A COURSE ON IRREGULAR SINGULARITIES OF
MEROMORPHIC CONNECTIONS
by
Jean-Baptiste Teyssier
Introduction
Let X be a smooth complex algebraic variety. Let i : D ãÑ X be a normal crossing
divisor in X and let η be a codimension 1 point in D. Let M be a meromorphic
connection on X with poles along D. Ignoring extension and ramification issues,
Levelt-Turrittin decomposition theorem asserts that the restriction of M to the formal neighbourhood of η splits as a direct sum of differential modules which are easy
to work with.
This decomposition may not hold at some other points of D, but when it does,
we sayp1q that M has good formal decomposition along D. A conjecture of Sabbah [Sab00], recently proved by Kedlaya [Ked10][Ked11] and Mochizuki [Moc09]
[Moc11] independently, asserts the existence of a chain of blow-ups p : Y ÝÑ X
above D such that p˚ M has good formal decomposition along p´1 pDq.
In a sense, this result is to meromorphic connections what Hironaka desingularization is to varieties. It has recently allowed ground-breaking progresses in our
understanding of D-modules. Let us mention an avatar of the decomposition theorem for semi-simple holonomic D-modules [Moc11], containing as a particular case
the decomposition theorem for an arbitrary semi-simple perverse sheaf, as conjectured by Kashiwara. Let us also mention the generalization of Grothendieck-Deligne
comparison theorem to arbitrary algebraic flat connections [Hie09], and Kashiwarad’Agnolo progress towards an irregular version of the Riemann-Hilbert correspondence
[DK13].
The goal of this course is to introduce various incarnations of irregularity (analytic, algebraic, formal, cohomological) and to explain the concepts at stake in the
statement of Kedlaya-Mochizuki theorem. In the first section, we explain the proof of
p1q See
section 3 for a precise definition.
2
J.-B. TEYSSIER
Malgrange formula [Mal71] for the irregularity number irrpP q of a differential operator P in one variable. In this situation, irrpP q measures the difference between the
actions of P on convergent power series and on formal power series.
In section 2, we give a cohomological interpretation of irrpP q when D{DP defines a
germ of meromorphic connection. In this situation, irregularity is the obstruction to
lift a formal solution of P pf q “ 0 to an analytic solution defined on a small punctured
disc.
Section 3 is devoted to the exegesis of Kedlaya-Mochizuki theorem. We also discuss
a fundamental result of André [And07].
In section 4, we give an application to the existence of periods for arbitrary flat
algebraic connections.
This text collects notes of a 6 hours course given by the author at ETH in October
2015. We thank J. Fresan and P. Jossen for the invitation, the audience for interesting
questions, as well as the FIM for optimal working conditions.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Irregularity in dimension one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Analysis of the action of P on CJxK . . . . . . . . . . . . . . . . . . . .
1.3. Analysis of the action of P on Ctxu . . . . . . . . . . . . . . . . . . . .
1.4. The irregularity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5. From differential operators to differential modules . . . . . .
2. A cohomological interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. A few examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. The main asymptotic existence theorem . . . . . . . . . . . . . . . .
2.5. Cohomological interpretation of irregularity . . . . . . . . . . . .
3. Kedlaya-Mochizuki theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Formal differential modules . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Statement of Kedlaya-Mochizuki theorem . . . . . . . . . . . . . .
4. Application to periods of algebraic flat connections . . . . . . . . . . . .
4.1. Irregularity in any dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Grothendieck-Deligne comparison theorem . . . . . . . . . . . . . .
4.3. Rapid decay homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
3
1. Irregularity in dimension one
1.1. Notations. — Let Ctxu be the space of germs at 0 P C of convergent power
series, and let CJxK be the ring of formal power series. We note
ˆ ˙k
à
d
D“
Ctxu
dx
kPN
d
the space of germs at 0 P C of finite order differential operators. The symbol dx
PD
is subjected to the relations
d
d
f “ f1 ` f
dx
dx
for every f P Ctxu. An operator P P D acts on Ctxu and on CJxK by usual differentiation. The main slogan of this section is
Slogan 1.1.1. — The irregularity of P measures the difference between the action of
P on Ctxu and the action of P on CJxK.
The aim of what follows is to give a precise meaning to 1.1.1. We write
ˆ ˙k
d
ÿ
d
ak
P “
dx
k“1
with ad ‰ 0 and we denote by KerpP, Ctxuq and CokerpP, Ctxuq (resp. KerpP, CJxK
and CokerpP, CJxKq) the kernel and cokernel of the action of P on Ctxu (resp. on
CJxK).
