∈ W_{−1}(H). By (i) x − N^{l}y ∈ M_{−l}(H) hence the inclusion follows. For (d”) use (5.13.d).

Now, just notice that the relations (a”–d”) determine the filtrations M(H) and M(H_{c}) completely. □ 5.21. Remark. The results of this section are compatible with those in [19], §1. E.g.: Saito in (1.9) [19] gives the relative monodromy filtration in the presence of a decomposition (cf. 1.9.1 loc. cit.); in our theorem (5.20) we do not use any supplementary restriction of these type. But the decomposition in (1.9) [loc. cit.] is allways satisfied in our case (and in general in any geometric situations). It is equivalent with our assumption (5.14.c), and in this paper it is proved in (9.6).

CHAPTER II. THE CASE OF ONE–DIMENSIONAL BASE SPACE.

§6. Notations and preliminaries

6.1. In this chapter we study the one–dimensional deformations of isolated complete intersection singularities. More precisely, we consider an analytic germ g : (Y, y) → (B, 0) such that (g^{−1}(0), y) is an icis, B is one–dimensional, and we will assume that the generic fiber of g is smooth. We fix a projective extension f : X → B as in section 4, and we will keep the notations of that section.

Let k = dim g^{−1}(0). Consider the (real) de Rham cohomology groups H^{k}(Y_{s}) and H^{k}(X_{s}) of the fibers Y_{s} and X_{s}. Moreover, let H_{c}^{k}(Y_{s}) be the cohomology group of the differential forms with compact support. One has a natural map j : H_{c}^{k}(Y_{s}) → H^{k}(Y_{s}) (defined by j[w]_{c} = [w]). Define the variation map V ar : H^{k}(Y_{s}) → H_{c}^{k}(Y_{s}) as follows. Let T_{geom} : (Y_{s}, ∂Y_{s}) → (Y_{s}, ∂Y_{s}) be the geometric monodromy of the germ g (defined modulo an isotopy) such that T_{geom} in a neighbourhood of ∂Y_{s} is the identity. Then V ar[w] = [T_{geom}^{∗}ω − ω]_{c}. These are connected with the invariants of the projective extension via the diagram:

[figure: H^{k}(X_{s}); H^{k}(X_{s}); H^{k}(X_{s}); H^{k}(Y_{s}); H_{c}^{k}(Y_{s}); H^{k}(H_{s}); (6.2); Var; j; T_{p} − I; q; i; q; =; ∩]

where T_{p} is the algebraic monodromy of the projective fibers, and i[ω]_{c} = [ω], q[ω] = [ω|Y_{s}].

Notice that the Poincaré maps η : H_{k}(Y_{s}) → H_{c}^{k}(Y_{s}) and η : H_{k}(Y_{s}, ∂Y_{s}) → H^{k}(Y_{s}) identify the groups H_{k}(Y_{s}; Z) and H_{k}(Y_{s}, ∂Y_{s}; Z) with some Z–lattices in H_{c}^{k}(Y_{s}) and H^{k}(Y_{s}) respectively, such that the maps j and V ar are defined over Z. Moreover, theorem (4.3) assures that i is one–to–one, and q is onto.

The monodromy operators T_{c} : H_{c}^{k}(Y_{s}) → H_{c}^{k}(Y_{s}); T : H^{k}(Y_{s}) → H^{k}(Y_{s}) and T_{p} : H^{k}(X_{s}) → H^{k}(X_{s}) are defined by [ω] → [T_{geom}^{∗}ω]. (6.2) is invariant with respect to the monodromy action, i.e:

(6.3) iT_{c} = T_{p}i, qT_{p} = T q, T_{c} ∘ V ar = V ar ∘ T,

in particular: T j = jT_{c}, j ∘ V ar = T − 1 and V ar ∘ j = T_{c} − 1.

Notice that if T_{Z} is the natural monodromy of H_{k}(Y_{s}, Z), then T_{c}η(c) = ηT_{Z}^{−1}(c), hence T_{Z}^{−1} ⊗ R = T_{c}. 6.4. There is a non–degenerate (−1)^{k}–symmetric real bilinear form on H^{k}(X_{s}) given by ([α], [β])_{X} = ∫_{X_{s}} α ∧ β. Similarly, < α, β >→ ∫_{Y_{s}} α ∧ β induces a perfect pair H_{c}^{k}(Y_{s}) ⊗ H^{k}(Y_{s}) → R. The following properties hold:

(6.5)  \{ \begin{matrix}(iα, β)_{X} =< α, qβ >; \\ < T_{c}α, T β >=< α, β > , (T_{p}α, T_{p}β)_{X} = (α, β)_{X}; \\ < V ar(α), β >= (−1)^{k+1} < V ar(β), T α >, \\ < α, j(β) >= (−1)^{k} < β, j(α) > . \end{matrix}

6.6. Let T = T_{s} ·T_{u} be the decomposition of T in its semisimple and nilpotent part. Take N := log T_{u}. Similarly define N_{p} = log T_{p,u} and N_{c} = log T_{c,u}. By the monodromy theorem there exists m such that T_{s}^{m} = 1. Then

N = \frac{1}{m} log T^{m} = (T^{m} − 1)[\frac{1}{m} ∑_{k≥0}(−1)^{k} \frac{(T^{m} − 1)^{k}}{k + 1} ].

Define:

V = V ar ∘ [1 + T + ⋯ + T^{m−1}] · \frac{1}{m} ∑_{k≥0}(−1)^{k} \frac{(T^{m} − 1)^{k}}{k + 1} .

Notice that V is independent of the choice of m and the following relations hold:

(6.7) i ∘ V ∘ q = N_{p}, j ∘ V = N and V ∘ j = N_{c} .

For emphasis: 6.8. Corollary. A projective extension f of g as in section 4 provides an ”algebraic extension” of H^{k}(Y) →^{V} H_{c}^{k}(Y) →^{j} H^{k}(Y) (cf. 5.10).

The relations (6.5) transform in:

(6.9) \{ \begin{matrix}< N_{c}α, β > + < α, Nβ >= (N_{p}α, β)_{X} + (α, N_{p}β)_{X} = 0 \\ < V α, β >= (−1)^{k+1} < V β, α > . \end{matrix}

6.10. If H is a vector space over C, and T : H → H, then H_{λ} denotes the generalised λ–eigenspace of T, and T_{λ} := T |H_{λ}. Then (H, T) = ⊕_{λ}(H_{λ}, T_{λ}).

Notice that all the invariants defined in this section are compatible with the spectral decomposition of the monodromy action. 6.11. In order to simplify the notation we will denote the projective and the Milnor fiber by X and Y (instead of X_{s} and Y_{s}) respectively; but we keep the notation X_{0} for the singular fiber f^{−1}(0).

§7. The mixed Hodge structure of H^{k}(Y). The weight filtration.

With the notations of section 6, one has:

7.1. Proposition. Consider the mixed Hodge structures on the cohomology groups H^{k}(X_{0}), H^{k}(X) and H^{k}(Y): H^{k}(X_{0}) carries Deligne’s MHS [5], H^{k}(X) the limit MHS constructed by Schmid [22], and H^{k}(Y) the MHS of the vanishing cohomology constructed by Steenbrink [25]. Then:

0 → H^{k}(X_{0}) → H^{k}(X) →^{q} H^{k}(Y) → 0

is an exact sequence of MHS’s. Proof. See [25] (2.6.f) and theorem (4.3). □

The above sequence induces another exact sequence of MHS’s:

(7.2) 0 → P^{k}(X_{0}) → P^{k}(X) →^{q} H^{k}(Y) → 0,

where P^{k}(X_{0}) := H^{k}(X_{0}) ∩ P^{k}(X).

