Oblique poles of $\int_X| {f}| ^{2λ}| {g}|^{2μ} \square$

0.28 MB·English

by  Daniel Barlet

OBLIQUE POLES OF |f|2λ|g|2µ (cid:3)
X
R
D. BARLET AND H.-M. MAIRE
9 Abstract. Existence of oblique polar lines for the meromorphic extension of the cur-
0 rent valued function |f|2λ|g|2µ(cid:3) is given under the following hypotheses: f and
0 g are holomorphic fuRnction germs in Cn+1 such that g is non-singular, the germ
2
S := {df ∧ dg = 0} is one dimensional, and g| is proper and finite. The main
S
n tools we use are interaction of strata for f (see [?]), monodromy of the local system
a Hn−1(u) on S for a given eigenvalue exp(−2iπu) of the monodromy of f, and the
J
monodromy of the cover g|S. Two non-trivial examples are completely worked out.
0
2
Introduction

]
G In the study of a holomorphic function f defined in an open neighbourhood of 0 ∈ Cn+1
A
with one dimensional critical locus S started in [?] and completed in [?], the main tool
h. was to restrict f to hyperplane sections transverse to S∗ := S \{0} and examine, for a
t given eigenvalue exp(−2iπu) of the monodromy of f, the local system Hn−1(u) on S∗
a
m formed by the corresponding spectral subspaces. Higher order polesof the current valued
[ meromorphic function |f|2λ(cid:3) at −u−m, some m ∈ N, aredetected bynon-extendable

1 sections of Hn−1(u) toRS. An important part of this local system remained unexplored
v in [?] and [?] because only the eigenvalue 1 of the monodromy Θ of the local system
0
Hn−1(u) has been considered, via the spaces H0(S∗,Hn−1(u)) and H1(S∗,Hn−1(u)).
7
0 In this paper, we will focus on the other eigenvalues of Θ. Let us introduce an auxiliary
3
function t with the following properties:
.
1
(1) the function t is non-singular near 0;
0
9 (2) the set Σ := {df ∧ dt = 0} is a curve ;
0 (3) the restriction t| : S → D is proper and finite ;
: S
iv (4) t|−S1(0) = {0} and t|S∗ is a finite cover of D∗ := D\{0}.
X Remarkthatcondition(4)mayalwaysbeacheivedbylocalizationnear 0 whenconditions
r (1), (2) and (3) are satisfied. These conditions are satisfied in a neighbourhood of
a
the origin if (f,t) forms an isolated complete intersection singularity (icis) with one
dimensional critical locus. But we allow also the case where Σ has branches in {f = 0}
not contained in S.
The direct image of the constructible sheaf Hn−1(u) supported in S by t will be denoted
by H; it is a local system on D∗. Let H be the fibre of H at t ∈ D∗ and Θ
0 0 0
its monodromy which is an automorphism of H . In case where S is smooth, it is
0
possible to choose the function t in order that t| is an isomorphism and Θ may be
S 0
identified with the monodromy Θ of Hn−1(u) on S∗. In general, Θ combines Θ and
0
Date: January 20, 2009.
2000 Mathematics Subject Classification. 32S40,58K55.
Key words and phrases. asymptotic expansion - fibre integrals - meromorphic extension.
1
2 D. BARLET ANDH.-M. MAIRE
the monodromy of the cover t|S∗.
If S =: ∪ S isthedecompositionof S intoirreduciblebranches, wehaveananaloguous
i∈I i
decomposition on S∗ of the local system Hn−1(u) = ⊕ Hn−1(u)i with Θ = ⊕ Θi.
i i
Take an eigenvalue exp(−2iπl/k) 6= 1 of Θ, with l ∈ [1,k − 1] and (l,k) = 1. We
define an anologue of the interaction of strata in this new context. The auxiliary non
singular function t is used to realize analytically the rank one local system on S∗ with
monodromy exp(−2iπl/k). To perform this we need the degree of t at the origin on the
irreducible branch S we are interested in to be relatively prime to k. Of course this is
i
the case when S is smooth and t transversal to S at the origin. Using then a k−th
root of t we can lift our situation to the case where we consider the invariant section of
˜
the complex of vanishing cycles of the lifted function f (see Theorem 3.2) and then use
already known results from [?].
