Géométrie algébrique / Algebraic Geometry

THE MIXED HODGE STRUCTURE

OF A COMPLETE INTERSECTION

WITH ISOLATED SINGULARITY

ANDRÁS NÉMETHI

Abstract — Consider a deformation g of an icis with discriminant Δ. Assume that Δ is a divisor with normal crossings, and let G be the fundamental group of its local complement. We announce the proof of the existence of a mixed Hodge structure on the vanishing cohomology of g compatible with the action of G, and of its equivariant decomposition in polarized mixed Hodge structures. Moreover, we find the local analog of Cattani–Kaplan’s result [1].

La structure de Hodge mixte d’une intersection complète avec singularité isolée.

Résumé — Soit g une déformation d’une icsi. Supposons que son discriminant Δ est un diviseur à croisements normaux et notons G le groupe fondamental du complémentaire local de Δ. Nous annonçons l’existence d’une structure de Hodge mixte (SHM) sur la cohomologie évanescente de g. Cette SHM est compatible avec l’action de G et admet une décomposition équivariante en structures de Hodge mixte avec polarization. Nous obtenons aussi une version locale des résultats de Cattani–Kaplan [1].

Version française abrégée— Fixons une déformation g : (Y, y) → (B, 0) d’une intersection complète avec singularité isolée (Y_{0}, y) = (g^{−1}(0), y) . Supposons que son discriminant Δ est un diviseur à croisements normaux et notons k = dim Y_{0}.

Dans le cas dim B = 1, J. Steenbrink a construit une structure de Hodge mixte sur le groupes de cohomologie H^{k}(Y) et H_{c}^{k}(Y) de la fibre Milnor Y de g [7, 8]. D’un autre côté, si f : (X , y) → (B, 0) est une application projective avec un discriminant qui est un diviseur à croisements normaux, W. Schmid a construit une SHM–limite (ie limite de structures de Hodge) sur la partie primitive P^{∗}(X) de la cohomologie de X (X étant la fibre générique) [6]. Cette SHM possède une forme bilinéaire naturelle avec une polarisation fournie par le cône de monodromie (au sens de Cattani–Kaplan [1, 2]). Beaucoup de travaux ont été consacrés aux propriétés de cette SHM (voir [1, 2] ainsi que leurs références).

Dans la présente note, nous annonçons le résultat suivant [5]. Théorème: Soient g et k comme ci–dessus. Le k–ième groupe de cohomologie de la fibre de Milnor Y possède une structure de Hodge mixte (H^{k}(Y), W, F) satisfaisant les propriétés suivantes:

a) Pour tout germe analytique, réduit et de dimension un (C, 0) ⊂ (B, 0) (avec (C, 0) ⊄ (Δ, 0)), la SHM sur la cohomologie évanescente de la déformation g|g^{−1}(C) : (g^{−1}(C) ∩ Y, y) → (C, 0) coincide avec (H^{k}(Y), W, F).

b) Pour toute extension projective f : (X , y) → (B, 0) de g, sans point critique hors de Y, on a la suite exacte de SHM:

0 → H^{k}(X_{0}) → H^{k}(X) → H^{k}(Y) → H^{k+1}(X_{0}) → H^{k+1}(X) → 0,

Dans cette suite exacte, X (resp. X_{0}) est la fibre générique (resp. la fibre centrale f^{−1}(0)) de la famille projective f; le groupe H^{∗}(X_{0}) est muni de la SHM de Deligne [4], H^{∗}(X) de la SHM–limite de Schmid [6] et H^{k}(Y) de la SHM donnée par (a).

c) La SHM (H^{k}(Y), W, F) est “compatible” avec l’action de G := π_{1}(B \ Δ, ∗).

Le groupe H_{c}^{k}(Y) possède une structure de Hodge mixte duale.

De plus, les résultats locaux que nous obtenons renforcent l’idée que toute propriété de la SHM–limite de Schmid doit avoir une version locale. Pour nos investigations locales, nous présentons le bon analogue du cône de monodromie. Dans notre cas local, nous devous non seulement considérer la représentation monodromique, mais aussi la variation V ar (voir §2). Les versions infinitésimales N et V de ces deux objets (voir §2 pour leurs définitions et propriétés) fournissent une représentation abélienne de R^{n} où r = dim(B). Et c’est le cône τ = {λ ∈ R^{r}; λ_{i} > 0; i = 1, …, r} qui joue le rôle du cône de monodromie.

Pour tout λ ∈ R^{r}, les opérateurs N(λ), N_{c}(λ) et V (λ) définissent une filtration par le poids W (λ) sur H^{k}(Y) et une filtration par le poids W_{c}(λ) sur H_{c}^{k}(Y). Plusieurs caractérisations de ces filtrations sont présentées dans la section 1. Nous définissons leurs versions relatives et nous démontrons l’unicité de ces dernières (c’est analogue à la définition de Deligne et à son résultat d’unicité dans [3]).