Notice that ( , )_{X} induces on P^{k}(X) a real non–degenerate form such that the MHS on P^{k}(X) is polarized by N|P^{k}(X) (cf. 2.10). Moreover, all the properties listed in section 6 are true for P^{k}(X) instead of H^{k}(X_{s}).

The above exact sequences are compatible with the monodromy action, and T |P^{k}(X_{0}) = identity. 7.3. Corollary.

⨁_{λ≠1} H^{k}(Y)_{λ} = ⨁_{λ≠1} P^{k}(X)_{λ} = ⨁_{λ≠1} H^{k}(X)_{λ} .

In particular, ⊕_{λ≠1}H^{k}(Y)_{λ} has a natural MHS, polarized by ⊕_{λ≠1}N_{λ}. The semi–simple part ⊕_{λ≠1}T_{s,λ} of the monodromy is a morphism of type (0, 0). The weight filtration is the monodromy weight filtration W (⊕_{λ≠1}N_{λ}, k), centered at k.

The next theorem shows that even for λ ≠ 1, the weight filtration of H^{k}(Y)_{1} is still topological: it depends only on the monodromy N_{1} = log T_{1} and the subspace ker V ar = ker V_{1}.

7.4. Theorem. The weight filtration of the MHS on the vanishing cohomology space H^{k}(Y)_{1} is W (H^{k}(Y)_{1}; N_{1}, V_{1}, k) . Proof. Use (6.8), (5.10) and (7.2). □ 7.5. Remark. (7.3) and (5.4) gives that (7.4) is true for H^{k}(Y) too, i.e. its weight filtration is exactly W (H^{k}(Y); N, V, k). 7.6. Example. Assume that (Y, y) is smooth. Then V ar (hence V_{1} too) is an isomorphism (cf. [1]). Therefore, by (5.4.a), the weight filtration of H^{k}(Y)_{1} is the monodromy weight filtration W (N_{1}; k + 1), centered at k + 1 (cf. [25]). 7.7. Remark. Starting from the exact sequence (of vector spaces) (7.1), and from the existence of Schmid’s limit mixed Hodge structure, one can even try to introduce a mixed Hodge structure on the vanishing cohomology of g. We want to emphasize that this does not replace Steenbrink’s constuction, because from this exact sequence does not follow that the induced MHS on the local Milnor fiber is independent on the choice of the projective extension. On the other hand, this approach can be very useful to provide nice properties of the local MHS from the global one.

§8. The mixed Hodge structure of H_{c}^{k}(Y). The weight filtration.

The mixed Hodge structure on H_{c}^{k}(T) was constructed by Steenbrink in [25]. It is dual of the MHS of H^{k}(Y) with respect to Q(−k) [loc. cit. 2.6.e]:

(8.1) \{ \begin{matrix}W_{l}H_{c}^{k}(Y) = [W_{2k−1−l}H^{k}(Y)]^{⊥}, and \\ F^{p}H_{c}^{k}(Y) = [F^{k−p+1}H^{k}(Y)]^{⊥}, \end{matrix}

where ⊥ is the orthogonality with respect to the perfect pair <, > (cf. 6.4).

Since the MHS of P^{k}(X) is auto–dual, (7.2) and (6.5) and (8.1) give the following exact sequence of MHS’s:

(8.2) 0 → H_{c}^{k}(Y) ↪^{i} P^{k}(X) → P^{k}(X)/H_{c}^{k}(Y) → 0.

Since the monodromy on the third term acts trivially, ⊕_{λ≠1}H_{c}^{k}(Y)_{λ} has a polarized MHS ( polarized by ⊕_{λ≠1}N_{c,λ}), which is dual of ⊕_{λ≠1}H^{k}(Y)_{λ}.

The parallel statement of (7.4) is the following: 8.3. Theorem. The weight filtration of H_{c}^{k}(Y)_{1} is W^{c}(H_{c}^{k}(Y)_{1}; N_{c,1}, V_{1}, k) . Proof. Use (5.10) and (8.2). □ 8.4. Example. Assume that (Y, y) is smooth. Then V_{1} is an isomorphism, hence (by 5.6.a) the weight filtration on H_{c}^{k}(Y)_{1} is W (N_{c,1}, k − 1).

§9. The decomposition of H^{k}(Y)_{˜ 1} and H_{c}^{k}(Y)_{˜ 1}.

9.1. In the sequel, ⊕_{h} denotes a direct sum of MHS’s which split over R. The functor ˜ (cf. 3.3) applied for the monomorphism of MHS’s P^{k}(X_{0}) ↪ P^{k}(X)_{1} gives P^{k}(X_{0})˜ ↪ P^{k}(X) _{˜ 1} (cf. 3.2). Notice that P^{k}(X_{0})˜ ⊂ ker N_{1}, hence one has:

9.2. Corollary of (3.4). There exist two polarized MHS’s, P_{long} and P_{short}, such that:

a) P^{k}(X) _{˜ 1} = P_{long} ⊕_{h} P_{short}, in particular: (P_{long})_{R} and (P_{short})_{R} are orthogonal with respect to S_{R}, and both are N_{p}–invariant subspaces.

b) ker N_{p}|P_{short} = P^{k}(X_{0})˜ .

In the sequel, we will use the notations N_{p}|P_{short} = N_{p,short} and N_{p}|P_{long} = N_{p,long}. The polarization form S_{p}(= ( , )_{X}) splits too: S_{p} = S_{p,long} ⊕ S_{p,short}.

The exact sequence (cf. 3.2)

0 → P^{k}(X_{0})˜ → P^{k}(X) _{˜ 1} → H^{k}(Y) _{˜ 1} → 0

and (9.2) gives the following. Notice that all the local invariants can be described in terms of the global ones!

9.3. Theorem.

a) There exist two sub–mixed Hodge structures H_{long} and H_{short} of H^{k}(Y) _{˜ 1}, both N–invariant, such that H^{k}(Y) _{˜ 1} = H_{long} ⊕_{h} H_{short}. Denote N|H_{long} = N_{long} and N|H_{short} = N_{short}.

b) One has the identification (H_{long}, N_{long}) = (P_{long}, N_{p,long}), hence the weight filtration of H_{long} is W (H_{long}; N_{long}, k), the form S_{long} = S_{p,long} is non–degenerate and N_{long} polarizes the MHS (H_{long}, S_{long}).

c) Moreover, (H_{short}, N_{short}) = (P_{short}/ ker N_{p,short}, N̂_{p,short}), where N̂_{p,short} is induced by N_{p,short}. Therefore, W (H_{short}; N_{short}, k + 1) is the weight filtration of H_{short}. By (2.11), H_{short} carries a polarized MHS : if S_{short}(x̂, ŷ); = S_{p,short}(x, Ny), then N_{short} polarizes the MHS (H_{short}, S_{short}).

