An algebraic proof of Deligne's regularity criterion for integrable connections

0.16 MB·English

by  Yves André

AN ALGEBRAIC PROOF OF DELIGNE’S REGULARITY
CRITERION FOR INTEGRABLE CONNECTIONS
7
0
0
YVESANDRE´
2

n
a ABSTRACT. Deligne’sregularitycriterionforanintegrableconnection
J
∇onasmoothcomplexalgebraicvarietyX saysthat∇isregularalong

0 theirreducibledivisorsatinfinityinsomefixednormalcompactification
3
of X if and only if the restrictionof ∇ to everysmoothcurveon X is

fuchsian (i. e. has only regularsingularitiesat infinity). The “only if”
]
G part is the difficult implication. Deligne’s proof is transcendental and
A usesHironaka’sresolutionofsingularities.
Following[1],wepresentapurelyalgebraicproofofthisimplication
.
h
whichdoesnotuseresolutionbeyondthecaseofplanecurves. Itrelies
t
a upon a study of the formal structure of integrable connections on sur-
m
faceswith(possiblyirregular)singularitiesalonga divisorwithnormal
[ crossings.

1
v
5
9
8 1. INTRODUCTION: FUCHSIAN CONNECTIONS.
1
0
1.1. Let X be a connected algebraic complex manifold, and let E be an
7
0 algebraicvectorbundleon X endowedwithan integrableconnection∇.
/
h When X is a curve, the dichotomy between regular and irregular sin-
t gularities (at infinity) goes back to the 19th century. The connection ∇ is
a
m said to be fuchsian if all singularities (at infinity) are regular, see Manin’s
: classicalpaper[8]. Thisis obviouslyabirationalnotion.
v
Xi In higher dimension, one may consider a normal compactification X¯ of
r X and lookat theirreduciblecomponentsZj of∂X¯ = X¯ \X ofcodimen-
a sion one in X¯. The local subring O of C(X) is a discrete valuation
X,Zj
ring with residue field O /m = C(Z ). One is then in the famil-
X,Zj X,Zj j
iar one-dimensional situation, over C(Z ) instead of C, and the notion of
j
regularity of ∇ along Z is defined in the usual way. Namely, consider a
j
germofvectorfield θ onX -viewedas aderivationofC(X)-whichpre-
j
serves m , but does not send it into m2 , and which acts trivially on
X,Zj X,Zj
theresiduefield. Then∇issaidtoberegularalongZ ifthereisanO -
j X,Zj
latticeinthegenericfiberE whichisstableunder∇(θ )(thiscondition
C(X) j
doesnotdepend on θ , cf. [2, I.3.3.4]).
j
Date:today.
1
2 YVESANDRE´
Inordertoobtainabirationalnotionoffuchsianity,oneisthenledtosay
that∇ isfuchsianifforany(X¯,Z ) as before, ∇ isregularat Z .
j j
At first look, this definition is rather forbidding, since it requires to con-
sider all divisorial valuations of C(X) at the same time. Fortunately, it
turns out that it suffices to consider only one normal compactification X¯.
Indeed, according to P. Deligne, one has the following characterizations of
fuchsianity.
1.1.1.Theorem. [5,II.4.4, 4.6]Thefollowingareequivalent:
i) ∇ is fuchsian,
ii) for some normal compactification X¯, ∇ is regular along all irre-
duciblecomponentsof∂X¯ ofcodimensiononein X¯,
iii) for any smooth curve C and any locally closed embedding h : C →
X,h∗∇ isfuchsian,
iv) foranysmoothY andanymorphismf : Y → X,f∗∇ isfuchsian,
v) for some dominant morphism f : Y → X with Y smooth, f∗∇ is
fuchsian.
