A SEMICONTINUITY RESULT FOR MONODROMY UNDER DEGENERATION

NICHOLAS M. KATZ

1. Introduction

We fix a a prime number l. We denote by E_{λ} a finite extension of ℚ_{l} inside a chosen algebraic closure ℚ̄_{l} of ℚ_{l}, by O_{λ} the ring of integers in E_{λ}, by 𝔽_{λ} its residue field, and by 𝔽̄_{λ} an algebraic closure of 𝔽_{λ}. We take as coefficient field A one of the fields on the following list: 𝔽_{λ}, 𝔽̄_{λ}, E_{λ}, or ℚ̄_{l}.

We work over a field k in which l is invertible. We are given a smooth connected k-scheme S/k, separated and of finite type, of dimension r ≥ 1. In S, we are given a reduced and irreducible closed subscheme Z, of some dimension d ≥ 0. We assume that an open dense set V_{1} ⊂ Z is smooth over k (a condition which is automatic if the ground field k is perfect).

On S, we are given a constructible A-sheaf F. Because F is constructible, its restriction to S − Z is constructible, so there exists a dense open set U in S − Z on which F is lisse. Similarly, the restriction of F to V_{1} is lisse, so there exists a dense open set V in V_{1} on which F is lisse. Let us denote by j the inclusion of U into S, and by i the inclusion of V into S. Thus we have a lisse A-sheaf j^{⋆}F on U, and a lisse A-sheaf i^{⋆}F on V.

In this generality, there is absolutely nothing one can say relating the monodromy of the lisse A-sheaf i^{⋆}F on V to the monodomy of the lisse A-sheaf j^{⋆}F on U. However, there is a class of constructible A-sheaves F on S for which these monodromies are related, namely those “of perverse origin”.

We say that a constructible A-sheaf F on S is of perverse origin if there exists a perverse A-sheaf M on S such that

F ≅ H^{−r}(M).

We say that any such perverse sheaf M gives rise to F.

The geometric interest of this notion is that, as we shall recall below (cf Corollaries 5 and 6), for any affine morphism f : X → S, and for any perverse A-sheaf M on X, the constructible A-sheaf

R^{−r}f_{!}M := H^{−r}(Rf_{!}(M))

Date: April 16, 2001.

on S is of perverse origin. In particular, suppose X/k is a local complete intersection, everywhere of dimension n + r. Then for any lisse A-sheaf G on X,

M := G[dim(X)] = G[r + n]

is perverse on X. Hence for any affine morphism f : X → S, the constructible A-sheaf

R^{n}f_{!}G = H^{−r}(Rf_{!}(M))

on S is of perverse origin .

Our main result is that for F of perverse origin, the monodromy of the lisse A-sheaf i^{⋆}F on V is “smaller” than the monodromy of the lisse A-sheaf j^{⋆}F on U.

To make this precise, let us pick geometric points u of U and v of V. We have monodromy homomorphisms

ρ_{U} : π_{1}(U, u) → Aut_{A}(F_{u})

and

ρ_{V} : π_{1}(V, v) → Aut_{A}(F_{v})

attached to j^{⋆}F on U and to i^{⋆}F on V respectively. We define compact subgroups

Γ_{U} := Image(ρ_{U}) ⊂ Aut_{A}(F_{u})

and

Γ_{V} := Image(ρ_{V}) ⊂ Aut_{A}(F_{v}).

Theorem 1. For S/k smooth and connected of dimension r ≥ 1 and for F a constructible A-sheaf on S of perverse origin, the group Γ_{V} is isomorphic to a subquotient of the group Γ_{U}. More precisely, there exists a compact group D, a continuous group homomorphism

D → Γ_{U},

a closed normal subgroup

I ⊲ D,

and an A-linear embedding

F_{v} ⊂ F_{u}^{I}

with the following property: if we view F_{u}^{I} as a representation of D/I, then the subspace

F_{v} ⊂ F_{u}^{I}

is D/I-stable, and under the induced action of D/I on F_{v}, the image of D/I in Aut_{A}(F_{v}) is the group Γ_{V}.