The dual statement is the following: 9.4. Theorem.

a) There exist two sub–MHS’s H_{c,long} and H_{c,short} of H_{c}^{k}(Y) _{˜ 1}, both N_{c}– invariant, such that H_{c}^{k}(Y) _{˜ 1} = H_{c,long} ⊕_{h} H_{c,short}. Denote N_{c}|H_{c,long} = N_{c,long} and N_{c}|H_{c,short} = N_{c,short}.

b) (H_{c,long}, N_{c,long}) = (P_{long}, N_{p,long}), its weight filtration is (H_{c,long}; N_{c,long}, k), it has a polarization form S_{c,long} := S_{p,long} and (H_{c,long}, S_{c,long}) is polarized by N_{c,long}.

c) (H_{c,short}, N_{c,short}) = (imN_{p,short}, N_{p,short}|imN_{p,short}), its weight filtration is (H_{c,short}; N_{c,short}, k−1). By (2.11) H_{c,short} carries a (−1)^{k−1}–symmetric polarization form defined by S_{c,short}(Nα, Nβ) := S_{p,short}(α, Nβ), and (H_{c,short}, S_{c,short}) is a MHS polarized by N_{c,short}.

The next result gives the decomposition of the maps V and j and the corresponding bilinear forms: 9.5. Corollary.

Consider the identifications given by (9.3) and (9.4). Then:

a) j : P_{long} ⊕ imN_{p,short} → P_{long} ⊕ P_{short}/ ker N_{p,short} decomposes as j = j_{long} ⊕ j_{short}, where:

\{ \begin{matrix}j_{long} = id_{P_{long}}, and \\ j_{short} = (P_{short} → P_{short}/ ker N_{p,short}) ∘ (imN_{p,short} ↪ P_{short}) . \end{matrix}

In particular, j˜ is of type (0, 0), j_{long} is an isomorphism and one has the following identifications of MHS’s:

(ker j)˜ = ker j_{short} = ker N_{c,short}.

b) V : P_{long} ⊕ P_{short}/ ker N_{p,short} → P_{long} ⊕ imN_{p,short} decomposes as V = V_{long} ⊕ V_{short}, where: V_{long} = N_{p,long} and V_{short} = N̂_{p,short}.

In particular, V ˜ is a morphism of type (−1, −1), V_{short} is an isomorphism, and one has the following identifications of MHS’s:

(ker V)˜ = ker V_{long} = ker N_{long}.

c) The perfect pair <, >: H_{c}^{k}(Y)_{1} ⊗ H^{k}(Y)_{1} → R has a decompositon in <, >_{long} ⊕ <, >_{short}, such that:

\{ \begin{matrix}<, >_{long}: P_{long} ⊗ P_{long} → R is exactly S_{p,long}; \\ <, >_{short}: imN_{p,short} ⊗ P_{short}/ ker N_{p,short} → R is given by: \end{matrix}
< N_{p,short}x, ŷ >_{short}= S_{p,short}(N_{p,short}x, y) = (N_{p,short}x, y)_{X}.

Notice that <, >_{long}= S_{long} = S_{c,long}, and:

[figure: P_{short}/ ker N_{p,short} ⊗ P_{short}/ ker N_{p,short}; imN_{p,short} ⊗ P_{short}/ ker N_{p,short}; imN_{p,short} ⊗ N_{p,short}; 1 ⊗ N̂_{p,short}; N̂_{p,short}; −S_{short}; <, >_{short}; −S_{c,short}; R; R; R; ∥; ∥]

In particular, S_{short} = − < V_{short}·, · > and S_{c,short} = − < ·, V_{short}^{−1}· >, i.e the polarization forms S_{short} and S_{c,short} are basically given by the (modified) variation map V_{short}. (See also 6.9.)

The above decomposition has the following very important consequence. It is formulated in terms of topological objects, but it is the consequence of the presence of a polarized mixed Hodge structure (cf. 5.8. as well).

9.6. Decomposition Theorem. The system (H^{k}(Y), H_{c}^{k}(Y), V, j) has a direct sum decomposition (H′ ⊕ H′′, H′_{c} ⊕ H′′_{c}, V′ ⊕ V′′, j′ ⊕ j′′) such that V′ and j′′ are isomorphisms. The same is true for (H^{k}(Y)_{1}, H_{c}^{k}(Y)_{1}, V_{1}, j_{1}).

The Decomposition Theorem allows us to represent the weight filtration in two different ways. More precisely, (5.7) (see also 5.8.ii) and (5.14) read as:

9.7. Corollary.

a) W H^{k}(Y) = W (H^{k}(Y); N, V, k) = W^{c}(H^{k}(Y); N, j, k + 1);

b) W H_{c}^{k}(Y) = W^{c}(H_{c}^{k}(Y); N_{c}, V, k) = W (H_{c}^{k}(Y); N_{c}, j, k − 1);

c) i) V W_{l}H^{k}(Y) ⊂ W_{l−2}H_{c}^{k}(Y), jW_{l}H_{c}^{k}(Y) ⊂ W_{l}H^{k}(Y), ii) V N^{l−1} : Gr_{k+l}^{W}H^{k}(Y) → Gr_{k−l}^{W}H_{c}^{k}(Y) is an isomorphism if l > 0, iii) N^{l}j : Gr_{k+l}^{W}H_{c}^{k}(Y) → Gr_{k−l}^{W}H^{k}(Y) is an isomorphism if l ≥ 0.

Moreover the properties i–ii–iii determine uniquely the weight filtrations. Actually, the maps V N^{l−1} and N^{l}j are isomorphisms of Hodge structures.

All the above properties are true for the generalized 1–eigenspaces and correspondent maps, instead of the whole cohomology groups. (Notice that the relations i–ii–iii were already proved in [25].) 9.8. Example. The two different representation of the weight filtration gives us nice relations for the operators N, N_{c}, V and j. For example, assume that k = 1. Then (9.7.a-b) has the following form:

a) W_{0}H^{1}(Y) = imN = imNj, W_{1}H^{1}(Y) = ker V = im j ∩ ker N.

b) W_{0}H_{c}^{1}(Y) = imV = ker j + imN_{c}(= ker j ⊕ imN_{c}), W_{1}H_{c}^{1}(Y) = ker N_{c} = ker jN_{c}.

Already in this very particular case (of k = 1), the above identities provide non–trivial restrictions for the maps V and j (besides N_{c} = V ∘j, N = j ∘V), which cannot be obtained by purely topological arguments.

§10. Applications. Connection with the MHS of the links.

In this section we are interested in the kernel and cokernel of the maps j, V and N_{c}. They provide the MHS of the corresponding real links. Their understanding will provide the main new result (10.6) of this section (but first we had to recall some well-known facts about the cohomology groups of links).

We start with the following exact sequences of MHS’s (cf. [25]):

(10.1) \begin{matrix}0 → H_{\{y\}}^{k}(Y_{0}) → H_{c}^{k}(Y)_{1} →^{j} H^{k}(Y)_{1} → H_{\{y\}}^{k+1}(Y_{0}) → 0 \\ 0 → H_{\{y\}}^{k+1}(Y) → H^{k}(Y)_{1} →^{V_{1}} H_{c}^{k}(Y)_{1} → H_{\{y\}}^{k+2}(Y) → 0 \end{matrix}

where: Y_{0} := X_{0} ∩ Y.