We say that a linear morphism Φ : E1 ÝÑ E2 between C-vector spaces has finite
index if Ker Φ and Coker Φ are finite dimensional. In that case, the number
χpΦq “ dim Ker Φ ´ dim Coker Φ
is called the index of Φ.
1.2. Analysis of the action of P on CJxK. — We prove the following
Proposition 1.2.1. — The action of P on CJxK has finite index, and
χpP, CJxKq “ Suppk ´ ord ak q
k
where ord ak denotes the maximal power of x dividing ak .
Proof. — The fact
ÿ that KerpP, CJxKq is finite dimensional boils down to the fact that
a solution f “ ak xk of P pf q “ 0 is determined by a finite number of ai ’s which is
independent of f .
Let us prove that CokerpP, CJxKq is finite dimensional. Set M “ Suppk ´ ord ak q.
Let us denote by A the set of k P N such that k ´ ord ak “ M . For every k, we have
ord ak ě k ´ M so we can write ak “ xk´M bk where bk P Ctxu and bk p0q ‰ 0 iff
k P A. We have
dk
ak pxq k xi “ ipi ´ 1q . . . pi ´ k ` 1qbk p0qxi´M ` terms of degree ą xi´M
dx
4
J.-B. TEYSSIER
Thus,
(1.2.2)
P pxi q “ p
ÿ
ipi ´ 1q . . . pi ´ k ` 1qbk p0qqxi´M ` terms of degree ą xi´M
kPA
The coefficient cpiq of xi´M is polynomial in i of degree Max A ě 0. If this degree
is 0, this means that A “ t0u and then c constant to b0 p0q ‰ 0. If not, c is non
constant. In both case, one can find i0 ą 0 such that cpiq ‰ 0 for every i ě i0 . For
i ě Maxpi0 , M q, one can write
(1.2.3)
P pcpiq´1 xi q “ xi´M ` terms of degree ą xi´M
So
P pcpiq´1 xi q “ xi´M ` αxi´M `1 ` terms of degree ą xi´M `1
The relation (1.2.3) for i ` 1 thus gives
P pcpiq´1 xi ´ cpi ` 1q´1 αxi`1 q “ xi´M ` terms of degree ą xi´M `1
By iterating this process, we see that for i ě Maxpi0 , M q, we have xi´M P Im P .
Thus, CokerpP, CJxKq is generated by the classes of 1, x, . . . , xi0 `M . So it is finite
dimensional.
The proof above gives slightly more than the finite dimensionality of CokerpP, CJxKq.
Let M :“ pxq be the maximal ideal of Ctxu. For every k ě M , we get from (1.2.2)
that P sends Mk to Mk´M , so induces a map
Pk : CJxK{Mk ÝÑ CJxK{Mk´M
The proof above implies that for k " 0, the map P : Mk ÝÑ Mk´M is surjective. In
particular, for k " 0
CokerpP, CJxKq » Coker Pk
Ş k
On the other hand, the sequence of spaces Ek :“ KerpP, CJxKq
M is a decreasing
Ş
sequence of finite dimensional C-vector spaces such that k Ek “ t0u. So Ek “ t0u
for k " 0. Thus
KerpP, CJxKq » Ker Pk
for k " 0. Choosing k big enough, we deduce
χpP, CJxKq “ χpPk q
“ dim CJxK{Mk ´ dim CJxK{Mk´M
“M
1.3. Analysis of the action of P on Ctxu. — We prove the following
Proposition 1.3.1. — The action of P on Ctxu has finite index, and
χpP, Ctxuq “ d ´ ord ad
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
5
For r ą 0 and k ě 0, let us denote by B k pDr q the space of C k functions on Dr
which are holomorphic on Dr . It is a Banach space for the norm
}f }k “ }f }8 ` }f 1 }8 ` ¨ ¨ ¨ ` }f pkq }8
where }f }8 “ SupxPDr |f pxq|. For r ą 0 small enough, P induces a continuous linear
map
Pr : B d pDr q ÝÑ B 0 pDr q
Let us prove the following
Lemma 1.3.2. — The operator Pr has finite index d ´ ord ad .
` d ˘d
Proof. — Let us write Pr “ ad dx
` Qr with Qr of order ă d and let us prove that
Pr has finite index equal to d ´ ord ad . Since multiplication by ad has finite index
` d ˘d
d
´ ord ad and dx
has finite index 1, the composite ad dx
has finite index d ´ ord ad .
So it is enough to prove that Qr is a compact operator. If pfn q P B d pDr qN is a
bounded sequence, then the sequences }Qr pfn q}8 and }Qr pfn q1 }8 are bounded. By
Arzela-Ascoli theorem, the set formed by the fn is relatively compact. Hence, Qr is
compact.