Note that we do not assume that ∂X¯ has normal crossings. The difficult
implicationisii) ⇒ iii). Theimplicationiii) ⇒ i)iscomparativelyeasyto
establish(cf. e.g. [2,I.3.4.7]),andtheotherimplicationsfollowveryeasily
from these two. The difficulty with ii) ⇒ iii) arises when the closure of
C in X¯ does not meet ∂X¯ transversally. A closely related difficulty, with
ii) ⇒ i), is to show that ∇ remains regularat theexceptional divisorwhen
oneblowsupasubvarietyof∂X¯.
1.2. Deligne’s proof (as contained in the erratum to [5]) is not elemen-
tary: it relies upon certain transcendental complex-analytic arguments on
onehand,anduponHironaka’sresolutionofsingularitiesontheotherhand.
Recently, inspired by part of Sabbah’s work on asymptotic analysis in
dimension2 [9], theauthorhasfoundapurelyalgebraicproofofDeligne’s
regularity criterion, as a consequence of a more general result about the
semicontinuityofthePoincare´-Katz rank [1].
The aim of this text is to explain this argument in the simplified form
needed for1.1.1, i.e. inthecontextofalgebraicconnectionswhich arereg-
ular at infinity. We do not use resolution of singularities beyond resolution
ofplanecurves.
We refer to [4] for a nice introduction to the background and the event-
ful story of this problem. Its difficulty is related to the fact that the stan-
dardtechniquesoflogarithmiclatticesbreakdowninthisalgebraiccontext;
alongtheway,itbecomesnecessarytogobeyondtheframeworkofregular
connections.
ANALGEBRAICPROOFOFDELIGNE’SREGULARITYCRITERIONFORINTEGRABLECONNECTIONS3
1.3. More precisely, we present an algebraic proof of the following local
refined form ofDeligne’sregularitycriterion1.
1.3.1. Theorem. Let X be a normal connected algebraic variety over C.
LetU beasmoothopensubsetofX,withcomplement∂X = X\U,andlet
Qbeaclosedpointof∂X. LetZ ,...,Z betheirreduciblecomponentsof
1 t
∂X of codimensiononeinX whichpassthroughQ.
LetC beasmoothconnectedcurve, P ∈ C beaclosedpoint,andh : C →
X bea morphismsuchthath(C \P) ⊂ U and h(P) = Q.
LetE beavectorbundleonU withanintegrableconnection∇. If(E,∇)is
regularalongeachZ ,thenthevectorbundleh∗(E,∇)onC \P isregular
j
atP.
1.4. Warning. What does regularity actually mean? Quot capita tot sen-
sus: in the literature, one can find analytic definitions (moderate growth of
solutions of meromorphic connections), algebraic definitions (such as the
items in 1.1.1, reducedness of the characteristic variety, and many others
[2]), as well as mixed definitions (of GAGA type: formal/algebraic versus
analyticpropertiesofconnections).
Theoccurrence ofsomanydifferent definitionsis an obvioussignofthe
richness of the concept, but at the same time an amazing source of confu-
sion. All of them are supposed to be equivalent in their common domains
of application, but this is often a matter of folklore or belief, as a complete
andprecisedictionaryisstilllacking(especiallyinthesensitivecasewhere
thepolardivisorhasnon-normalcrossings). Astrangesituation,indeed,for
asupposedlywell-understoodclassical notion!
It is thus essential to keep in mind the definition which we have adopted
rightatthebeginninginordertounderstandwhatwedointhispaper: com-
paring,inapurelyalgebraicway,afewoftheexistingalgebraicdefinitions
of regular connections. The wit is that the core of the paper deals with
irregularconnections.
2. REDUCTION TO THE PLANE CASE.
2.1. In this section, we reduce 1.3.1 to the case where X is the projective
plane2.
We may assume dimX ≥ 2, otherwise the theorem is essentially triv-
ial. Replacing X by a quasi-projective neighborhood of Q and taking the
1thisstatementappearsin[2,I.5.4],buttheargumentgiventhereworksonlyincasethe
polardivisorhasnormalcrossings. Indeed,aswaspointedoutbyJ.Bernstein,lemma5.5
ofop.cit. onwhichitreliesdoesnotholdingreatergenerality.