The decompositions of j and V_{1} described in section 9, give the characterization of the R–split MHS’s associated with the above local cohomology groups in terms of (P^{k}, N_{p}). Already this presentation gives a series of properties of the above MHS’s, (for example the corresponding vanishing results for Gr_{l}^{W} ’s).

Recall, that one has the following isomorphism of MHS’s [25]:

(10.2) H_{{y}}^{i}(Y_{0}) = H^{i−1}(K_{Y_{0}}) and H_{{y}}^{i+1}(Y) = H^{i}(K_{Y}); i = k, k + 1,

where K_{Z} is the (real) link of the analytic space Z : K_{Z} = Z ∩ ∂B, where B is a sufficiently small ball. 10.3. Remarks.

a) If (Y, y) is smooth, (or at least is Q–homology manifold), then V is an isomorphism (over Z in the first case, and over Q, in the second case). Then (9.5.b) implies that P_{long} = 0, hence all ”long–type”–invariants vanish.

In this case, by the results of section 9, one has

H^{k}(X_{0}) = ker(N_{p} : H^{k}(X) → H^{k}(X)) .

This is exactly the ”invariant cycle theorem” for the projective family f : X → B.

b) If (Y, y) is not smooth then one has the following isomorphism:

\frac{ker[N_{p} : H^{k}(X) → H^{k}(X)]}{˜ H^{k}(X_{0})}˜  = \frac{ker N_{p}|P^{k}(X) _{˜ 1}}{P^{k}(X_{0})}˜  = ker V ˜ = H_{\{y\}}^{k+1}(Y)˜ .

For the proof of this “generalized invariant cycle theorem” in the general case (i.e. when f is not a projective family with “good homological properties” (cf. 4.3)), see [20], or the Appendix of [7], written by R. García López and J. Steenbrink.

c) Consider once more the isomorphism H_{{y}}^{k+1}(Y)˜ = ker N_{long} of MHS’s which split over R. Via (3.11), H_{{y}}^{k+1}(Y)˜ determines completely H_{long} = H_{c,long}. This supports the idea that the ”long–type” invariants are provided by the germ (Y, y), and the ”short–type” invariants by the map–germ g.

10.4. Now consider an icis (Z, y) and a germ g_{1} : (Z, y) → (C, 0) such that (Y, y) := (g_{1}^{−1}(0), y) is an icis. Moreover, take another germ g_{2} : (Y, y) → (C, 0) such that (Y_{0}, y) = (g_{2}^{−1}(0), y) is icis too. We will distinguish between the invariants of g_{1} and g_{2} by putting the germ in paranthesis after the corresponding invariants.

Now, from (10.1) one has:

(10.5) \begin{matrix}H_{\{y\}}^{k+1}(Y)˜ = ker V_{1}(g_{2}) = ker N_{long}(g_{2}) \\ H_{\{y\}}^{k+1}(Y)˜ = ker j(g_{1}) = ker N_{c,short}(g_{1}). \end{matrix}

10.6. Theorem. One has the following isomorphism of polarized mixed Hodge structures (which split over R):

H_{c,long}(g_{2}) = H_{c,short}(g_{1}) .

Notice also that N_{long}(g_{2}) = N_{c,long}(g_{2}) (isomorphism in the same sense). Proof. (10.5) and (3.11) give that H_{long}(g_{2}) = H_{c,short}(g_{1}). But H_{long}(g_{2}) = H_{c,long}(g_{2}) by (9.3)–(9.4). □ 10.7. Example. Assume that (Z, y) is a Q–homology manifold (e.g. it is smooth). Then H_{c,long}(g_{1}) = 0 (cf. 10.3.a), hence:

H_{c,long}(g_{2}) = H_{c}^{k+1}( Milnor fiber of g_{1})_{1}˜ (as polarized MHS’s).

10.8. Example. Let (Z, y) be as in (10.7). Then the decompositions in section 9, and the characterization of the weight filtration on each component give the following. Let #_{l} be the number of Jordan blocks with eivenvalue 1 and size l of the monodromy operator associated with a germ. Then

#_{l}(g_{1}) = dim Gr_{k+1−l}^{W}H^{k}(K_{Y}),
#_{l}(g_{2}) = #_{l}(g_{1}) + dim Gr_{k−l}^{W}H^{k−1}(K_{Y_{0}}).

In particular, #_{k+1}(g_{1}) = #_{k+1}(g_{2}). 10.9. Example. Let (Y, y) be a two–dimensional icis, and g : (Y, y) → (C, 0) an isolated curve singularity. Then #_{2}(g) = dim Gr_{0}^{W} H^{1}(K_{Y}) = dim Gr_{2}^{W} H_{c}^{1}(g) . In [16] it is proved that #_{2}(g) = rank H_{1}(Γ_{Y}), where Γ_{Y} is a resolution graph of Y and rank H_{1}(Γ_{Y}) is the number of independent cycles of Γ_{Y}. Moreover, dim H^{1}(K_{Y}) = #_{2}(g)+2G, where G is the sum of the geometric genus of the exceptional curves in Γ_{Y}. Now, by section 9, Gr_{l}^{W} H^{1}(K_{Y}) = 0 if l ≠ 0, 1, hence: Gr_{1}^{W} H^{1}(K_{Y}) = 2G . Moreover, H_{c}^{1}(g) = ker j ⊕ H_{c,long}, hence: dim H_{c}^{1}(g) = dim ker j + 2 · #_{2}(g) + 2G .

§11. The Hodge numbers and the signature.

11.1. Consider the mixed Hodge structures H_{c}^{k}(Y) and H^{k}(Y) associated with the germ g : (Y, y) → (C, 0). Let T_{c,s} (and T_{s}) be the semi–simple parts of monodromy operators T_{c} and T respectively. Define for any λ on the unit circle:

h_{c,λ}^{pq} = dim{ker(T_{c,s} − λ : Gr_{F}^{p} Gr_{p+q}^{W}H_{c}^{k}(Y) → Gr_{F}^{p} Gr_{p+q}^{W}H_{c}^{k}(Y))}

and similarly h_{λ}^{pq}, for any p and q. Let h^{pq} = ∑_{λ} h_{λ}^{pq} and h_{c}^{pq} = ∑ h_{c,λ}^{pq}. The dualities and the results of the last section give:

a) If h_{λ}^{pq} ≠ 0 or if h_{c,λ}^{pq} ≠ 0 then 0 ≤ p ≤ k, 0 ≤ q ≤ k.

b) h_{1}^{pq} = h_{long}^{pq} + h_{short}^{pq}, h_{c,1}^{pq} = h_{c,long}^{pq} + h_{c,short}^{pq}; h_{short}^{pq} ≠ 0 then 1 ≤ p ≤ k, and 1 ≤ q ≤ k; h_{c,short}^{pq} ≠ 0 then 0 ≤ p ≤ k − 1 and 0 ≤ q ≤ k − 1.

c) for any (p, q) and λ: h_{λ}^{pq} = h_{c,λ̄}^{k−p,k−q}, h_{long}^{pq} = h_{c,long}^{k−p,k−q}, h_{short}^{pq} = h_{c,short}^{k−p,k−q}.