Before giving the proof of the theorem, we must develop some basic properties of sheaves of perverse origin.

2. Basic properties of sheaves of perverse origin

Throughout this section, S/k is smooth and connected, separated and of finite type, of dimension r ≥ 1, and F is a constructible A-sheaf on S of perverse origin.

Recall that for M a perverse A-sheaf on S, its ordinary cohomology sheaves H^{i}(M) vanish for i outside the interval [-r, 0]. This is obvious for simple objects from their explicit description as middle extensions, and it follows for the general case because any perverse sheaf is a successive extension of finitely many simple objects, cf. [BBD, 2.1.11 and 4.3.1].

Recall that attached to any object K in D_{c}^{b}(S, A) are its perverse cohomology sheaves ^{p}H^{i}(K): these are perverse A-sheaves on S, all but finitely many of which vanish. Their behavior under shifts is given by

pH^{i}(K[j]) = ^{p}H^{i+j}(K).

A distinguished triangle gives rise to a long exact sequence of perverse cohomology sheaves.

Given two integers a ≤ b, an object K in D_{c}^{b}(S, A) is said to lie in ^{p}D^{[a,b]} if its perverse cohomology sheaves ^{p}H^{i}(K) vanish for i outside the closed interval [a, b]. Similarly, an object K in D_{c}^{b}(S, A) is said to lie in ^{p}D^{≥a} (respectively in ^{p}D^{≤b}) if its perverse cohomology sheaves ^{p}H^{i}(K) vanish for i < a (respectively for i > b). An object of ^{p}D^{[a,a]} is precisely an object of the form M[−a] with M perverse. For a < b, any object K of ^{p}D^{[a,b]} is a successive extension of its shifted perverse cohomology sheaves ^{p}H^{i}(K)[−i], i ∈ [a, b]. More precisely, any object K of ^{p}D^{[a,b]} sits in a distinguished triangle

pH^{a}(K)[−a] → K → ^{p}τ_{≥a+1}(K) →

with the third term ^{p}τ_{≥a+1}(K) in ^{p}D^{[a+1,b]}.

Lemma 2. Let K be an object of ^{p}D^{[a,b]}. Then its ordinary cohomology sheaves H^{i}(K) vanish for i outside the interval [a − r, b].

Proof. We proceed by induction on b − a. If K lies in ^{p}D^{[a,a]}, then K is M[−a] with M perverse. The ordinary cohomology sheaves of M, H^{i}(M), vanish for i outside the interval [−r, 0]. So those of M[−a] vanish outside [a − r, a]. To do the induction step, use the distinguished triangle

pH^{a}(K)[−a] → K → ^{p}τ_{≥a+1}(K) →

above, and its long exact cohomology sequence of ordinary cohomology sheaves.

Corollary 3. Let K be an object of ^{p}D^{≥a}. Then its ordinary cohomology sheaves H^{i}(K) vanish for i < a − r.

Proposition 4. Let K be an object of ^{p}D^{≥0} on S. Then its −r’th ordinary cohomology sheaf H^{−r}(K) is of perverse origin.

Proof. We have a distinguished triangle

M → K → ^{p}τ_{≥1}(K) →

whose first term M :=^{p} H^{0}(K) is perverse, and whose last term ^{p}τ_{≥1}(K) lies in ^{p}D^{≥1}. From the long exact cohomology sequence for ordinary cohomology sheaves, we find

H^{−r}(M) ≅ H^{−r}(K).

Corollary 5. For any affine morphism f : X → S, and for any perverse A-sheaf M on X, the constructible A-sheaf

R^{−r}f_{!}M := H^{−r}(Rf_{!}(M))

on S is of perverse origin.

Proof. Indeed, one knows [BBD, 4.1.1] that for an affine morphism f, Rf_{⋆} maps^{p}D^{≤0} on X to ^{p}D^{≤0} on S. Dually, Rf_{!} maps ^{p}D^{≥0} on X to ^{p}D^{≥0} on S [BBD, 4.1.2]. So for M perverse on X, Rf_{!}M lies in ^{p}D^{≥0} on S, and we apply to it the previous result.