2thenexttwolemmasarejointworkwihF.Baldassarri.
4 YVESANDRE´
normalizationofitsZariski closurein someprojectiveembedding,wemay
assumethatX isprojective.
2.1.1. Lemma. Let X ⊂ PN be a closed normal connected subvariety of
C
dimensiond ≥ 2. Let U, Z , Q, h : C → X,P beas in1.3.1.
j
For any sufficiently large integer δ, there exists an irreducible complete
intersection Y ⊂ PN of dimension N − d + 2 and multidegree (δ,...,δ)
C
suchthat:
(i) Y containsh(C) (with reduced induced structure)and cuts U transver-
sallyatitsgenericpointη ;
h(C)
(ii) Y ∩ X is an normal connected surface and Y cuts U \ (h(C) \ Q)
transversally;
(iii) in a neighborhood of Q, Y cuts each Z \ Q transversally, and does
j
notcutanyirreduciblecomponentof∂X ofcodimension> 1inX,northe
singularlocusof Z := ∪Z , except inQ.
j
Proof. This is more or less standard, but, for lack of adequate reference,
wegivesomedetail. Forshort,wechangealittlenotationandnowwriteC
for the closure of h(C) in X, with reduced structure (an integral, possibly
singular, curve). Let π : P˜ −→ PN be the blow-up centered at C, and let
E ⊂ P˜ theexceptionaldivisor.
We set I = Ker(O −→ O ). Then Im(π∗I → O ) = O(−E)
C PN C C P˜
and, for δ >> 0, π∗(O (δ))⊗O(−E) is very ample: a basis of sections
PN
definesanembeddingofP˜ intoPM. Ontheotherhand,sinceπisbirational
and PN is normal, one has π O = O . It follows that the two natural
∗ P˜ PN
arrows
I → π (Im(π∗I → O )) = π O(−E) → π O = O
C ∗ C P˜ ∗ ∗ P˜ PN
are inclusions of ideals of O . Let D ⊂ C be the closed subscheme
PN
corresponding to π O(−E). If D 6= C, D would be punctual and 1 ∈
∗
O wouldcorrespondtoafunctiononP˜ \(π−1(D))red vanishingon
PN\Dred
E\(π−1(D))red,acontradiction. HenceD = C andthereforeπ O(−E) =
∗
I . From theprojectionformula,onededuces
C
π (π∗(O (δ))⊗O(−E)) ∼= O (δ)⊗I ∼= I (δ),
∗ PN PN C C
whence
Ker(H0(PN,O (δ)) → H0(C,O (δ))) = H0(PN, I (δ))
PN C C
= H0(P˜,π∗(O (δ))⊗O(−E)),
PN
andthelinearsystemofhypersurfacesofdegreeδinPN containingC gives
riseto alocallyclosedembedding
PN \C ֒→ PM = P(H0(PN,I (δ))) ,
C
ANALGEBRAICPROOFOFDELIGNE’SREGULARITYCRITERIONFORINTEGRABLECONNECTIONS5
with Zariski closure P˜. The canonical bijection between hyperplanes H
of PM and hypersurfaces H of degree δ in PN containing C, is such that
the intersection H∩(PN \C) (in PM) equals H \C. So, the intersection
of X \C with a general complete intersection Y of multidegree (δ,...,δ)
(1 ≤ s ≤ d−1 entries) in PN containing η , is the intersection of X \C
C
with a general linear subvariety Y of codimension s in PM. By [3, Exp.
XI, Thm. 2.1. (i)], Y cuts X \ C (resp. the smooth part of Z \ (C ∩Z))
transversally and intersects properly any irreducible component of ∂X \
(C ∩ ∂X). Since s < d, Bertini’s theorem shows that the intersection of
Y with the strict transform of X in PM is normal and connected. On the
other hand, since η is a simplepoint of X, it is well-knownthat a general
C
complete intersection of s hypersurfaces of degree δ in PN containing η ,
C
intersectsX transversallyat thispoint.