Withthehelpofelementarypropertiesofmeromorphicfunctionsoftwovariablesdetailed
in paragraph 2, we deduce from interaction of strata above the existence of oblique polar
lines for the meromorphic extension of |f|2λ|t|2µ(cid:3). This result is new and consists in
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a first step toward the comprehension oRf the polar structure of such an extension.
1. Polar structure of |f|2λ (cid:3)
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Theorem 1.1. Bernstein & Gelfand. For m and p ∈ N∗, let Y be an open subset
in Cm, f : Y → Cp a holomorphic map and X a relatively compact open set in Y .
Then there exists a finite set P(f) ⊂ Np such that, for any form φ ∈ Λm,mC∞(X) with
c
compact support, the holomorphic map in {ℜλ > 0}×···×{ℜλ > 0} given by
1 p
(1.1) (λ ,...,λ ) 7→ |f |2λ1 ...|f |2λpφ
1 p Z 1 p
X
has a meromorphic extension to Cp with poles contained in the set
{ha | λi+l = 0}.
a∈P([f),l∈N∗
Proof. For sake of completeness we recall the arguments of [?].
Using desingularization of the product f ...f , we know [?] that there exists a holomor-
1 p
phic manifold Y˜ of dimension m and a holomorphic proper map π : Y˜ → Y such that
˜
the composite functions f := f ◦π are locally expressible as
j j
(1.2) f˜(y) = yak1 ...yakmu (y),1 6 k 6 p,
k 1 m k
where ak ∈ N and u is a holomorphic nowhere vanishing function. Because π−1(X) is
j k
relatively compact, it may be covered by a finite number of open set where (1.2) is valid.
For ϕ ∈ Λm,mC∞(X) and ℜλ ,...,ℜλ positive, we have
c 1 p
|f |2λ1 ...|f |2λpφ = |f˜|2λ1...|f˜|2λpπ∗φ.
Z 1 p Z 1 p
X π−1(X)
OBLIQUE POLES OF R |f|2λ|g|2µ (cid:3) 3
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Using partition of unity and setting µ := a1λ +···+apλ , 1 6 k 6 m, we are reduced
k k 1 k p
to give a meromorphic extension to
(1.3) (µ ,...,µ ) → |y |2µ1...|y |2µmω(µ,y),
1 m Z 1 m
Cm
where ω isa C∞ formoftype (m,m) withcompact support in Cm valued inthespaceof
entire functions on Cm. Of course, (1.3) is holomorphic in {ℜµ > −1,...,ℜµ > −1}.
1 m
The relation
(µ +1).|y |2µ1 = ∂ (|y |2µ1.y )
1 1 1 1 1
implies by partial integration in y
1
−1
|y |2µ1 ...|y |2µmω(µ,y) = |y |2µ1.y .|y |2µ2 ...|y |2µm∂ ω(µ,y).
Z 1 m µ +1 Z 1 1 2 m 1
Cm 1 Cm
Because ∂ ω is again a C∞ form of type (m,m) with compact support in Cm valued in
1
the space of entire functions on Cm, we may repeat this argument for each coordinate
y ,...,y and obtain
2 m
|y |2µ1 ...|y |2µmω(µ,y) =
Z 1 m
Cm
(−1)m
|y |2µ1.y .|y |2µ2.y ...|y |2µm.y .∂ ...∂ ω(µ,y).
(µ +1)...(µ +1) Z 1 1 2 2 m m 1 m
1 m Cm
The integral on the right hand side is holomorphic for ℜµ > −3/2,...,ℜµ > −3/2.
1 m
Therefore the function (1.3) is meromorphic in this domain with only possible poles in
the union of the hyperplanes {µ +1 = 0},...,{µ +1 = 0}.