d) For any (p, q) one has: h_{λ}^{pq} = hqp ; λ̄ = h_{λ}^{k−q,k−p}, if λ ≠ 1; h_{long}^{pq} = h_{long}^{qp} =

h_{long}^{k−p,k−q}; h_{short}^{pq} = h_{short}^{qp} = h_{short}^{k+1−p,k+1−q}. 11.2. Now, consider the bilinear form:

I : H_{c}^{k}(Y_{s}) ⊗ H_{c}^{k}(Y_{s}) → R

defined by I(α, β) :=< α, jβ >=< α, qiβ >= (iα, iβ)_{X}. Then I is (−1)^{k}– symmetric, and it corresponds (by the Poincaré duality) to the intersection form on H_{k}(Y_{s}). Obviously, I(T_{c}α, T_{c}β) = I(α, β), hence I is compatible with the spectral decomposition ⊕_{λ≠1}(H_{c}^{k})_{λ} ⊕ (H_{c}^{k})_{long} ⊕ (H_{c}^{k})_{short}. By (9.4), it is non–degenerate on ⊕_{λ≠1}(H_{c}^{k})_{λ} ⊕ (H_{c}^{k})_{1,long}. Let μ_{0} be the dimension of the null–space of I. If k is even, let μ_{±} = max{dim K|K ⊂ H_{c}^{k}(Y_{s}), I|K is (±)–definite }. Then μ = μ_{0} + μ_{+} + μ_{−} is exactly the Milnor number of g.

For k even, define the signature by σ = μ_{+} − μ_{−}. For isolated hypersurface singularities (μ_{0}, μ_{+}, μ_{−}) is computed in terms of Hodge numbers by Steenbrink [24]. See also [25]. 11.3. Proposition. The invariants μ, μ_{0} and σ, in terms of {h_{c}^{pq}}, have the following expression:

a) μ = ∑_{p,q} h_{c}^{pq};

b) \begin{matrix}μ_{0} & = ∑_{p+q≤k−1} h_{c}^{pq} − ∑_{p+q≥k+1} h_{c}^{pq} = ∑_{p+q≤k−1} h_{c,1}^{p,q} − ∑_{p+q≥k+1} h_{c,1}^{pq} \\ & = ∑_{p+q≤k−1} h_{c,short}^{p,q} − ∑_{p+q≥k+1} h_{c,short}^{pq}; \end{matrix}

c) if k is even then define σ_{l} = ∑_{p+q=l}(−1)^{q}h_{c}^{pq}. Then

σ = σ_{k} + 2 ∑_{l<k} σ_{l}.

Notice that σ_{l} = 0 if l is odd. Proof. It is enuogh to verify any of the above relations for the invariant subspaces (H_{c}^{k})_{≠1}, (H_{c}^{k})_{long} and (H_{c}^{k})_{short}. In the first two cases μ_{0} = 0 and σ = ∑_{k} σ_{k} (because the Hodge signature formula can be applied; i.e the computation of σ is identically the same). By the symmetry of the Hodge numbers one obtains the result. The verification of μ_{0} for (H_{c}^{k})_{short} is an easy consequence of the Lefshetz decomposition and the signature computation is similar as in the hypersurface case (see [24]). For the convenience of the reader we prove the signature formula.

Since H_{c,short}^{k} = imN_{p,short}, we can assume that N_{p,short} has no Jordan blocks of size 1. Then ( imN_{p,short})^{⊥} = ker N_{p,short} ⊂ imN_{p,short}, hence σ = σ(imN_{p,short}/ ker N_{p,short}) is equal to σ(H_{p,short}) by a well–known property of the signature. Let b^{pq} = dim H^{pq}Gr_{p+q}^{W}H_{p,short} and σ_{p,l} = ∑_{p+q=l}(−1)^{q}b^{pq}. Then σ_{p,l} = σ_{l} if l ≤ k and for l > k one has σ_{p,l} = σ_{p,2k−l}. Therefore:

σ = ∑_{l≤k} σ_{p,l} + ∑_{l>k} σ_{p,l} = ∑_{l≤k} σ_{p,l} + ∑_{l<k} σ_{p,l} = σ_{k} + 2 ∑_{l<k} σ_{l}. □

CHAPTER III. THE GENERAL CASE.

§12. The mixed Hodge structure on H^{k}(Y_{s}) and H_{c}^{k}(Y_{s}).

12.1. The basic object of this chapter is a deformation g : (Y, y) → (B, 0) of an icis (Y_{0}, y) := (g^{−1}(0), y) such that the discriminant locus Δ of g is a (local) divisor with normal crossings. Notice that, in general, (Y, y) is not an icis. Set k := dim g^{−1}(0).

Such an object appears in the following context. Consider an icis (Z, z) and a deformation h : (Z, z) → (S, 0) of the central fiber (h^{−1}(0), z), which is again an icis. Let D be the discriminant locus of h. Consider the embedded resolution π : (S̃, D̃) → (S, D) of D and fix a point d̃ ∈ D̃ together with B, one of its sufficiently small neighbourhoods. Then the pull–back π^{∗}(h) over B provides an example of our basic object. Actually, there is no hope to obtain a limit mixed Hodge structure (compatible with the monodromy action of π_{1}(S \ D, ∗)) associated with the deformation h. On the other hand, for the germ g, we prove the existence of a canonical mixed Hodge structure and we expect that (after a suitable reorganization of the data) it has similar properties as the polarized mixed Hodge structures provided by a multi–dimensional nilpotent orbit. In the next sections we list some (local) properties of this type. 12.2. Let g be as above. Fix a “homologically good” projective extension f of g as in section 4. Let Y_{s} and X_{s} be the generic fibers (cf. 4.2) and X_{0} the central fiber of f.

Assume that (C, 0) ⊂ (B, 0) is the germ of a one–dimensional reduced analytic space. Let n_{C} : (B_{C}, 0) → (C, 0) be the normalization of (C, 0) and take the pull–back n_{C}^{∗}g : (Y_{C}, y_{C}) → (B_{C}, 0) of g. Then n_{C}^{∗}g determines a one–dimensional deformation of the icis (Y_{0}, y). In particular, by [24], (see also section 7) the group H^{k}(Y_{s}) carries a mixed Hodge structure (W_{C}H^{k}(Y_{s}), F_{C}H^{k}(Y_{s})), associated with the deformation n_{C}^{∗}g. The exact sequence (7.2) (associated with n_{C}^{∗}g) is:

0 → P^{k}(X_{0}) → P^{k}(X_{s}) →^{q} H^{k}(Y_{s}) → 0.

The cohomology groups and maps in the sequence are independent of the choice of C. In the sequel we analyse how the corresponding mixed Hodge structures depend on C.

Fix C and consider the limit MHS (P^{k}(X_{s}), W_{C}, F_{C}) associated with n_{C}^{∗}f. Assume that Δ ⊂ {z_{1} ⋯ z_{n} = 0}, where (z_{1}, ⋯, z_{n}) is a local coordinate system if (B, 0). Let T_{p,i}(i = 1, ⋯, n) be the monodromy (acting on P^{k}(X_{s})) induced by an oriented single loop around the divisor {z_{i} = 0}, and N_{p,i} the logarithm of its unipotent part. Then the nilpotent operator N_{p,C} associated with the family n_{C}^{∗}f (cf. 6.6), is N_{p,C} = ∑ λ_{i}N_{p,i}, where λ_{i} is the intersection multiplicity m_{0}(C, z_{i}) > 0. In particular, N_{p,C} is in the monodromy cone {∑ λ_{i}N_{p,i} : λ_{i} > 0, i = 1, ⋯, n} of the projective family f, therefore W_{C} is independent of the choice of C: it is exactly the weight filtration of the limit MHS associated with the projective family f : f^{−1}(B^{∗}) → B^{∗} (here B^{∗} := B − {z_{1} ⋯ z_{n} = 0}) (cf. [2]).