Applying these considerations for s = d − 2, one obtains (i), (ii) and
(iii). (cid:3)
2.1.2.Corollary. Thereexistsanormalquasi-projectiveirreducibleneigh-
borhood X′ of Q in Y ∩ X, containing an open subset of h(C), such that
U′ := X′∩U issmooth,thedistinctirreduciblecomponentsofcodimension
oneof∂X′ = X′\U′passingthroughQ,arepreciselyZ ∩X′,...,Z ∩X′.
1 t
(cid:3)
If (E,∇) is regular along each Z1,...Zt, so is its pullback (E,∇)|U′
alongZ ∩X′,...,Z ∩X′ (cf. [2, I.3.4.4]). Thisreduces theorem1.3.1to
1 t
the case where X is a normal surface, or even a projective normal surface
(bythesameargumentas inthebeginningofthissection).
2.1.3. Lemma. In the notation of lemma 2.1.1, let us further assume that
d = 2. There exists a morphism g : X −→ P2 , which is finite in a neigh-
C
borhoodV ofQ,suchthatg(h(C)∩V)isnotcontainedinthebranchlocus,
and such that, for any irreduciblecomponent T of ∂X of dimension 1 with
Q ∈/ T,g(Q) ∈/ g(T).
Proof. Let G(N − 3,PN) be the Grassmannian of linear subvarieties of
PN of codimension 3, and let G be its dense open subset consisting of
linearsubvarietieswhich donotintersect X.
Foranyγ ∈ G,X maybeconsideredasaclosedsubvarietyoftheblow-
upP˜ ofPN atγ, and theprojectionwithcenterγ
γ
p : P˜ → P2
//γ γ
inducesamorphism
g : X → P2.
γ
6 YVESANDRE´
LetΛ ⊂ G(N−2,PN)×G(N−3,PN)theincidencesubvariety(locus
of (λ,α) such that λ contains α), and let p ,p be the natural projections.
2 3
Notice that p is a fibration with fiber P2 and admits a section above G:
3
there is a unique λ ∈ G(N − 2,PN) of PN of codimension 2 passing
γ
through Q and containing γ. Then γ varies, the λ form (via p ) a dense
γ 2
opensubsetofG(N −2,PN).
On the other hand, λ may be identified with the fiber of p above
γ //γ
g (Q) ∈ P2,and λ ∩X withg−1(g (Q)).
γ γ γ γ
By Bertini, one deduces that there is an open dense subset G′ ⊂ G such
thatforanyγ ∈ G′, g is finitein aneighborhoodofQ.
γ
Moreover, if there exists λ ∈ p−1(γ) which intersects X transversally
3
andcutsh(C)∩V (resp. whichpassesthroughQandavoidstheT’s),then
g (h(C) ∩ V) is not contained in the branch locus of g (resp. g (Q) ∈/
γ γ γ
g (T)).
γ
One deduces that there is an open dense subset G′′ ⊂ G′ such that for
anyγ ∈ G′′, g = g satisfies theconditionsinthelemma. (cid:3)
γ
In the situation of theorem 1.3.1 with X a projective surface, g induces
an e´talemorphism
X′ = V \(g−1(B)∪g−1(g(∂X)))) −→ P2 \(B ∪g(∂X))
|V |V C
whichisfiniteoveritsimage. UptoreplacingC byasuitableneighborhood
ofP, h(C \P) ⊂ U ∩X′.
Moreover, using [2, I.3.2.5] and 2.1.3, one sees that the push-forward
g∗((E,∇)|U∩X′) on g(U ∩ X′) is regular along each 1-dimensional irre-
ducible components of P2 \ g(U ∩ X′) passing through g(Q) (which are
C
eithertheg(Z )’sorelsearecontainedin B).
j
Ontheotherhand,iftheconnection(g◦h)∗g∗((E,∇)|U∩X′)onC \P is
regularat P,thesameistrueforitssubconnectionh∗((E,∇)|U∩X′).