1 m
Iteration of these arguments concludes the proof. (cid:3)
Remark 1.2. An alternate proof of Theorem 1.1 has been given for p = 1 by Bernstein
[?], Bj¨ork [?], Barlet-Maire [?], and by Loeser [?] and Sabbah [?] in general.
In case where f ,...,f define an isolated complete intersection singularity (icis), Loeser
1 p
andSabbahgavemorover thefollowing informationontheset P(f) ofthe”slopes”ofthe
polar hyperplanes in the meromorphic extension of the function (1.1): it is contained in
the set of slopes of the discriminant ∆ of f, which in this case is an hypersurface in Cp.
More precisely, take the (p−1)−skeleton of the fanassociated to the Newton polyhedron
of ∆ at 0 and denote by P(∆) the set of directions associated to this (p−1)−skeleton
union {(a ,...,a ) ∈ Np | a ...a = 0}. Then
1 p 1 p
P(f) ⊆ P(∆).
In particular, if the discriminant is contained in the hyperplanes of coordinates, then
there are no polar hyperplanes with direction in (N∗)p .
The results of Loeser and Sabbah above have the following consequence for an icis which
is proved below directly by elementary arguments, after we have introduced some termi-
nology.
Definition 1.3. Let f ,...,f be holomorphic functions on an open neighbourhood X
1 p
of the origin in Cm. We shall say that a polar hyperplane H ⊂ Cp for the meromorphic
extension of |f |2λ1 ...|f |2λp(cid:3) is supported by the closed set F ⊂ X, when H is not
X 1 p
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4 D. BARLET ANDH.-M. MAIRE
a polar hyperplane for the meromorphic extension of |f |2λ1...|f |2λp(cid:3). We shall
X\F 1 p
say that a polar direction is supported in F if any polRar hyperplane with this direction
is supported by F.
Proposition 1.4. Assume f ,...,f are quasi-homogeneous for the weights w ,...,w ,
1 p 1 p
of degree a ,...,a . Then if there exists a polar hyperplane direction supported by the
1 p
origin for (1.1) in (N∗)p it is (a ,...,a ) and the corresponding poles are at most simple.
1 p
In particular, for p = 2, and if (f ,f ) is an icis, all oblique poles have direction (a ,a ).
1 2 1 2
Proof. Quasi-homogeneity gives f (tw1x ,...,twmx ) = takf (x), k = 1,...,p.
k 1 m k
Let Ω := m(−1)j−1w x dx ∧···∧ dx ∧···∧ dx so that dΩ = ( w )dx.
1 j j 0 j m j
¿From EuPler’s relation,because fλk is quasi-homogeneous of degree a Pλ :
k d k k
dfλk ∧Ω = a λ fλk dx,
k k k k
dxδ ∧Ω = hw | δixδdx, ∀δ ∈ Nm.
Take ρ ∈ C∞(Cm); then, with 1 = (1,...,1) ∈ Np and ε ∈ Nm:
c
d(|f|2λxδx¯ερΩ∧ dx¯) = (ha | λi+hw | δ+1i)|f|2λxδx¯ερdx∧ dx¯+|f|2λxδx¯εdρ∧Ω∧ dx¯.
Using Stokes’ formula we get
(ha | λi+hw | δ +1i) |f|2λxδx¯ερdx∧ dx¯ = − |f|2λxδx¯εdρ∧Ω∧ dx¯.
Z Z
For ρ = 1 near 0, dρ = 0, near 0. Therefore the right hand side has no poles supported
by the origin. Now the conclusion comes from the Taylor expansion at 0 of the test
function. (cid:3)
The question of whether a polar hyperplane is effectively present in the meromorphic
extension of the function (1.1) for a least one φ has been addressed in case p = 1 under
the name ”contribution effective” in a sequence of papers by D. Barlet [?], [?], [?] etc ...
For p > 1 no general geometric conditions are known to produce poles with direction in
(N∗)p. In the following paragraphs, we examine the case p = 2.