Dans le section 2, nous présentons les analogues locaux des principaux résultats de l’article [1] de Cattani et Kaplan. Les propriétés de polarisation de ces SHM sont considérées dans la section 3.

§1. Filtrations and relative filtrations.— If N_{p} is a nilpotent endomorphism of the vector space H_{p}, then let W (H_{p}; N_{p}, k) be the monodromy weight filtration associated with N_{p} and centered at k [3, 4, 7, 8, 1].

Let H and H_{c} be finite dimensional Q–vector spaces. Consider two linear maps V : H → H_{c} and j : H_{c} → H; and fix an integer k. Assume that N := j ∘ V is nilpotent (or equivalently N_{c} := V ∘ j is nilpotent).

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

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

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

(1.2)  \{ \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}

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 (1.1). 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 the sequel we present three characterizations of the above filtrations. 1.3. 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 an ”algebraic extension” of type:

[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.

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).

1.4. Second characterization. Let V and j be as in (1.4). The filtrations (1.1) and (1.2) 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}

1.5. Third characterization. Let V and j be as in (1.4).

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), 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.

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

In the sequel we introduce the relative version of the above definitions. For the classical case see [3] and [9].

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) .

1.6. 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) .

§2. The MHS on the vanishing cohomology of an icis.— Let g : (Y, y) → (B, 0) be a deformation of the icis (g^{−1}(0), 0) such that its generic fiber is smooth. Assume that its discriminant locus Δ is a divisor with normal crossings. Let Y be its Milnor fiber and k = dim Y.

2.1. Theorem. Let g : (Y, y) → (B, 0) be as above. Then the (middle) cohomology group of its Milnor fiber carries a mixed Hodge structure (H^{k}(Y), 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) coincide with (H^{k}(Y), W, F).

b) For any projective extension f : (X , x) → (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) → H^{k}(Y) → H^{k+1}(X_{0}) → H^{k+1}(X) → 0,

where X (resp. X_{0}) denotes the generic (resp. the central) fiber of the projective family f, H^{∗}(X_{0}) carries Deligne’s MHS, H^{∗}(X) Schmid’s limit MHS, and H^{k}(Y) the MHS (H^{k}(Y), W, F) given by (a).

c) The MHS (H^{k}(Y), W, F) is ”compatible” with the action of π_{1}(B \ Δ, ∗), in the sence which will be clarified in the sequel.

The space H_{c}^{k}(Y) carries a dual MHS (duality with respect to Q(−k)).

Let G := π_{1}(B \ Δ, ∗) and consider the monodromy representations: ρ : G → Aut(H^{k}(Y)), ρ_{c} : G → Aut(H_{c}^{k}(Y)); and the variation map V ar : G → Hom(H^{k}(Y) → H_{c}^{k}(Y)). Consider also the natural map j : H_{c}^{k}(Y) → H^{k}(Y). Let <, > denote the perfect pair <, >: H_{c}^{k}(Y) ⊗ H^{k}(Y) → R. Then the representations are compatible with this bilinear form. We define the corresponding infinitesimal isometries and variations as follows. Let m be an integer such that ρ(g)^{m} = 1 for any g ∈ G. Define 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} .

Notice that these maps are independent of the choice of m.

2.2. Proposition. The operators N, N_{c} and V satisfy the following properties (for g, h ∈ G):

a) j ∘ V (g) = N(g), V (g) ∘ j = N_{c}(g),

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

c) 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}.

d) < N_{c}(g)α, β > + < α, N(g)β >= 0, and < V (g)α, β >= (−1)^{k+1}· < V (g)β, α > .

The relations (a–d) are compatible with respect to the spectral decomposition of the spaces and maps.

e) The above morphisms are compatible with the mixed Hodge structures introduced in (2.1): N, N_{c} and V are morphisms of type (−1, −1), and j is a morphism of type (0, 0).

Now, fix a standard set of generators of G = Z^{r}. If Δ = {z_{1} · · · · · z_{r}} (r = dim B) in a suitable local coordinate system, then let e_{i} (i = 1, …, r) be the class of a positively oriented loop around the divisor {z_{i} = 0}. If λ = (λ_{1}, ⋯, λ_{r}) ∈ Z^{r}, then one has (cf. 2.2):

(2.3) N(λ) = ∑_{i=1}^{r} λ_{i}N(e_{i}), N_{c}(λ) = ∑_{i=1}^{r} λ_{i}N_{c}(e_{i}) and V (λ) = ∑_{i=1}^{r} λ_{i}V (e_{i}).

We extend the action of G to an action of R^{r} via the relations (2.3). Notice that the properties (2.2) still hold for the extended R^{r} action as well.