Therefore, in order to prove the theorem, one may substitute to X any
Zariskineighborhoodofg(Q)in theprojectiveplane.
2.1.4. Corollary. Statement 1.3.1 holds in general if it holds when X is an
affineneighborhoodof theorigin Q in thecomplex plane and ∂X = Z :=
Z ∪...∪Z . (cid:3)
1 t
2.2. By the classical embedded resolution of plane curves, there is a se-
quenceofblow-ups
π : X′ = X → ··· → X = X
N 0
suchthatπ−1(h(C)∪Z)(withitsreducedstructure)hasstrictnormalcross-
ings. We denote by C′,Z′ the strict transforms of h(C),Z respectively;
j j
theyaresmoothcurves onX′, and hliftstoa morphismC → C′.
ANALGEBRAICPROOFOFDELIGNE’SREGULARITYCRITERIONFORINTEGRABLECONNECTIONS7
Weset U′ = X′\π−1(Z). WedenotebyE ,i = 1,...,s,theirreducible
i
componentsoftheexceptionaldivisorπ−1(Q), andset E0 = E ∩U′.
i i
Wedenoteby(E′,∇′) theinverseimageof(E,∇)on U′.
2.2.1.Proposition. (E′,∇′) isregularalongeveryE .
i
Via 2.1.4, it is clear that 1.3.1 follows from this assertion, whose proof
willoccupythenextthreesections.
3. POINCARE´-KATZ RANK AND THE TURRITTIN-LEVELT
DECOMPOSITION.
3.1. LetK beafieldofcharacteristic0,andletM beadifferentialmod-
K
ule over K((x)), i.e. a vector space over K((x)) of finite dimension µ
endowedwithan action∇(θ ) ofθ = x d satisfyingtheLeibnizrule.
x x dx
3.1.1. Example. K = C(E0) ∼= C(y), x is a local coordinate on X′ such
i
that x = 0 defines E0, y is a local coordinate on E , and M is thegeneric
i i K
fiber of E′ tensored with C(E0)((x)) over the subfield C(X′), endowed
i
withtheinduced actionof∇(θ ).
x
According to the Turrittin-Levelt theorem, there is a finite extension
K′/K andapositiveintegeresuchthat,puttingx′ = x1/e,onehasacanon-
icaldecompositionofdifferentialmodules
(3.1) MK ⊗K′((x′)) = ⊕Lϕj ⊗K′((x′)) Rj
where
- R is regular, i.e. admits a basis in which the matrix of ∇(θ ) has entries
j x
inK′[[x′]],
-L = K′((x′))with∇(θ )(1) = ϕ ∈ 1K′[1](andtheϕ ’saredistinct).
ϕj x j x′ x′ j
Wedenoteby µ thedimensionofthedifferentialmoduleR .
j j
Moreover, if one gathers together the summands L ⊗R according to
ϕj j
the slope, i.e. the negative of the (fractional) degree of ϕ , the resulting
j
coarser decompositiondescends to K((x)) and gives the slope decomposi-
tion
(3.2) M = ⊕ M .
K σ∈Q K,(σ)
We denote by µ the dimensionof the differential moduleM , so that
(σ) K,(σ)
µ = µ .
P (σ)
3.2. The highest slope is the Poincare´-Katz rankof M, denoted by ρ(M)
orρ forshort. Ifϕ is ofslopeρ, wewriteitintheform
j
ϕ = ϕ .x−ρ +h.o.t., ϕ ∈ K′.
j j,−ρ j,−ρ
8 YVESANDRE´
If ρ > 0, the coefficients ϕ may be calculated as follows: one
j,−ρ
takes a cyclic vector m for M . Then the matrix of ∇(θ ) in the ba-
K x
sis(m,xρ∇(θ )(m),...,x(µ−1)ρ∇(θ )µ−1(m)) ofM ⊗K′((x′))isofthe
x x K
form x−ρH where H(x) ∈ M (K[[x′]]) and the ϕ ’s are the non-zero
µ j,−ρ
eigenvaluesofH(0)(cf. e.g. [1, §2]).