Corollary 6. Suppose X/k is a local complete intersection, everywhere of dimension n + r. For any lisse A-sheaf G on X, and any affine morphism f : X → S, the constructible A-sheaf R^{n}f_{!}G on S is of perverse origin .

Proof. Because X/k is a local complete intersection, everywhere of dimension n + r, given any lisse A-sheaf G on X, the object

M := G[dim(X)] = G[r + n]

is perverse on X. [See [Ka-PES II, Lemma 2.1] for the case when G is the constant sheaf A, and reduce to this case by observing that if K is perverse on X and G is lisse on X, then G ⊗_{A} K is perverse on X.] Now apply the previous result to M.

Proposition 7. Let F be of perverse origin on S. For any connected smooth kscheme T/k, and for any k-morphism f : T → S, the pullback f^{⋆}F is of perverse origin on T.

Proof. We factor f as the closed immersion i of T into T ×_{k} S by (id, f), followed by the projection pr_{2} of T ×_{k} S onto S. So it suffices to treat separately the case when f is smooth, everywhere of some relative dimension a, and the case when f is a regular closed immersion, everywhere of some codimension b. Pick M perverse on S giving rise to F. In the first case, K := f^{⋆}M[a] is perverse on T [BBD, paragraph above 4.2.5], dim(T) = r + a, and

f^{⋆}F = f^{⋆}H^{−r}(M) = H^{−r−a}(f^{⋆}M[a]) = H^{−r−a}(K)

is thus of perverse origin on T. In the second case, K := f^{⋆}M[−b] lies in ^{p}D^{[0,b]} (apply [BBD, 4.1.10(ii)] b times Zariski locally on T, and observe that the property of lying in ^{p}D^{[0,b]} can be checked Zariski locally, since it amounts to the vanishing of certain perverse cohomology sheaves), dim(T) = r − b, and

f^{⋆}F = f^{⋆}H^{−r}(M) = H^{b−r}(f^{⋆}M[−b]) = H^{b−r}(K)

is thus, by the previous Proposition, of perverse origin on T.

Proposition 8. Let F be of perverse origin on S. Given a connected smooth kscheme T/k of dimension a, with function field k(T) and generic point η := Spec(k(T)), and given a smooth k-morphism f : S → T with generic fibre S_{η}/k(T), the restriction F_{η} := F|S_{η} is of perverse origin on S_{η}/k(T).

Proof. Indeed, for M perverse on S giving rise to F, M_{η}[−a] is perverse on S_{η}/k(T) and gives rise to F_{η}.

Proposition 9. Let F be of perverse origin on S. For j : U → S the inclusion of any dense open set on which F is lisse, the canonical map

F → j_{⋆}j^{⋆}F

is injective.

Proof. Let M be a perverse A-sheaf on S which gives rise to F. We know [BBD, 4.3.1] that the category of perverse A-sheaves on S is an abelian category which is both artinian and noetherian, so every object is a successive extension of finitely many simple objects. We proceed by induction on the length of M.

If M is simple, then in fact we have F ≅ j_{⋆}j^{⋆}F. To see this, we distinguish two cases. The first case is that M is supported in an irreducible closed subscheme W of S with dim(W) ≤ r − 1. In this case its ordinary cohomology sheaves H^{i}(M) vanish for i outside the closed interval [1 − r, 0]. Thus F = 0 in this case, so the assertion trivially holds. The second case is that M is the middle extension of its restriction to any dense open set on which it is lisse. In this case M is j_{⋆!}(j^{⋆}F[r]), and from the explicit description [BBD, 2.1.11] of middle extension we see that

H^{−r}(M) = j_{⋆}j^{⋆}F.

In the general case, we pick a simple subobject M_{1} of M, and denote

M_{2} := M/M_{1}.

We put

F_{i} := H^{−r}(M_{i})

for i = 1, 2. Then the short exact sequence

0 → M_{1} → M → M_{2} → 0

leads to a left exact sequence

0 → F_{1} → F → F_{2}.