We are left to show how 1.3.2 implies 1.3.1. Taking the germ at 0 induces an injection
(1.3.3)
Ker Pr ÝÑ KerpP, Ctxuq
and KerpP, Ctxuq is the union of all the Ker Pr . As a subspace of KerpP, CJxKq,
KerpP, Ctxuq is finite dimensional. So (1.3.3) is an isomorphism for r small enough.
We are left to prove that the same holds for cokernels. Let us first assume that for
r ď r1 small enough, the restriction morphism
Coker Pr1 ÝÑ Coker Pr
(1.3.4)
is injective. Then 1.3.2 shows that (1.3.4) is an isomorphism for r ď r1 small enough.
From
Coker Pr1
o
'
CokerpP, Ctxuq
7
Coker Pr
we deduce that Coker Pr » CokerpP, Ctxuq for r ! 1, which finishes the proof.
Let us prove that (1.3.4) is injective for r ă r1 small enough. Let g P B 0 pDr1 q
such that g “ Pr pf q on Dr with f P B d pDr q. We have to prove that f extends to
fr P B d pDr1 q. Take r1 such that ad does not vanish on Dr1 zt0u. By Cauchy theorem,
f extends uniquely and holomorphically to Dr1 .
6
J.-B. TEYSSIER
Let us still denote by f this extension. Then, fr is uniquely
determined
Ş by f , so it is enough to extend f to a lens
L :“ Dr1 Dpx0 , q where x0 P Sr1 and ą 0 is small enough.
Let us denote by B 0 pLq (resp. B 1 pLq) the space of continuous (resp. C 1 ) functions on L which are holomorphic on the
interior of L. B 0 pLq is a Banach space for } ¨ }8 . To construct
fr on L is equivalent to find u P B 1 pLqd satisfying
" 1
u “ Au ` v
(1.3.5)
upx1 q “ w
where
˛
¨
˛
0
0
1
0
˚ .. ‹
˚ ..
‹
..
..
˚ . ‹
˚ .
‹
.
.
‹
˚
˚
‹
‹
˚
˚
‹
.
A“˚ .
‹ , v “ ˚ ... ‹
.
‹
˚
˚
‹
˝ 0 ‚
˝ 0
‚
¨¨¨
0
1
g{ad
´a0 {ad ¨ ¨ ¨ ´ad´2 {ad ´ad´1 {ad
where x1 is a point in the interior of L, and where w is the column vector
pf px1 q, . . . , f pd´1q px1 qq. If L is chosen small enough, the operator
¨
B 0 pLq
ÝÑ
u
ÝÑ
B 0 pLq
ˆ
˙
żx
z ÝÑ w `
pAu ` gq
x1
is contracting. By Banach fixed point theorem, it has a unique fixed point, which
automatically satisfies (1.3.5).
1.4. The irregularity space. — We can now give a precise meaning to 1.1.1.
Define Q :“ CJxK{Ctxu. Applying RHompD{DP, ¨ q to the exact sequence of Dmodules
/ Ctxu
/ CJxK
/Q
/0
0
gives a distinguished triangle
(1.4.1)
RHompD{DP, Ctxuq
/ RHompD{DP, CJxKq
/ RHompD{DP, Qq
As a by-product of the proof of the finite dimensionality of KerpP, CJxKq in 1.2.1, we
have CokerpP, Qq » 0. So the long exact sequence associated to (1.4.1) reads
(1.4.2)
0
KerpP, Ctxuq
KerpP, CJxKq
CokerpP, Ctxuq
CokerpP, CJxKq
0
KerpP, Qq
Thus, the obstruction that the analytic and formal Kernel and Cokernel of P coincide
lies in the non vanishing of KerpP, Qq. Putting together 1.2.1 and 1.3.1 gives the
following theorem, due to Malgrange [Mal71]
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
7
Theorem 1.4.3. — The space KerpP, Qq is finite dimensional, and we have
dimC KerpP, Qq “ Supk pk ´ ord ak q ´ pd ´ ord ad q
We define the irregularity number of P as
irrpP q :“ dimC KerpP, Qq
Definition 1.4.4. — We say that P is regular if irrpP q “ 0.
d
´ 1qq “ 1. On
Example 1.4.5. — Malgrange
formula predicts that irrpD{Dpx2 dx
ÿ
n`1
the other hand, f :“ n!x
satisfies
x2
df
´ f “ ´x “ 0 in Q
dx
d
So Kerpx2 dx
´ 1, Qq » Cf .