3.3. Let us now assume that K is the function field of a smooth affine
curve SpecA, and that M comes from a differential module M over
K
A((x)) (as in example 3.1.1, where A = O(E0) is a localization of C[y]).
i
LetA′ bethenormalizationofAin K′.
Itisnottrueingeneralthatintheabovedecompositions,onemayreplace
K and K′ by A and A′ respectively (this is the well-known problem of
turning points). However, this becomes true if one restricts to a suitable
dense open subset of the curve (see [1, §3] for a detailed analysis of this
point).
More precisely, there exists f ∈ A \ {0} such that A′[1] is finite etale
f
overA[1], ϕ ∈ A′[1]((x′)), and the decompositions(3.1) and (3.2) come,
f j f
respectively,from decompositions
1
(3.3) M ⊗A′[f]((x′)) = ⊕Lϕj ⊗A′[f1]((x′)) Rj
1
(3.4) M ⊗A[ ]((x)) = ⊕ M .
σ∈Q (σ)
f
3.4. In general, ϕ ∈/ A′((x′)), but one always has ϕ ∈ A′, due to the
j j,−ρ
aboveinterpretationofthesecoefficients(cf. 3.2).
Denoting by µ the rank of R 3, a simple Galois argument then shows
j j
that ϕ(x) := (xρ −ϕ )µj lies in A[x]. For ρ > 0, this allows to define
Q j,−ρ
the effective divisor D = (ϕ(x)) on SpecA[x], which is finite of degree
µ .ρ overSpecA. Ifρ = 0,we setD = 0.
(ρ)
3.5. WhenM comesfromanintegrableconnection,totheeffectthatthere
is an action ∇(θ ) of θ = y d commuting with ∇(θ ), all the above de-
y y dy x
compositionsareautomaticallystableunder∇(θ ),i.e. aredecompositions
y
ofintegrableconnections.
3.5.1. Example. Let us come back to example 3.1.1, and take M =
E ⊗ O(E0)((x)). In the sequel, we shall have to deal with all the
i
E ’s simultaneously, so we emphasize the index i and write ρ for the
i i
3not to be confused with the dimensions µ introduced in 3.1. Notice that µ =
(σ) (ρ)
µ wherethesumsrunsoverallj’ssuchthatL isofslopeρ,thePoincare´-Katzrank
P j ϕj
ofM.
ANALGEBRAICPROOFOFDELIGNE’SREGULARITYCRITERIONFORINTEGRABLECONNECTIONS9
Poincare´-Katz rank of E along E , and µ , ϕ , µ , ϕ , D , instead
i (ρi) i,j i,j i,j,−ρi i
ofµ , ϕ , µ , ϕ , D.
(ρ) j j j,−ρ
Geometrically,forρ > 0, thedivisor
i
(3.5) Di = (Y(xρi −ϕi,j,−ρi)µi,j)
has an intrinsic meaning as a divisor on the normal bundle NE0X′ ∼=
i
SpecA[x] (the point is that theterm oflowerdegree of ϕ has an intrinsic
i,j
meaning independent of the choice of the transversal derivation at E0 by
j
means of which one identifies NE0X′ and SpecA[x]). It is finite of degree
i
µ .ρ over E0.
(ρi) i i
4. FORMAL DECOMPOSITIONS AT CROSSINGS.
4.1. LetusnowconsideracrossingpointQ′ofE withanothercomponent
i
of π−1(Z) (which is either another component E or one of the Z ’s). Let
j j
x,y beetalecoordinatesatQ′ suchthatE isdefinedbyx = 0andtheother
i
componentby y = 0.