In the construction of F_{C} (see [22, 23]), the choice of the local parameter of B_{C} influences F_{C}: F_{C} is well–defined only modulo exp(σ_{C}), where σ_{C} is the “complex monodromy cone” {∑ λ_{i}N_{i} : λ_{i} ∈ C}. Now, the existence of the limit MHS of the family f : f^{−1}(B^{∗}) → B^{∗} assures that F_{C} is independent of C (modulo the action of exp σ_{C}): it is exactly the Hodge filtration of the limit MHS of f. (Notice that this ambiguity is a natural one, it does not come from the unsufficiency of the theory.)

Now, for any C as above, consider the MHS (H^{k}(Y_{s}), qW_{C}, qF_{C}). By section 7, this is exactly the MHS of the vanishing cohomology (W_{C}H^{k}(Y_{s}), F_{C}H^{k}(Y_{s})) of n_{C}^{∗}g, constructed by Steenbrink. Since W_{C} is independent of C, one has that qW_{C} is also independent of C. Moreover, qF_{C} is independent of C as well (modulo the action of exp(∑ λ_{i}N_{i}, λ_{i} ∈ C), where N_{i} ∈ End (Y^{k}(Y_{s})) is defined similarly as N_{p,i}).

Therefore if we consider the limit MHS (P^{k}(X_{s}), W, F) of the projective family f : f^{−1}(B^{∗}) → B^{∗} (constructed by Schmid in [22]), and we define the weight, respectively the Hodge filtration on H^{k}(Y_{s}) by W H^{k}(Y_{s}) = qW P^{k}(X_{s}) and F H^{k}(Y_{s}) = qF P^{k}(X_{s}), then (H^{k}(Y_{s}), W, F) is a mixed Hodge structure which coincides with Steenbrink’s mixed Hodge structure associated with the deformation n_{C}^{∗}g for any C. This assures one more thing. Since Steenbrink’s MHS depends only on the (local) germ n_{C}^{∗}(g) (it is constructed in terms of its embedded resolution), and not on the extension n_{C}^{∗}f, the MHS (H^{k}(Y_{s}), W, F) (defined above in terms of f) is independent of the choice of the extension f.

Therefore, we proved the following result: 12.3. Theorem. Let g : (Y, y) → (B, 0) be a deformation of an icis (g^{−1}(0), y) such that its discriminant Δ is a divisor with normal crossings. Then the (middle) cohomology group of its Milnor fiber carries a mixed Hodge structure (H^{k}(Y_{s}), W, F) which has the following properties:

a) For any reduced analytic one–dimensional space–germ (C, 0) ⊂ (B, 0) (with (C, 0) ⊄ (Δ, 0)), the MHS on the vanishing cohomology of the deformation g|g^{−1}(C) : (g^{−1}(C) ∩ Y, y) → (C, 0) coincides with (H^{k}(Y_{s}), W, F) (modulo the action of the local complex monodromy cone).

b) For any projective extension f : (X , y) → (B, 0) of g such that f has no critical points outside Y (not necessarily ”homologically good”) one has an exact sequence of MHS’s:

0 → H^{k}(X_{0}) → H^{k}(X_{s}) → H^{k}(Y_{s}) → H^{k+1}(X_{0}) → H^{k+1}(X_{s}) → 0,

H^{∗}(X_{0}) carries Deligne’s MHS, H^{∗}(X_{s}) Schmid’s limit MHS and H^{k}(Y_{s}) the MHS (H^{k}(Y_{s}), W, F).

c) The MHS (H^{k}(Y_{s}), W, F) is “compatible” with the action of π_{1}(B^{∗}, ∗), in the sense which will be clarified in the next section (together with the polarization properties).

The second part b) follows from the first part a) and the one–dimensional case [25] (2.6.7).

Obviously, by duality, the inclusion H_{c}^{k}(Y_{s}) →^{i} H_{c}^{k}(X_{s}) provides on H_{c}^{k}(Y_{s}) a mixed Hodge structure with similar (or dual) properties. The formulation of this dual statement is left to the reader. Notice that the relations (8.1) still hold.

Moreover, all the invariants h_{λ}^{pq}, h_{c,λ}^{pq}, μ_{0}, μ_{±}, μ, σ can be computed by reducing the computation to the one–dimensional case, therefore all the results of section 11 are valid in this general case too.

§13. The action of π_{1}(B^{∗}).

13.1. Let g : (Y, y) → (B, 0) be a deformation of the icis (g^{−1}(0), 0), and f its ”homologically good” projective extension (cf. section 4). For simplicity, we will assume that Δ = {z_{1} ⋯ z_{n} = 0} for local coordinates (z_{1}, …, z_{n}) suitably chosen (i.e if the discriminant is a proper subset of {z_{1} ⋯ z_{n} = 0}, we define Δ as {z_{1} ⋯ z_{n} = 0}). We will simplify the notations: Y = Y_{s}, X = X_{s}(s ∉ Δ) and G := π_{1}(B \ Δ, ∗). In particular G ≈ Z^{n}.

The exact sequences:

(13.2)  \{ \begin{matrix}0 → P^{k}(X_{0}) → P^{k}(X) →^{q} H^{k}(Y) → 0 \\ 0 → H_{c}^{k}(Y) ↪^{i} P^{k}(X) → P^{k}(X)/H_{c}^{k}(Y) → 0 \end{matrix}

are equivariant with respect to the monodromy action of G in such a way, that the action is trivial on P^{k}(X_{0}), and on P^{k}(X)/H_{c}^{k}(Y) respectively. If for a character χ of G, U_{χ} denotes the generalized χ–eigenspace of a G–module U, then one has:

13.3. Proposition. If χ ≠ 1 then H^{k}(Y)_{χ} ≈ P^{k}(X)_{χ} ≈ H^{k}(X)_{X} (isomorphism of G–modules). In particular ⊕_{χ≠1}H^{k}(Y)_{χ} has a natural polarized MHS induced by the projective extension. Similar result holds for ⊕_{χ≠1}H_{c}^{k}(Y)_{χ}.