By induction, we know that

F_{i} ↪ j_{⋆}j^{⋆}F_{i}

for i = 1, 2. A simple diagram chase shows that F ↪ j_{⋆}j^{⋆}F, as required.

3. Proof of the theorem

Proof. If the theorem is true for one choice of geometric points u of U and v of V, it is true for any other choice. So we may assume that u lies over the generic point of U, and that v lies over the generic point of V. By Proposition 7, we may at will shrink S to any dense open set S′ ⊂ S which meets Z, then replace U and V by their intersections with S′. This changes neither Γ_{U} nor Γ_{V}.

Denote by Z_{1} ⊂ Z the closed subset Z − V. Shrinking S to S − Z_{1} we reduce to the case when Z is smooth in S, and F is lisse on Z. Pulling F back to the blowup of S along Z, allowable by Proposition 7, changes neither Γ_{U} nor Γ_{V}, and reduces us to the case where Z is a connected smooth divisor in S.

We now focus on the relative dimension r of S/k. We first treat the case r = 1. In this case, S is a smooth connected curve over k, and Z is a closed connected subscheme of S which is etale over k. Thus Z is a closed point Spec(L) of S, with L/k a finite separable extension. Deleting from S the finitely many closed points other than Z at which F is not lisse, we may further assume that F is a sheaf of perverse origin on S which is lisse on S − Z.

We now come to the essential point, that denoting by j the inclusion S − Z ⊂ S, the canonical map

F → j_{⋆}j^{⋆}F

is injective (by Proposition 9).

Let us spell out what this mean concretely (compare [Mil, II.3.12 and II.3.16]). Denote by K the function field of S, by

η : Spec(K) → S

the (inclusion of the) generic point of S, by K̄ an algebraic closure of K, by

η̄: Spec(K̄) → S

the (inclusion of the) corresponding geometric generic point of S, and by K^{sep} ⊂ K̄ the separable closure of K inside K̄. The stalk Fη̄ is the representation of Gal(K^{sep}/K) obtained from viewing F|S − Z as a representation of π_{1}(S − Z, η̄) and composing with the canonical surjection

Gal(K^{sep}/K) → π_{1}(S − Z, η̄).

View the closed point Z as a discrete valuation v of K, and extend v to a valuation v̄ of K̄, with valuation ring Ov̄ ⊂ K̄. The residue field of Ov̄ is an algebraic closure of L, so a geometric generic point z̄ of Z. Inside Gal(K^{sep}/K), we have the corresponding decomposition group D := Dv̄, which contains as a normal subgroup the inertia group I := Iv̄. We have a short exact sequence of groups

0 → I → D → Gal(L^{sep}/L) → 0.

The stalk Fz̄ of F at z̄ is the representation of D/I ≅ Gal(L^{sep}/L) ≅ π_{1}(Z, z̄) given by F|Z. The stalk (j_{⋆}j^{⋆}F)z̄ of j_{⋆}j^{⋆}F at z̄ is the representation of D/I ≅ Gal(L^{sep}/L) ≅ π_{1}(Z, z̄) on the subspace of I-invariants in the restriction to D of the representation of Gal(K^{sep}/K) on Fη̄:

(j_{⋆}j^{⋆}F)z̄ ≅ (Fη̄)^{I}

For F any constructible sheaf on S which is lisse on S − Z, the injectivity of

F ↪ j_{⋆}j^{⋆}F

means precisely that at the single point Z we have an A-linear D-equivariant inclusion

Fz̄ ↪ (j_{⋆}j^{⋆}F)z̄ ≅ (Fη̄)^{I}.

So to conclude the proof of the theorem in the case r = 1, we have only to take I ⊲D as our groups, and use the composite homomorphism

D ↪ Gal(K^{sep}/K) → π_{1}(S − Z, η̄) → Aut_{A}(Fη̄)

to map D to

Γ_{U} := Image(π_{1}(S − Z, η̄) → Aut_{A}(Fη̄)).