1.5. From differential operators to differential modules. —
Definition 1.5.1. — We define a Ctxurx´1 s-differential module as the data of a
finite dimensional Ctxurx´1 s vector space endowed with a C-linear map ∇ : M ÝÑ M
satisfying the Leibniz rule
∇pf mq “ f 1 m ` f ∇m
The Leibniz rule turns M into a module over D. Set d :“ dimCtxurx´1 s M . Let
A P Md pCtxurx´1 sq be the matrix of ∇ in a basis e. Let e1 be another basis of M
and let P be the matrix of coefficients of e1 in e. Then, the matrix of ∇ in e1 is
P ´1 P 1 ` P ´1 AP
One can prove [Sab93, 4.2.8] that as a D-module, M » D{DP where P P D. In
particular, irrpM q is well-defined. We have
Theorem 1.5.2. — The following conditions are equivalent
(1) irrpM q “ 0.
(2) M admits a basis in which the entries of the matrix of x∇ have no poles.
Note that description p2q is quite concrete and subtle at the same time, since one
requires that there exists a basis such that etc. For a randomly given basis, the
poles of the matrix of ∇ may be arbitrary high even if M is regular. Also note that
condition p2q is algebraic so it can be generalized to other differential fields.
2. A cohomological interpretation
2.1. Motivation. — Let P P D be a finite order differential operator. As seen in
the previous section, if P is regular, the canonical comparison map
KerpP, Ctxuq ÝÑ KerpP, CJxKq
8
J.-B. TEYSSIER
p2q
is an isomorphism. In general it is not, as
shows: take
ř the nfollowing example
d
2 d
P “ x dx ` p3x ´ 1q dx ` 1. Then the serie n!x is a formal solution for P .
This is however not the end of the story. Suppose that D{DP is a differential
module as defined in 1.5.1, and choose a direction d. Then, the main asymptotic
existence theorem allows to lift a formal solution F to an analytic solution fd defined
on a small sector containing d. The collection of fd gives rise to a 1-cocycle with value
in the sheaf of solutions of P pf q “ 0 with rapid decay at 0. If this cocycle vanishes,
one can produce an analytic lift to F defined on a small punctured disc. We will in
see in 2.5.1 how this vanishing question relates to irrpP q. The main upshot will be
the
Slogan 2.1.1. — Irregularity is the obstruction to lift a formal solution of P pf q “ 0
to an analytic solution defined on a small punctured disc.
r of D
2.2. Asymptotic expansions. — Let us define the real oriented blow up D
1
˚
1
as the closure in D ˆ S of the graph of the map D ÝÑ S given by x ÝÑ x{|x|. The
r ÝÑ D induced by first projection is an isomorphism above D˚ . Let
morphism π : D
˚
r be the associated open immersion. On the other hand p0, eiθ q P t0u ˆ S 1
j : D ÝÑ D
is the limit of peiθ {n, eiθ qnPN˚ , so π ´1 t0u “ t0u ˆ S 1 . In what follows, we identify
π ´1 t0u with S 1 and note θ for p0, eiθ q.
r inherits the structure of a C 8 -manifold with boundary S 1 . The
One can show that D
r
topology of D is that induced by D ˆ S 1 , so θ P S 1 has a fundamental system of
r X Dr ˆ pa, bq where pa, bq is the open interval
neighbourhood of the form Ur pa, bq :“ D
in S 1 between a and b. So if we define the sector
Sr pa, bq “ tx P Dr˚ with x{|x| P pa, bqu
we have set theoretically
Ur pa, bq “ pa, bq
(2.2.1)
ğ
Sr pa, bq
In particular, the space of germs of j˚ OD˚ at θ is the inverse limit of the ΓpSr pa, bq, OD q
where r and the amplitude of pa, bq become smaller and smaller.
p2q which
I owe to P. Jossen.
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
9
ř
Definition 2.2.2. — We say that f P ΓpSr pa, bq, OD q has c “ ně0 cn xn P CJxK as
asymptotic expansion at 0 if for every N ą 0 and every closed sector W in Sr pa, bq,
one can find CpN, W q ą 0 such that
(2.2.3)
Nÿ
´1
cn xn | ď CpN, W q|x|n
|f pxq ´
n“0
for every x P W .
Note that in case c as in 2.2.2 exists, it is unique. We denote it by Jpf q.
Definition 2.2.4. — Let A be the subsheaf of i´1 j˚ OD˚ whose space of germs at
θ P S 1 is the space of f P pj˚ OD˚ qθ admitting an asymptotic expansion in a small
enough sector centred at θ.