As we will see, the MHS H^{k}(Y)_{1} is more complicated. 13.4. Consider the monodromy representations: ρ : G → Aut(H^{k}(Y)), ρ_{c} : G → Aut(H_{c}^{k}(Y)) and ρ_{p} : G → Aut(P^{k}(X)); and the variation map V ar : G → Hom(H^{k}(Y) → H_{c}^{k}(Y)). They satisfy the following properties (see [1]): 13.5. Properties: For elements g and h of G one has:

[figure: P^{k}(X); P^{k}(x); P^{k}(X); H^{k}(Y); H_{c}^{k}(Y); H^{k}(Y); ρ_{p}(g) − I; V ar(g); =; j; q; i; q; (a)]

b) Var(g) ∘ j = ρ_{c}(g) − 1, j ∘ Var(g) = ρ(g) − 1 ; ρ_{p}(g) ∘ i = i ∘ ρ_{c}(g), ρ(g) ∘ q = q ∘ ρ_{p}(g) and ρ(g) ∘ j = j ∘ ρ_{c}(g);

c) ρ_{c}(g)Var(h) = Var(h)ρ(g), Var(gh) = ρ_{c}(g)Var(h) + Var(g) = ρ_{c}(h)Var(g) + Var(h);

d) The representations are compatible with the pairings ( , )_{X} and < , > (defined similarly as in section 6), i.e.:

(ρ_{p}(g)α, ρ_{p}(g)β)_{X} = (α, β)_{X}, < ρ_{c}(g)α, ρ(g)β >=< α, β >,

< Var(g)α, β >= (−1)^{k+1} < Var(g)β, ρ(g)α > .

13.6. Similarly as in the one–dimensional case, we define the infinitesimal isometries and variations. Let m be an integer such that ρ_{p}(g)^{m} = 1 for any g ∈ G. Define N_{p}(g) := \frac{1}{m} log ρ_{p}(g^{m}), N(g) = \frac{1}{m} log ρ(g^{m}) and N_{c}(g) = \frac{1}{m} log ρ_{c}(g_{m}). Moreover, let

V (g) := Var(g) ∘ [1 + ρ(g) + ⋯ + ρ(g)^{m−1}]\frac{1}{m} ∑_{k≥0}(−1)^{k} \frac{[ρ(g^{m}) − 1]^{k}}{k + 1} .

Then these maps are independent of the choice of m. 13.7. Proposition. The operators N, N_{c}, N_{p} and V satisfy the following properties (for g, h ∈ G):

a) i ∘ V (g) ∘ q = N_{p}(g); b) j ∘ V (g) = N(g), V (g) ∘ j = N_{c}(g), in particular :

N_{p} ∘ i = i ∘ N_{c}, N ∘ q = q ∘ N_{p} and N ∘ j = j ∘ N_{c};

c) N(gh) = N(g) + N(h), similarly for N_{c} and N_{p}; V (gh) = V (g) + V (h);

d) N_{c}(g)V (h) = V (h)N(g) = N_{c}(h)V (g) = V (g)N(h), N(g)N(h) = N(h)N(g), similarly for N_{c} and N_{ρ}.

e) < N_{c}(g)α, β > + < α, N(g)β >= 0, (N_{p}(g)α, β)_{X} + (α, N_{p}(g)β)_{X} = 0, < V (g)α, β >= (−1)^{k+1} < V (g)β, α > .

All these relations are compatible with respect to the spectral decomposition of the spaces. Proof. Everything follows from the one–dimensional case, expect c) and d). For c) consider the representation ρ̃: G → Aut(H^{k}(Y) ⊕ H_{c}^{k}(Y)) defined by

ρ̃(g) = ( \begin{matrix}ρ(g) & 0 \\ V ar(g) & I \end{matrix} ) .

Then ρ̃ is a representation of the (abelian) group G. Notice that

g → \frac{1}{m} log ρ̃(g^{m}) = ( \begin{matrix}N(g) & 0 \\ V (g) & 0 \end{matrix} )

is an (additive) representation. For d) use (13.5.c) and the definitions. □ 13.8. We fix a standard set of generators of G = Z^{n}: e_{i} is a class of a positively oriented loop around the divisor {z_{i} = 0}. If g = (λ_{1}, …, λ_{n}) ∈ Z^{n}, then one has N(g) = ∑_{i=1}^{n} λ_{i}N(e_{i}) and V (g) = ∑_{i=1}^{n} λ_{i}V (e_{i}) (cf. 13.7.c).

We extend the action of G to an action of R^{n} via the relations:

N(λ_{1}, ⋯, λ_{n}) := ∑ λ_{i}N(e_{i}) and V (λ_{1}, ⋯, λ_{n}) := ∑ λ_{i}V (e_{i}),

and similarly for N_{c} and N_{p}. Obviously, all the properties of (13.7) remain valid for the extended R^{n}–action.

It is an interesting problem to study the family of operators {N(λ), N_{c}(λ), V (λ)}_{λ∈τ̄}, where τ denotes the cone:

τ = {λ = (λ_{1}, ⋯, λ_{n}) ∈ R^{n}; λ_{i} > 0 for all i = 1, ⋯, n} ,

and τ̄ denotes its closure in R^{n}.

Notice that {N_{p}(λ)}_{λ∈τ} is exactly the monodromy cone σ associated with the projective family f (cf. 2.7). This gives the following addendum to (13.3):

The MHS ⊕_{χ≠1} H^{k}(Y)_{χ} (respectively ⊕_{χ≠1}H_{c}^{k}(Y)_{χ}) is polarized by the monodromy cone {N(λ)}_{λ∈τ} (respectively by {N_{c}(λ)}_{λ∈τ}).

The geometric meaning of τ ∩ Z^{n} is the following (cf. section 12): if we consider the germ of a curve (C, 0) ⊂ (B, 0) which is not contained in Δ, then the infinitesimal monodromy and variation associated with g|g^{−1}(C) : (g^{−1}(C)∩Y, y) → (C, 0) is (N(g), V (g)) where g = ∑ λ_{i}e_{i} and λ_{i} = m_{0}(C, z_{i}).

§14. The weight filtrations associated with the faces of the cone.

For any fixed J ⊂ {1, 2, …, n}, define τ_{J} = {λ = (λ_{1}, …, λ_{n}) ∈ τ̄, λ_{j} > 0 if and only if j ∈ J} ⊂ τ̄. Notice that τ_{{1,2,...,n}} = τ (cf. 13.8).

The main result of this section is the following local analog of a result of Cattani–Kaplan [2]: 14.1. Theorem.

a) For any J ⊂ {1, 2, …, n}, there exist increasing filtrations W (τ_{J}) = W (J) and W_{c}(τ_{J}) = W_{c}(J) of H^{k}(Y) and H_{c}^{k}(Y) respectively, such that for any λ ∈ τ_{J} one has:

W (H^{k}(Y); N(λ), V (λ), k) = W (J),
W^{c}(H_{c}^{k}(Y); N_{c}(λ), V (λ), k) = W_{c}(J).

b) For τ_{J} = τ, the following supplementary identities hold for any λ ∈ τ_{J}:

W^{c}(H^{k}(Y); N(λ), j, k + 1) = W ({1, 2, …, n}),
W (H_{c}^{k}(Y); N_{c}(λ), j, k − 1) = W_{c}({1, 2, …, n}).

Moreover,

V (λ)N(λ)^{l−1} : Gr_{k+l}^{W (τ)}H^{k}(Y) → Gr_{k−l}^{W (τ)}H_{c}^{k}(Y)

is an isomorphism for l > 0, and

N(λ)^{l}j : Gr_{k+l}^{W (τ)}H_{c}^{k}(Y) → Gr_{k−l}^{W (τ)}H^{k}(Y)

is an isomorphism for l ≥ 0.

Actually, W (τ) and W_{c}(τ) are the weight filtrations of the mixed Hodge structures of H^{k}(Y) and H_{c}^{k}(Y) respectively (defined in §12). c) For any λ ∈ τ̄ and J ⊂ {1, 2, …, n} one has:

V (λ)W_{l}(J) ⊂ W_{c,l}(J) and jW_{c,l}(J) ⊂ W_{l}(J).