2. Existence of polar oblique lines
In this paragraph, we consider two holomorphic functions f,g : Y → C, where Y is an
open subset in Cm and fix a relatively compact open subset X of Y . Without loss of
generality, we assume 0 ∈ X. We study the possible oblique poles of the meromorphic
extension of the current valued function
(2.1) (λ,µ) 7→ |f|2λ|g|2µ(cid:3).
Z
X
The following elementary lemma is basic.
Lemma 2.1. Let M be a meromorphic function in C2 with poles along a family of lines
with directions in N2. For (λ ,µ ) ∈ C2, assume
0 0
(i) {λ = λ } is a polar line of order 6 k of M,
0 0
(ii) {µ = µ } is not a polar line of M,
0
(iii) λ 7→ M(λ,µ ) has a pole of order at least k +1 at λ .
0 0 0
OBLIQUE POLES OF R |f|2λ|g|2µ (cid:3) 5
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Then there exists (a,b) ∈ (N∗)2 such that the function M has a pole along the (oblique)
line {aλ+bµ = aλ +bµ }.
0 0
Proof. If M does not have an oblique pole through (λ ,µ ), then the function (λ,µ) 7→
0 0
(λ−λ )k0M(λ,µ) is holomorphic near (λ ,µ ). Therefore, λ 7→ M(λ,µ ) has at most a
0 0 0 0
pole of order k at λ . Contradiction. (cid:3)
0 0
It turns out that to check the first condition in the above lemma for the function (2.1),
it is sufficient to examine the poles of the meromorphic extension of the current valued
function
(2.2) λ 7→ |f|2λ(cid:3).
Z
X
Such a simplification does not hold for general meromorphic functions. For example,
λ+µ
(λ,µ) 7→
λ2
has a double pole along {λ = 0} but its restricition to {µ = 0} has only a simple pole
at 0.
Proposition 2.2. If the meromorphic extension of the current valued function (2.2) has
a pole of order k at λ ∈ R , i.e., it has a principal part of the form
0 −
T T
k 1
+···+ ,
(λ−λ )k λ−λ
0 0
at λ , then the meromorphic extension of the function (2.1) has a pole of order
0
(2.3) k := max{0 6 l 6 k | suppT 6⊆ {g = 0}}
0 l
along the line {λ = λ }.
0
Proof. As a consequence of the Bernstein identity (see [?]), there exists N ∈ N such that
the extension of |f|2λφ in {ℜλ > λ −1} can be achieved for φ ∈ Λm,mCN(X). Our
X 0 c
hypothesis implieRs that this function has a pole of order 6 k at λ . Because |g|2µφ is
0
of class CN for ℜµ large enough, the function
λ 7→ |f|2λ|g|2µφ
Z
X
has a meromorphic extension in {ℜλ > λ −1} with a pole of order 6 k at λ . We have
0 0
proved that (2.1) has a pole of order 6 k along the line {λ = λ }.
0
Near λ , the extension of |f|2λφ writes
0 X
R
hT ,φi hT ,φi
k 1
+···+ +....
(λ−λ )k λ−λ
0 0
Hence that of |f|2λ|g|2µφ looks
X
R
hT |g|2µ,φi hT |g|2µ,φi
k 1
+···+ +....
(λ−λ )k λ−λ
0 0
If suppT ⊆ {g = 0}, then the first term vanishes for ℜµ large enough, because T is
k k
of finite order (see the beginning of the proof). So the order of the pole along the line
{λ = λ } is 6 k .
0 0
6 D. BARLET ANDH.-M. MAIRE
Take x ∈ suppT such that g(x ) 6= 0 and V a neighborhood of x in which g does
0 k0 0 0
not vanish. From the definition of the support, there exists ψ ∈ Λm,mC∞(V) such that
c
hT ,ψi =6 0. With φ := ψ|g|−2µ ∈ Λm,mC∞(V), we get
k0 c
hT |g|2µ,φi = hT ,ψi =6 0.
k0 k0
Therefore, the extension of (2.1) has a pole of order k along the line {λ = λ }. (cid:3)
0 0
Corollary 2.3. For (λ ,µ ) ∈ (R )2, assume
0 0 −
(i) the extension of the current valued function (2.2) has a pole of order k at λ ,
0
(ii) µ is not an integer translate of a root of the Bernstein polynomial of g,
0
(iii) λ 7→ Pf(µ = µ , |f|2λ|g|2µ(cid:3)) has a pole of order l > k where k is defined
0 X 0 0 0
in (2.3) at λ . R
0
Then the meromorphic extension of the current valued function (2.1) has at least l −k
0 0
oblique lines, counted with multiplicities, through (λ ,µ ).