In the following we study the operators {N(λ), N_{c}(λ), V (λ)}_{λ∈τ̄}, where τ denotes the cone τ := {λ ∈ R^{r}; λ_{i} > 0 for all i = 1, …, r}, and τ̄ denotes its closure in R^{r}.

The following result is the local version of a result of Cattani–Kaplan [1], and it characterizes the weight filtrations associated with the faces of the cone.

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

2.4. Theorem.

a) For any J ⊂ {1, 2, …, r}, 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, …, r}),

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

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.

c) For any λ ∈ τ̄ and J ⊂ {1, 2, …, r} one has:

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

Besides , for any J, J′ and λ ∈ J one has: V (λ)W_{l}(J ∪ J′) ⊂ W_{c,l−2}(J ∪ J′).

d) If J, J′ ⊂ {1, 2, …, r}, J ∪J′ = {1, 2, …, r}, 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)).

All the above properties are compatible with the spectral decomposition.

§3. The polarization.— Now, we list our polarization properties.

Let χ be a character of G. If U is a G–module, let U_{χ} denote its generalized χ–eigenspace.

3.1. Proposition. The subspace ⊕_{χ≠1}H_{c}^{k}(Y)_{χ} has a natural polarized MHS, polarized by the cone {N_{c}(λ)}_{λ∈τ}. The bilinear (polarization) form is induced by the intersection form < ·, j· >. Similar result holds for ⊕_{χ≠1}H^{k}(Y)_{χ}, actually they are dual with respect to Q(−k). Moreover, ⊕_{χ≠1}j_{χ} is an isomorphism of MHS’s.

The case χ = 1 is more complicated. First, we recall a construction: Deligne associated with any mixed Hodge structure H = (H, W, F) another MHS H˜ = (H, W, F̃), which splits over R (i.e. its generalized Hodge decomposition is compatible with the conjugation) and has the property that Gr^{W} H = Gr^{W} H˜ (for details see, for example, [2]).

3.2. Theorem. For any λ ∈ τ the system (H^{k}(Y)_{1}˜ , H_{c}^{k}(Y)_{1}˜ , V (λ), j) has a direct sum decomposition (H′ ⊕ H′′, H′_{c} ⊕ H′′_{c}, V′(λ) ⊕ V′′(λ), j′ ⊕ j′′) such that:

a) V′′(λ) and j′ are isomorphisms.

b) H′, H′′, H′_{c} and H′′_{c} are mixed Hodge structures which split over R.

c) H′_{c} (resp. H′′_{c}) is dual of H′ (resp. of H′′) with respect to Q(−k).

d) H′ has a natural polarization (−1)^{k}–symmetric form induced by (the Poincaré dual) of the intersection form < ·, j′· > (as in 3.1), and it is polarized by the infinitesimal monodromy N′(λ) = j′ ∘ V′(λ). Its weight filtration is exactly W (H′; N′(λ), k).

e) H′′ has a natural (−1)^{k+1}–symmetric polarization form given by − < V′′(λ)·, · >, and it is polarized by the monodromy N′′(λ) = j′′ ∘ V′′(λ). Its weight filtration is W (H′′; N′′(λ), k + 1).

References

[1] E. Cattani and A. Kaplan, Polarized mixed Hodge structures and the local monodormy of a variation of Hodge structure, Invent. math., 67, 1982, p. 101-115.

[2] E. Cattani, A. Kaplan and W. Schmid, Degeneration of Hodge structures, Annals of Math., 123, 1986, p. 457-535.

[3] P. Deligne, La conjecture de Weil. II, Publ. Math I.H.E.S., 52, 1980, p. 137-252.

[4] P. Deligne, Théorie de Hodge, II. Publ. Math. I.H.E.S., 40, 1972, p. 5-57. III. Publ. Math. I.H.E.S., 44, 1974, p. 5-77.

[5] A. Némethi, The mixed Hodge structure of an ICIS, I., preprint, November 1994.

[6] W. Schmid, Variation of Hodge Structure: The Singularities of the Period Mapping, Invent. math., 22, 1973, p. 211-319.

[7] J.H.M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Nordic Summer School/NAVF, Symposium in Mathematics, Oslo, August 5-25, 1976, p. 525-563.

[8] J.H.M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proceedings of Symposia in Pure Mathematics, 40, 1983, Part 2, p. 513-536.

[9] J.H.M. Steenbrink and S. Zucker, Variation of mixed Hodge structure I., Invent. math., 80, 1985, p. 489-542.

The Ohio State University, Columbus, OH 43210, U.S.A.

Running title: The mixed Hodge structure of an ICIC.

Correction des épreuves: András Némethi Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210-1174, U.S.A.

Télécopie: (19-1) 614-292-1479