Theformalizationof(E′,∇′)atQ′thengivesrisetoadifferentialmodule
M˜ overC[[x,y]][ 1 ](withmutuallycommutingactionsofθ and θ ).
xy x y
Weshallsaythat(E′,∇′)hasniceformalstructureatQ′4if,afterramifi-
cationalongxy = 0, thereisadecomposition
(4.1) M˜ = ⊕L˜ ⊗R˜
ϕ˜k,ψ˜k k
where
- R˜ is regular, i.e. admits a basis in which the matrix of ∇˜(θ ) and ∇˜(θ )
j x y
haveentries inC[[x,y]],
-L˜ = C[[x,y]][ 1 ]with
ϕ˜k,ψ˜k xy
1
∇˜(θ )(1) = ϕ˜ , ∇˜(θ )(1) = ψ˜ ∈ C[[x,y]][ ], θ (ψ˜ ) = θ (ϕ˜ ).
x k y k x k y k
xy
Withoutloss of generality, one may then assume that the pairs (ϕ˜ ,ψ˜ ) are
k k
distinctmoduloC[[x,y]](whichensures unicityofthedecomposition).
4.2. For K = C((y)), let us compare a decomposition (4.1) with the
Turritin-Leveltdecompositionofthedifferentialmodule
MK := M˜ ⊗C[[x,y]][ 1 ] K((x)).
xy
Onehas acanonical decomposition
C[[x,y]][ 1 ] 1 1 1 1 1 1 1
xy ∼= C[[y]][ ] ⊕ C[ , ] ⊕ C[[x]][ ]
C[[x,y]] x x xy x y y y
4this is equivalentto sayingthatQ′ is semi-stable for(E′,∇′) in the sense of [1], but
weakerthansayingthat(E′,∇′)hasagoodformalstructureinthesenseof[9].
10 YVESANDRE´
andtheprojectionontothefirst twotermscan bewritten
C[[x,y]][ 1 ] 1 1
xy
̟ : → K[ ].
C[[x,y]] x x
Thedecomposition(4.1)ofM˜ thusgivesrise,bytensorisationwithK((x)),
toadecompositionofM whichrefines(3.1)(takingintoaccounttheunic-
K
ityofthelatter):
(4.2) Lϕj ⊗Rj = M (L˜ϕ˜k,ψ˜k ⊗R˜k)⊗K((x)).
k,̟(ϕ˜k)=ϕj
4.3. Whereas it is not always the case that (E′,∇′) has nice formal struc-
tureat Q′, thiscan befixedby blowingup:
4.3.1. Proposition. After blowing up finitely many times some crossing
points, (E′,∇′) acquires a nice formal structure at every crossing point of
theinverseimageof Z.
This was first proved by C. Sabbah [9, III, 4.3.1], using a generalization
in dimension2 of the nilpotentorbit method of Babbitt-Varadarajan. In [1,
5.4.1], we have given a simpler and more straightforward proof (which is
neverthelesstoolongto berepeated here).
5. PROOF OF 2.2.1.
We assume from now on, as we may by 4.3.1, that (E′,∇′) has a nice
formal structure at all crossing points of π−1(Z) lying on the exceptional
divisor.
5.1. Let P(N X′) be the projectivization of the normal bundle N X′,
Ei Ei
and let (∞) be the section at infinity over E . Taking Zariski closure,
i
the effective divisor Di on NE0X′ gives rise to an effective divisor D¯i on
i
P(N X′),whichisfiniteoverE (ofdegreeµ .ρ ). Itisclearfromequa-
Ei i (ρi) i
tion(3.5)that D¯ doesnotmeet (∞)aboveE0.
i i
5.1.1.Lemma. i) AboveE ∩Z′, D¯ doesnot meet (∞).
i j i
ii) Above Q′ = Ei ∩ Ei′, the intersection multiplicity of D¯i and (∞) is
≤ µ(ρi).ρi′.
Proof. Letususecoordinatesx,y asin4.1. Replacingthembyx1/e,y1/e if
necessary, one may assume that one has a decomposition (4.1). One reads
onequation(3.5)thattheintersectionmutiplicityof D¯ and (∞) aboveQ′
i
is
max(0,− X µi,j.ordyϕj).
j,ordxϕj=−ρi