Still denoting by CJxK the constant sheaf on S 1 with value CJxK, asymptotic expansion defines a morphism of sheaves on S 1
(2.2.5)
J : A ÝÑ CJxK
The following is known as the Borel-Ritt lemma
Lemma 2.2.6. — The morphism (2.2.5) is surjective.
We denote by A0 the kernel of J. The sections of A0 are called holomorphic
functions with rapid decay at 0.
2.3. A few examples. —
(1) Complex analysis shows that ΓpS 1 , Aq » Ctxu and ΓpS 1 , A0 q » 0.
(2) Let ϕ “ an x´n ` ¨ ¨ ¨ ` a1 x´1 P x´1 Crx´1 s with ai P C and an ‰ 0. Then
eϕ P j˚ OD˚ and one can ask for which direction θ the function eϕ falls in Aθ . We
iθ
iθ
have to analyse the growth of r ÝÑ |eϕpre q | “ |eRe ϕpre q | when r Ñ 0.
iθ
– If Repan e´inθ q ą 0, the function |eϕpre q ´ P | diverges to 8 when r Ñ 0 for
every polynomial P , so (2.2.3) cannot hold on a close sector containing θ.
– If Repan e´inθ q “ 0, for any angle θ` ą θ close enough to θ, we have
`
Repan e´inθ q ą 0, so again (2.2.3) cannot hold on a close sector containing θ.
– Suppose Repan e´inθ q ă 0, that is θ belongs to a segment of type
ˆ
˙
π
2kπ
π
2kπ
Spk, n, θ0 q :“ θ0 ´
`
, θ0 `
`
2n
n
2n
n
where 0 ď k ď n ´ 1 and where θ0 P r0, 2πr is the argument of an . Then (2.2.3)
holds with c “ 0 for any close sector W in a sector of type
ˆ
˙
π
2kπ
π
2kπ
Sr pk, n, θ0 q :“ Sr θ0 ´
`
, θ0 `
`
2n
n
2n
n
Ů
So eϕ defines a section of A over k Spk, n, θ0 q.
10
J.-B. TEYSSIER
(3) Take f P ΓpSr pa, bq, OD q with rapid decay at 0. Choose θ P pa, bq, ρ ă r and let
γ : r0, 1s ÝÑ C be the segment joining 0 to ρeiθ . The function
ż
f pζq
g : z ÝÑ
dζ
ζ
γ ´z
is well defined on Dρ zr0, ρeiθ r.
Fact 2.3.1. — The function g admits
˙
ÿ ˆż
´n´1
gp :“
f pζqζ
dζ xn P CJxK
γ
ně0
as asymptotic expansion at 0 on Dρ zr0, ρeiθ r. In particular, g induces an element of
ΓpS 1 ztθu, Aq.
Proof. — Let W be a close sector in Dρ zr0, ρeiθ r and let N ě 0. We have M :“
inf s0,ρeiθ rˆW |1 ´ x{ζ| ą 0. By definition of f , one can find C ě 0 such that |f pζq| ď
C|ζ|N `1 for ζ P r0, ρeiθ r. Thus
ˇ
ˇ
˙ ˇˇ ˇż
Nÿ
´1 ˆż
ˇ
ˇ f pζqpx{ζqN ˇ ρC N
ˇ
´n´1
nˇ
ˇ
ˇď
gpxq
´
f
pζqζ
dζ
x
“
dζ
|x|
ˇ
ˇ ˇ
ˇ
ˇ
ˇ
M
γ ζp1 ´ x{ζq
γ
n“0
(4)
ÿ Let f P ΓpSr pa, bq, OD q admitting an asymptotic expansion at 0. Set Jpf q “
cn xn . We prove the
ně0
Fact 2.3.2. — The function f 1 admits Jpf q1 as asymptotic expansion at 0.
Proof. — Let W Ă Sr pa, bq be a closed sector, and N ě 0. Let W 1 Ă Sr pa, bq be a
closed sector whose radius and amplitude is strictly bigger than that of W . There
exists δ ą 0 such that for any x P W zt0u, the closed disc Dpx, δ|x|q centred at x with
radius δ|x| is included in W 1 . For x P W zt0u, define γx : t ÝÑ x ` δ|x|e2iπt . Cauchy
formula reads
ż
f pζq
f pyq “
dζ
ζ
γx ´ y
for every y P Dpx, δ|x|q. Derivating and evaluating at x gives
ż
f pζq
f 1 pxq “
dζ
pζ
´ xq2
γx
The same holds for Jpf qN :“
Nÿ
´1
cn xn . So
n“0
1
|f pxq ´
Jpf q1N pxq|
1
ď
pδ|x|q2
ď 2π
ż
|f pζq ´ Jpf qN pζq|dζ
γx
CpN ` 1, W 1 q N ´1
|x|
δ
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
11
As a corollary of 2.3.2, we obtain that P acts on A and A0 .