For general r ≥ 1, we argue as follows. Recall that F is lisse on an open dense set U ⊂ S − Z, and is lisse on Z. Pick a closed point z in Z. Shrinking S to a Zariski open neighborhood of z in S, we reduce to the case where there exist r functions s_{i}, i = 1, …, r on S which define an etale k-morphism S → 𝔸_{k}^{r} and such that Z is defined in S by the single equation s_{r} = 0. Then the map S → 𝔸_{k}^{r−1} defined by s_{i}, i = 1, …, r − 1 is smooth of relative dimension 1, and makes Z etale over 𝔸_{k}^{r−1}. Denote by F the function field of 𝔸_{k}^{r−1}, and make the base change of our situation (S, Z)/𝔸_{k}^{r−1} from 𝔸_{k}^{r−1} to Spec(F). This is allowable by Proposition 8. We obtain a situation (S_{F} , Z_{F})/F in which S_{F} is a connected smooth curve over F, and Z_{F} is a nonvoid connected closed subscheme of S_{F} which is etale over F. Thus Z_{F} is a closed point Spec(L) of S_{F}, with L a finite separable extension of F. The r = 1 case of the theorem applies to this situation over F. Its truth here gives the theorem for our situation (S, Z)/𝔸_{k}^{r−1}. Indeed, the connected normal schemes U and U_{F} have the same function fields, and, being normal, their fundamental groups are both quotients of the absolute galois group of their common function field. So the groups Γ_{U} and Γ_{U_{F}} coincide. Similarly for V = Z, the groups Γ_{V} and Γ_{V_{F}} coincide.

4. Application to Zariski closures of monodromy groups

As an immediate corollary of the theorem, we obtain:

Corollary 10. Hypotheses and notations as in the theorem, denote by N_{U} the rank of the lisse A-sheaf j^{⋆}F on U, and by N_{V} the rank of the lisse A-sheaf i^{⋆}F on V.

(1) We have the inequality of ranks

N_{V} ≤ N_{U}.

(2) Suppose in addition that A is ℚ̄_{l}. Denote by G_{U} the ℚ̄_{l}- algebraic group which is the Zariski closure of Γ_{U} in Autℚ̄_{l}(F_{u}) ≅ GL(N_{U}, ℚ̄_{l}), and denote by G_{V} the ℚ̄_{l}- algebraic group which is the Zariski closure of Γ_{V} in Autℚ̄_{l}(F_{v}) ≅ GL(N_{V} , ℚ̄_{l}). Then the algebraic group G_{V} is a subquotient of G_{U}.

In particular, we have (2a) if G_{U} is finite (or equivalently if Γ_{U} is finite) then G_{V} is finite (or equivalently

Γ_{V} is finite), (2b) dim(G_{V}) ≤ dim(G_{U}), (2c) rank(G_{V}) ≤ rank(G_{U}).

5. Appendix: when and where is a sheaf of perverse origin lisse?

Proposition 11. Hypotheses and notations as in the theorem, the sheaf F of perverse origin on S is lisse, say of rank N, if and only if its stalks F_{s} at all geometric points s of S have constant rank N.

Proof. It is trivial that if F is lisse on S, then its stalks have constant rank. Suppose now that F on S is of perverse origin, and that all its stalks have constant rank N. We must show that F is lisse on S.

It suffices to show that F is lisse on an open set V ⊂ S whose complement S−V has codimension ≥ 2 in S. Indeed, by Zariski-Nagata purity, if we denote by j : V → S the inclusion, the lisse sheaf j^{⋆}F on V extends uniquely to a lisse sheaf E on S. For any lisse sheaf E on S, and any dense open set V ⊂ S, we have E ≅ j_{⋆}j^{⋆}E. But j^{⋆}E ≅ j^{⋆}F, so we find that E ≅ j_{⋆}j^{⋆}F. In particular, j_{⋆}j^{⋆}F is lisse on S, and hence all its stalks have constant rank N. The injective (by Proposition 9) map

F ↪ j_{⋆}j^{⋆}F

must be an isomorphism, because at each geometric point the stalks of both source and target have rank N. Thus we find

F ≅ j_{⋆}j^{⋆}F ≅ E,

which shows that F is lisse on S.