Besides , for any J, J′ ⊂ {1, 2, …, n} and λ ∈ J one has:

V (λ)W_{l}(J ∪ J′) ⊂ W_{c,l−2}(J ∪ J′).

d) If J, J′ ⊂ {1, 2, …, n}, J ∪ J′ = {1, 2, …, n}, and λ ∈ τ_{J′}, then the pair of filtrations associated with V (λ) and j relative to the pair (W (J), W_{c}(J)) exists and it is exactly the pair (W (τ), W_{c}(τ)) (cf. 5.19-5.20).

All the above properties are true for the generalized 1–eigenspaces (H^{k}(Y)_{1}, H_{c}^{k}(Y)_{1}) (and induced maps) instead of (H^{k}(Y), H_{c}^{k}(Y)).

Proof. Let W (σ_{J}) be the filtration of H^{k}(X) corresponding to J (cf. 2.8). If we define W (J) as qW (σ_{J}), and W_{c}(J) as i^{−1}W (σ_{J}), then the first part (a) follows from (2.8) and (5.10). For (b), we succeed as follows. Fix λ ∈ τ. Then N_{p}(λ) polarizes (P^{k}(X), W, F) and P^{k}(X_{0}) ⊂ ker N_{p}(λ). Then (3.4) provides a decomposition of P^{k}(X)˜ , therefore (9.6) and (9.7) are valid for V (λ) too. The first part of (c) follows from the abelianity of the action and (5.14.c). This reference shows also that for any λ ∈ τ_{J∪J′} one has V (λ)W_{l}(J ∪ J′) ⊂ W_{c,l−2}(J ∪ J′). Now, the last assertion of (c) follows by continuity.

For (d), we have to verify the properties (i-ii-iii) of (5.20). (i) follows from (c). We will prove (ii), the part (iii) follows by dual arguments.

Let λ ∈ τ. Then λ can be represented as λ′ + λ′′, where λ′ ∈ τ_{J′} and λ′′ ∈ τ_{J}. Then V (λ) and V (λ′) have the same action on Gr^{W (J)}. Therefore, it is enough to prove the assertion for λ ∈ τ.

Let λ ∈ τ. Consider the ”extension” (H_{p}, N_{p}) = (H^{k}(X), N_{p}(λ)), and for any J take the filtration W (σ_{J}) = W (J)H_{p} of H_{p}.

Consider the diagram:

[figure: Gr_{i}^{W (J)}H_{p}; Gr_{i}^{W (J)}H_{p}; Gr_{i}^{W (J)}H_{p}; Gr_{i}^{W (J)}H; Gr_{i}^{W (J)}H_{c}; Gr_{i}^{W (J)}H; Gr_{i}^{W (J)}N_{p}(λ) − I; Gr_{i}^{W (J)}V (λ); =; Gr_{i}^{W (J)}j; Gr_{i}^{W (J)}q; Gr_{i}^{W (J)}i; Gr_{i}^{W (J)}q]

Then Gr_{i}^{W (J)}q is onto and Gr_{i}^{W (J)}i is one-to-one.

In particular, (Gr_{i}^{W (J)}H_{p}; Gr_{i}^{W (J)}N_{p}(λ)) is an “algebraic extension” of (Gr_{i}^{W (J)}H, Gr_{i}^{W (J)}H_{c}; Gr_{i}^{W (J)}V (λ), Gr_{i}^{W (J)}j). Therefore, by (5.10):

W (Gr_{i}^{W (J)}H; Gr_{i}^{W (J)}N(λ), Gr_{i}^{W (J)}V (λ), i) = (Gr_{i}^{W (J)}q)W (Gr_{i}^{W (J)}H_{p}; Gr_{i}^{W (J)}N_{p}(λ), i).

By (2.8) one has:

W (Gr_{i}^{W (J)}H_{p}; Gr_{i}^{W (J)}N_{p}(λ), i) is induced on Gr_{i}^{W (J)}H_{p} by W (τ)H_{p}.

Notice that q(W (τ)H_{p}) = W (τ)H. Now, the above identities assure that one has to show only the following:

(Gr_{i}^{W (J)}q)(filtration induced on Gr_{i}^{W (J)}H_{p} by W (τ)H_{p}) =
= filtration induced on Gr_{i}^{W (J)}H by W (τ)H.

This is equivalent to:

(14.2) q(\frac{W_{i}(J)H_{p} ∩ W_{l}(τ)H_{p}}{W_{i−1}(J)H_{p} ∩ W_{l}(τ)H_{p}} ) = \frac{qW_{i}(J)H_{p} ∩ qW_{l}(τ)H_{p}}{qW_{i−1}(J)H_{p} ∩ qW_{l}(τ)H_{p}} .

Notice that by an easy set–theoretical argument, the set on the left hand side of (14.2) is a subset of the set on the right hand side. We have to show the invers inclusion.

Let K = ker q = H^{k}(X_{0}) ⊂ H_{p}. Then it is enough to show that:

(14.3) (W_{i}(J)H_{p} + K) ∩ (W_{l}(τ)H_{p} + K) ⊂ W_{i}(J)H_{p} ∩ W_{l}(τ)H_{p} + K

for any i and l.

Now, consider the mixed Hodge structure (H_{p}; W (τ)H_{p}, F). Then K ⊂ H_{p} is a sub–mixed Hodge structure. Obviously, (14.3) is true if and only if it is true for F̃ instead of F, hence in the sequel we replace H_{p} by H_{p}˜ . Let {I^{pq}}_{pq} be the generalized Hodge decomposition of H_{p}˜ . Let E_{t} = ⊕_{p+q=t}I^{pq} be the corresponding splitting of (H_{p}˜ , W (τ)):

(14.4) W_{l}(τ)H_{p} = ⨁_{t≤l} E_{t}.

Since K = ⊕_{p,q}I^{pq} ∩ K, one has:

(14.5) K = ⊕_{t}(E_{t} ∩ K).

We need one more relation:

(14.6) W_{i}(J)H_{p} = ⊕_{t}[E_{t} ∩ W_{i}(J)H_{p}] for any i and J.

Indeed, for any α ∈ τ one has [Y, N_{p}(α)] = −2N_{p}(α), where Y is the semi– simple endomorphism defined by Y |E_{t} = (t − k)id_{E_{t}}. Then, by continuity, for any α ∈ τ_{J} also holds: [Y, N_{p}(α)] = −2N_{p}(α). Then {N_{p}(α), Y } can be extended to an sl_{2}–representation, and W (J)H_{p} = W (H_{p}; N_{p}(α), k). Therefore, (14.6) holds.

Now, (14.3) follows from (14.4-14.5.-14.6). □

14.7. Problem. Prove or disprove the assertion (14.1.d) without the assumption J ∪ J′ = {1, 2, …, n}.

14.8. Remark. The main difference between (14.1.d) and (14.7) is that the inclusion K ⊂ (H_{p}; W (τ)H_{p}, F) is a morphism of mixed Hodge structures, in particular W_{l}K = K ∩ W_{l}(τ)H_{p}. For the other faces τ_{J} ⊂ τ̄ \ τ this property, in general, is not true.

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