2.4. The main asymptotic existence theorem. — In this section, P P D such
that D{DP is a differential module in the sense of 1.5.1. The main asymptotic
development theorem is the following
Theorem 2.4.1. — Let θ P S 1 and let g P Aθ . Suppose that g=P(F) with F P CJxK.
Then, one can find f P Aθ such that Jpf q “ F and g “ P pf q.
This theorem is a consequence of the following
Theorem 2.4.2. — The morphism of sheaves P : A0 ÝÑ A0 is surjective.
Let us prove that 2.4.2 implies 2.4.1. We have a diagram
0
/ A0
/ Aθ
J
θ
/ CJxK
/0
0
/ A0
θ
/ Aθ
J
/ CJxK
/0
where the vertical maps are given by the action of P . By Borel-Ritt lemma 2.2.6,
the lines of this diagram are exact. The first square is trivially commutative. The
commutativity of the second square comes from 2.3 p4q. The first vertical morphism
is surjective by 2.4.2. One gets the sought after lift by a diagram chase.
r
2.5. Cohomological interpretation of irregularity. — Recall that i : S 1 ÝÑ D
0
r
is the inclusion of the boundary of D. We denote by KerpP, A q the kernel of the
action of P on A0 . The goal of this section is to prove the following
Theorem 2.5.1. — Let P P D be a germ of differential operator such that D{DP is
a differential module in the sense of 1.5.1. Then
irrpP q “ dim H 1 pS 1 , KerpP, A0 qq
Theorem 2.5.1 is a consequence of the following
Lemma 2.5.2. — The canonical map H 1 pS 1 , A0 q ÝÑ H 1 pS 1 , Aq is zero.
Let us first see why 2.5.2 implies 2.5.1. From the Borel-Ritt lemma 2.2.6, we have
an exact sequence of sheaves on S 1
/ A0
0
(2.5.3)
/A
/ CJxK
/0
From 2.5.2 and 2.3 p1q, it induces an exact sequence
/ Ctxu
0
/ CJxK
/ H 1 pS 1 , A0 q
/0
Hence H 1 pS 1 , A0 q » Q. From 2.4.2, the following sequence of sheaves
0
/ KerpP, A0 q
/ A0
P
/ A0
/0
12
J.-B. TEYSSIER
is exact. So the associated exact sequence in cohomology gives the short exact sequence
`
˘
/ H 1 S 1 , KerpP, A0 q
/Q P /Q
/0
0
and 2.5.1 is proved. We are left to prove 2.5.2.
From [Har77, III 4. Ex 4.4], we know that for any sheaf of abelian groups F on
S 1 , the canonical morphism
1
1
1
lim
ÝÑ Ȟ pU, Fq ÝÑ H pS , Fq
is an isomorphism, where the limit is taken over open covers U of S 1 . Elements
1
1
in lim
ÝÑ Ȟ pU, Fq can be represented by a cocycle defined on a finite cover of S by
successive intervals. So we are left to prove that
1
0
1
lim
ÝÑ Ȟ pU, A q ÝÑ lim
ÝÑ Ȟ pU, Aq
is zero, where the limits are taken over finite covers by successive intervals. We
consider the case where U is made of two intervals pa1 , b1 q and pa2 , b2 q, the general
case being a consequence of this one. Then, Ȟ1 pU, A0 q is generated by classes of the
form pf, 0q, p0, gq P A0 pa1 , b2 q ˆ A0 pa2 , b1 q. By symmetry, we are left to prove that
for f P A0 pa1 , b2 q, the class of pf, 0q in Ȟ1 pU, Aq is 0. At the cost of shrinking the
amplitude of the intervals of U (which amounts to refining the cover), we can suppose
that f P ΓpSr pa1 , b2 q, OD q for some r ą 0.
Take ρ ă r and define on Sρ pa1 ` , b1 q
ż
g1 : x ÝÑ
f pζq
dζ
γ1 ζ ´ x
and on Sρ pa2 , b2 ´ q
ż
g2 : x ÝÑ ´
f pζq
dζ
γ2 `γ3 ζ ´ x
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
13
If V :“ tpa1 ` , b1 q, pa2 , b2 ´ qu, example 2.3 p3q shows pg1 , g2 q P Cˇ0 pV, Aq, where
pCˇ‚ , dČ q denotes the Cech complex construction. Thus
(2.5.4)
dČ pg1 , g2 q “ ppg1 ´ g2 q|pa1 `,b2 ´q , pg1 ´ g2 q|pa2 ,b1 q q
By Cauchy formula, the first term in the right hand side of (2.5.4) is f|pa1 `,b2 ´q .