We now show that an F of perverse origin on S which has constant rank N must be lisse. Thanks to the above discussion, we may remove from S any closed set of codimension 2 or more. Thus we may assume that F is lisse on an open set U ⊂ S, inclusion denoted j : U → S, and that that the complement S − U is a disjoint union of finitely many irreducible divisors Z_{i}. Denote by η the generic point of S. At the generic point z_{i} of Z_{i}, the local ring O_{S,z_{i}} is a discrete valuation ring. For suitable geometric points η̄ and z̄_{i} lying over η and z_{i} respectively, we have the inertia and decomposition groups I_{i} and D_{i}. We have an injective (by Propostion 9) D_{i}-equivariant map

Fz̄_{i} ↪ (j_{⋆}j^{⋆}F)z̄_{i} ≅ (Fη̄)^{I_{i}} ⊂ Fη̄.

As both Fz̄_{i} and Fη̄ have the same rank N, all the displayed maps must be isomorphisms. Therefore I_{i} acts trivially on Fη̄. Thus j^{⋆}F is a lisse sheaf on U which is unramified at the generic point of each Z_{i}. So by Zariski-Nagata purity, j^{⋆}F extends to a lisse sheaf E on S. Exactly as in the paragraph above, we see that

F ≅ j_{⋆}j^{⋆}F ≅ E,

which shows that F is lisse on S.

Proposition 12. Hypotheses and notations as in the theorem, let F be of perverse origin on S. The integer-valued function on S given by

s ↦ rank(F_{s})

is lower semicontinuous, i.e., for every integer r ≥ 0, there exists a reduced closed subscheme S_{≤r} ⊂ S such that a geometric point s of S lies in S_{≤r} if and only if the stalk F_{s} has rank ≤ r. If we denote by N the generic rank of F, then S = S_{≤N}, and S − S_{≤N−1} is the largest open set on which F is lisse.

Proof. Once we show the lower semicontinuity of the rank, the second assertion is immediate from the preceeding proposition.

To show the lower semicontinuity, we first reduce to the case when k is perfect. Indeed, for k^{per} the perfection of k, and S_{1} := S⊗_{k}k^{per}, the natural map π : S_{1} → S is a universal homeomorphism, S_{1}/k^{per} is smooth and connected, and π^{⋆}F is of perverse origin on S_{1}/k^{per}. Thus it suffices to treat the case when k is perfect.

Because F is a constructible sheaf, its rank function is constructible. So to show lower semicontinuity, it suffices to show that the rank decreases under specialization. Thus let Z ⊂ S be an irreducible reduced closed subscheme, with geometric generic point η̄_{Z}. We must show that at any geometric point z ∈ Z, we have

rank(Fη̄_{Z}) ≤ rank(F_{z}).

Because the field k is perfect, we may, by de Jong [de Jong, Thm. 3.1], find a smooth connected k-scheme Z_{1} and a proper surjective k-morphism f : Z_{1} → Z. By Proposition 7, f^{⋆}F is of perverse origin on Z_{1}. By Corollary 10, (1), applied on Z_{1}, we get the asserted inequality of ranks.

Acknowledgement This work began as a diophantine proof of Parts (2a) and (2c) of the Corollary, in the special case when the ground field k is finite, and when the

10 NICHOLAS M. KATZ

sheaf F of perverse origin on S is R^{n}f_{!}G for f : X → S a smooth affine morphism everywhere of relative dimension n, with G a lisse sheaf on X. I owe to Deligne both the idea of formulating the theorem in terms of subquotients , and the idea that it applied to sheaves of perverse origin.

References

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51-93. [Ka-PES II] , Katz, N., Perversity and Exponential Sums II, pages 205-252 in Barsotti Symposium

in Algebraic Geometry, (ed. Cristante and Messing), Academic Press, 1994. [Ka-SE] Katz, N., Sommes Exponentielles, rédigé par G. Laumon, Astérisque 79, 1980.

[Mil] Milne, J.S., Étale Cohomology, Princeton University Press, 1980.

[SGA4] A. Grothendieck et al - Séminaire de Géométrie Algébrique du Bois-Marie 1963/64 SGA 4, Tome III, Springer Lecture Notes in Math. 305, 1973.