0 0
Example 2.4. m = 3, f(x,y,z) = x2 +y2+z2, g(x,y,z) = z.
Example 2.5. m = 4, f(x,y,z) = x2 +y2+z2 +t2, g(x,y,z,t) = t2.
OBLIQUE POLES OF R |f|2λ|g|2µ (cid:3) 7
X
Example 2.6. m = 3, f(x,y,z) = x2 +y2, g(x,y,z) = y2+z2.
In this last example, Corollary 2.3 does not apply because for λ = −1 we have k = l .
0 0 0
Existence of an oblique polar line through (−1,0) is obtained by computation of the
extension of λ 7→ Pf(µ = 1/2, |f|2λ|g|2µ(cid:3)).
X
R
3. Pullback and interaction
In this paragraph, we give by pullback a method to verify condition (iii) of Corollary 2.3
when g is a coordinate. As a matter of fact the function λ 7→ |f|2λ|g|2µ0(cid:3) is only
X
known by meromorphic extension (via Bernstein identity) when Rµ is negative; it is in
0
general difficult to exhibit some of its poles.
We consider therefore only one holomorphic function f : Y → C, where Y is an open
subset in Cn+1 and fix a relatively compact open subset X of Y . The coordinates in
Cn+1 will be denoted by x ,...,x ,t. Let us introduce also the finite map
1 n
(3.1) p : Cn+1 → Cn+1 such that p(x ,...,x ,τ) = (x ,...,x ,τk)
1 n 1 n
for some fixed integer k. Finally, put f˜:= f◦p : X˜ → C where X˜ := p−1(X).
Proposition 3.1. With the above notations and λ ∈ R suppose
0 −
(a) the extension of the current valued function (2.2) has a pole of order 6 1 at λ ,
0
(b) λ 7→ |f˜|2λ(cid:3) has a double pole at λ .
X˜ 0
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8 D. BARLET ANDH.-M. MAIRE
Then there exists l ∈ [1,k − 1] such that the extension of the current valued function
λ 7→ |f|2λ|t|−2l/k(cid:3) has a double pole at λ .
X 0
R
Proof. Remark that the support of the polar part of order 2 of |f˜|2λ(cid:3) at λ is con-
X˜ 0
tained in {τ = 0} because we assume (a) and p is a local isomorpRhism outside {τ = 0}.
After hypothesis (b) there exists ϕ ∈ Λn+1,n+1C∞(X˜) such that
c
A := P λ = λ , |f˜|2λϕ 6= 0.
2 0 Z
(cid:0) X˜ (cid:1)
Consider the Taylor expansion of ϕ along {τ = 0}
ϕ(x,τ) = τjτ¯j′ϕj,j′(x)∧dτ ∧dτ¯+o(|τ|N)
j+Xj′6N
where N is larger than the order of the current defined by P on a compact set K
2
containing the support of ϕ. Therefore
A = P2 λ = λ0,Z |f˜|2λτjτ¯j′ϕj,j′(x)χ(|τ|2k)∧dτ ∧dτ¯
j+Xj′6N (cid:0) X˜ (cid:1)
where χ has support in K and is equal to 1 near 0. Because A does not vanish there
exists (j,j′) ∈ N2 with j +j′ 6 N such that
Aj,j′ := P2 λ = λ0,Z |f˜|2λτjτ¯j′ϕj,j′(x)χ(|τ|2k)∧dτ ∧dτ¯ 6= 0.