Since γ1 and ´γ2 ´γ3 are homotopic, the second term in the right hand side of (2.5.4)
is 0. We conclude by noticing that V refines U.
3. Kedlaya-Mochizuki theorem
3.1. Formal differential modules. — Let k be a field of characteristic 0.
Definition 3.1.1. — A kppxqq-differential module M is the data of a finite dimensional kppxqq-vector space endowed with a k-linear operator ∇ : M ÝÑ M satisfying
the Leibniz rule
∇pf mq “ f 1 m ` f ∇m
for every f P kppxqq and every m P M .
A basic example of such a module is a regular module (take 3.1.1 p2q as a definition
of regularity). Another example is as follows: take ϕ P x´1 krx´1 s and define E ϕ as the
kppxqq-differential module whose underlying space is kppxqq with ∇ : f ÝÑ f 1 ` f ϕ1 .
Such a module is called an exponential module.
Regular modules and exponential modules are essentially the building blocs of all
kppxqq-differential modules, due to the theorem of Levelt-Turrittin [Sv00, 3.1].
Theorem 3.1.2. — There exists a finite extension k 1 {k and an integer n ě 1 such
that
à
(3.1.3)
k 1 ppx1{n qq bkppxqq M »
E ϕ b Rϕ
ϕPx´1{n krx´1{n s
where the Rϕ are regular k 1 ppx1{n qq-differential modules and where the induced connection ∇1{n on the left hand side of (3.1.3) is ∇1{n pf b mq “ f 1 b m ` f b ∇m.
For k 1 and n realizing (3.1.3), the set of ϕ contributing to (3.1.3) is unique, as well
as decomposition (3.1.3). In case k “ C and M “ Cppxqq bCtxurx´1 s M0 where M0 is
a Ctxurx´1 s module, one can show
(3.1.4)
irrpM0 q “
ÿ
ϕPx´1{n Crx´1{n s
ord ϕ
rk Rϕ
n
Formula (3.1.4) is algebraic and makes sense for arbitrary k. Thus, we have a welldefined irregularity number for a kppxqq-differential module M .
14
J.-B. TEYSSIER
3.2. Statement of Kedlaya-Mochizuki theorem. — The analytico-formal approach to irregularity introduced in section 1 generalizes to holonomic D-modules in
any dimension, see 4.1. For a normal crossing divisor D, The cohomological approach
of section 2 generalizes to holonomic D-modules into a relation between the De Rham
complex with rapid decay along D and the irregularity sheaf along D. See [Sab00,
1.1.19]. What about the formal approach 3.1 for meromorphic connections in any
dimension? It requires first a generalization of Levelt-Turrittin decomposition. Let
us proceed by analogy with the one dimensional case to see what is the best to be
expected.
dim 1
dim ą 1
kJxK
kJx1 , . . . , xn K
kppxqq :“ kJxKrx´1 s
´1
Rn,m :“ kJx1 , . . . , xn Krx´1
1 , . . . , xm s
kppxqq-differential module
Rn,m -differential module: free Rn,m module of finite
type M endowed with commuting k-linear maps ∇i :
M ÝÑ M , i “ 1, . . . , n satisfying the Leibniz rule
Bf
∇i pf mq “ Bx
m ` f ∇i m for every f P Rn,m and
i
m P M.
Regular module
M admits a basis in which the matrix of xi ∇i has
no poles for every i “ 1, . . . , n.
Module E ϕ for ϕ P kppxqq
For ϕ P Rn,m admissible, that is ϕ “ 0 or ϕ “
u{xa1 1 ¨ ¨ ¨ xamm with u P Rn,0 such that up0q ‰ 0, the
Bf
Bϕ
differential module E ϕ :“ pRn,m , ∇i f “ Bx
` f Bx
q
i
i
Unramified Levelt-Turrittin
decomposition
Admissible decomposition: there exists an isomorphism of Rn,m -differential modules
à
M»
E ϕ b Rϕ
ϕ admissible
where the Rϕ are regular.
Levelt-Turrittin
tion
decomposi-
Ramified admissible decomposition: there exists k 1 {k
finite extension and an integer n P N such that if
´1{n
´1{n
Rn,m pk 1 , nq :“ k 1 Jx1 , . . . , xn Krx1 , . . . , xm s, the
module Rn,m pk 1 , nq bRn,m M admits an admissible
decomposition.
Note that
Ramified admissible decompositions may not exist !!