(cid:0) X˜ (cid:1)
˜
The change of variable τ → exp(2iπ/k)τ that leaves f invariant, shows that
Aj,j′ = exp(2iπ(j − j′)/k)Aj,j′. Hence Aj,j′ = 0 for j − j′ 6∈ kZ. We then get the
existence of (j,j′) ∈ N2 verifying j′ = j +kν with ν ∈ Z and Aj,j′ 6= 0.
The change of variable t = τk in the computation of Aj,j′ gives
P λ = −λ , |f|2λ|t|2(j−k+1)/kt¯νϕ (x)χ(|t|2)∧dt∧dt¯ 6= 0.
2 0 Z j,j+kν
(cid:0) X (cid:1)
This ends the proof with −l = j −k + 1 if ν > 0 and with −l = j′ −k +1 if ν < 0.
Notice that l < k in all cases. Necessarily l 6= 0 because from hypothesis (a), we know
that the extension of the function (2.2) does not have a double pole at λ . (cid:3)
0
Theorem 3.2. For Y open in Cn+1 and X relatively compact open subset of Y , let
f : Y → C be holomorphic and g(x,t) = t. Assume (f,g) satisfy properties (1) to (4)
of the Introduction. Moreover suppose
(a) |f|2λ(cid:3) has a at most a simple pole at λ −ν , ∀ν ∈ N;
0
(b) Re2πiλ0 is a eigenvalue of the monodromy of f acting on the Hn−1 of the Milnor fiber
of f at the generic point of a connected component S∗ of S∗, and there exist a non
i
zero eigenvector such its monodromy around 0 in S is a primitive k−root of unity,
i
with k > 2;
(c) the degree d of the covering t : S∗ → D∗ is prime to k;
i i
(d) e2πiλ0 is not an eigenvalue of the monodromy of f˜ acting on the Hn−1 of the Milnor
fiber of f˜ at 0 , where f˜(x,τ) = f(x ,...,x ,τk).
1 n
OBLIQUE POLES OF R |f|2λ|g|2µ (cid:3) 9
X
Then there exists an oblique pole of |f|2λ|g|2µ(cid:3) through (λ −j,−l/k), some j ∈ N,
0
and some l ∈ [1,k−1]. R
Remark that condition (b) implies that λ 6∈ Z because of the result of [?].
0
Proof. Notice first that (λ,µ) 7→ |f|2λ|g|2µ(cid:3) has a simple pole along {λ = λ }. Indeed
0
the support of the residue currenRt of λ 7→ |f|2λ(cid:3) at λ contains S∗ where t does not
0 i
vanish and Proposition 2.2 applies. R
Denote by z a local coordinate on the normalization of S . The function t has a zero of
i
order d on this normalization and hence d is the degree of the cover S∗ → D∗ induced
i i i
by t. Without loss of generality, we may suppose t = zdi on the normalization of S .
i
Lemma 3.3. Let k,d ∈ N∗ and put
Y := {(z,τ) ∈ D2/τk = zd},Y∗ := Y \{0}.
d d d
If k and d are relatively prime, then the first projection pr : Y∗ → D∗ is a cyclic cover
1,d d
of degree k.
Proof. Let us prove that the cover defined by pr is isomorphic to the cover defined by
1,d
pr , that may be taken as definition of a cyclic cover of degree k.
1,1
After B´ezout’s identity, there exist a,b ∈ Z such that
(3.2) ak +bd = 1.
Define ϕ : Y → C2 by ϕ(z,τ) = (z,zaτb). From (3.2), we have ϕ(Y ) ⊆ Y and clearly
d d 1
pr ◦ϕ = pr .
1,1 1,d
The map ϕ is injective because
ϕ(z,τ) = ϕ(z,τ′) =⇒ τk = τ′k and τb = τ′b,
hence τ = τ′, after (3.2). It is also surjective: take (z,σ) ∈ Y∗; the system
1
τb = σz−a, τk = zd, when σk = z,
has a unique solution because the compatibility condition σkz−ak = zbd is satisfied. (cid:3)
End of proof of Theorem 3.2. Take the eigenvector with monodromy exp(−2iπl/k)
with (l,k) = 1 given by condition (b). Its pullback by p becomes invariant under the
monodromy of τ because of the condition (c) and the lemma given above. After (d),
this section does not extend through 0. So we have interaction of strata (see [?]) and a
double pole for λ 7→ |f˜|2λ(cid:3) at λ −j with some j ∈ N. It remains to use Proposition
0
3.1 and Corollary 2.3R. (cid:3)
4. Interaction of strata revised
In this paragraph notations and hypotheses are those of the Introduction. Here, the
function t is the last coordinate. We suppose that the eigenvalue exp(−2iπu) of the
monodromy of f is simple at each point of S∗. Therefore, this eigenvalue is also simple
for the monodromy acting on the group Hn−1 of the Milnor fibre of f at 0. In order to
compute the constructible sheaf Hn−1(u) on S we may use the complex (Ω [f−1],δ ),
X u
that is the complex of meromorphic forms with poles in f−1(0) equipped with the dif-
ferential δ := d−udf∧ along S. This corresponds to the case k = 1 in [?].
u f 0
10 D. BARLET ANDH.-M. MAIRE
We use the isomorphisms
(4.1) rn−1 : hn−1 → Hn−1(u) over S and τ : hn−1 → hn over S∗,
1
where hn−1 [resp. hn] denotes the (n − 1)−th [resp. n−th] cohomology sheaf of the
complex (Ω [f−1],δ ).
X u
In order to look at the eigenspace for the eigenvalue exp(−2iπl/k) of the monodromy
Θ of the local system Hn−1(u) on S∗, it will be convenient to consider the complex
of sheaves
l dt
Γ := Ω•[f−1,t−1],δ − ∧
l u
k t
(cid:0) (cid:1)
which is locally isomorphic along S∗ to (Ω [f−1],δ ) via the choice of a local branch
X u
of tl/k and the morphism of complexes (Ω [f−1],δ ) → Γ given by ω 7→ tl/k.ω
X u l
which satisfies
l dt
δ (tl/kω)− ∧tl/kω = tl/kδ (ω).
u u
k t
Butnoticethatthiscomplex Γ isalsodefinedneartheorigin. Ofcourse,aglobalsection
l
σ ∈ H0(S∗,hn−1(Γ )) gives, via the above local isomorphism, a multivalued global
l
section on S∗ of the local system Hn−1(u) ≃ hn−1 with monodromy exp(−2iπl/k)
(as multivalued section).
So a global meromorphic differential (n−1)−form ω with poles in {f.t = 0} such
that dω = udf ∧ω+ l dt∧ω defines such a σ, and an element in H with monodromy
f k t 0
exp(−2iπl/k).
We shall use also the morphism of complexes of degree +1
τˇ : Γ → Γ
1 l l
given by τˇ (σ) = df ∧σ. It is an easy consequence of [?]that inour situation τˇ induces
1 f 1
an isomorphisme τˇ : hn−1(Γ ) → hn(Γ ) on S∗, because we have assumed that the
1 l l
eigenvalue exp(−2iπu) for the monodromy of f is simple along S∗.
Our first objective is to build for each j ∈ N a morphism of sheaves on S∗
(4.2) r : hn−1(Γ ) → Hn (O ),
j l [S] X
via the meromorphic extension of |f|2λ|t|2µ(cid:3). Here Hn (O ) denotes the subsheaf
X [S] X
of the moderate cohomology with Rsupport S of the sheaf Hn(O ). It is given by the
S X
n−th cohomology sheaf of the Dolbeault-Grothendieck complex with support S :
Hn (O ) ≃ Hn(H0(Db0,•), d′′).
[S] X S X
Let w bea (n−1)−meromorphic formwithpolesin {f = 0},satisfying δ (w) = l dt∧w
u k t
on an open neighbourghood U ⊂ X \{t = 0} of a point in S∗. Put for j ∈ N :
df
r (w) := Res λ = −u,Pf(µ = −l/k, |f|2λ|t|2µf¯−j ∧w∧(cid:3)) .
j Z f
(cid:0) U (cid:1)
These formula define d′−closed currents of type (n,0) with support in S∗ ∩U.