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
15
To understand where the obstruction comes from, let us consider a R2,1 -differential
module M . We have R2,1 “ kJx2 Kppx1 qq. We denote by s the closed point of
Spec kJx2 K and by η its generic point. The restriction of M at η is
Mη “ kppx2 qqppx1 qq bkJx2 Kppx1 qq M
We are here in a one-dimensional situation with residue field kppx2 qq. In particular,
the Levelt-Turrittin decomposition is available. Ignoring ramification and extension
issues, Mη decomposes as
à
(3.2.1)
Mη »
E ϕ b Rϕ
´1
ϕPx´1
1 kppx2 qqrx1 s
The coefficients of the ϕ lies in kppx2 qq. They may have poles.
On the other hand, to say that M0 has an admissible decomposition is to say that
à
E ψ b Rψ
(3.2.2)
M»
ψ admissible
a
u{x1 ψ
where ψ “
with u P kJx1 , x2 K such that up0q ‰ 0. If such a decomposition
exists, the unicity of the exponential factors appearing in (3.1.2) gives that the set
´1
tϕ P x´1
1 kppx2 qqrx1 s contributing to (3.2.1)u
is equal to the set
tψ P kJx2 Kppx1 qq contributing to (3.2.2)u
So again ignoring extension and ramification issues, if M admits an admissible decomposition, we have the following
Condition 3.2.3. — The coefficients of a ϕ contributing to the Level-Turrittin decomposition of Mη have no poles at 0, and if ϕ ‰ 0, the coefficient of smallest degree
of ϕ does not vanish at 0.
The reciproque cannot hold for the following heuristical reason: even if the right
hand-side of (3.2.1) makes sense as a potential admissible decomposition for M , it may
happen that the isomorphism (3.2.1) cannot be chosen to be defined over kJx2 Kppx1 qq.
This obstruction is hard to control in practice, since the isomorphism (3.2.1) is not
explicit and there are many choices for it. However, as a consequence of a theorem of
André [And07]
Theorem 3.2.4. — The following conditions are equivalentp3q
(1) M and End M admits an admissible decomposition.
(2) Condition (3.2.3) is satisfied for M and End M .
This result is breathtaking: it makes the non explicitness problem of the isomorphism in (3.2.1) disappear. Let us note that theorem 3.2.4 is used in Mochizuki’s
proof [Moc09] of Sabbah’s conjecture for surfaces.
Let us now pass to a global situation. Suppose to simplify that k “ k. Let X be
p3q For
a finer statement valid in the ramified case with general coefficients, see [And07].
16
J.-B. TEYSSIER
a smooth algebraic variety over k and let D be a normal crossing divisor in X. By
point of X, we mean a close point.
Definition 3.2.5. — A flat meromorphic connection on X with poles along D is
the data of a locally free OX p˚Dq-module of finite rank M endowed with a k-linear
operator
∇ : M ÝÑ Ω1X p˚Dq b M
satisfying the Leibniz rule
∇pf mq “ df b m ` f ∇m
for every f P OX p˚Dq and m P M. Furthermore, if for i “ 1, . . . , n we denote by
∇i : M ÝÑ M the k-linear morphism induced by ∇ and the contraction by BxB i , we
require that the ∇i commute with each other.
pX,A » kJx1 , . . . , xn K where x1 ¨ ¨ ¨ xm “ 0 is an equation of D
For a A P D, we have O
in a neighbourhood of A. Restricting M to this formal neighbourhood yields a Rn,m xA . The module M
xA may not have an admissible
differential module denoted by M
decomposition.
Definition 3.2.6. — We say that M has good formal decomposition at A P D if
{
xA and EndpMq
M
A admit a ramified admissible decomposition.
The complements in D of the set of good formal decomposition points is called the
turning point locus of M.
We say that M has good formal decomposition if the turning point locus of M is
empty.
As shown in [Ked11], the turning point locus of M is a close subset of D. If D is
smooth, André proves that it is either empty or of pure codimension 1 in D. Working
at good formal decomposition points is very pleasant since exponential modules and
regular modules are easy to handle. One can often reduce to this situation, thanks to
Kedlaya-Mochizuki theorem:
Theorem 3.2.7. — Let M be a flat meromorphic connection on X with poles along
D. There exists a finite composite π : Y ÝÑ X of blow-up above D such that π ˚ M
has good formal decomposition.
Example 3.2.8. — Take X “ A2k with coordinate px, yq and M “ E x{y . The origin
r ÝÑ X be the blow-up of X at the origin. In the
is a turning point for M. Let π : X
chart U0 where x “ uv et y “ v, the pole locus of π ˚ E x{y is v “ 0, and
pπ ˚ E x{y q|U0 » E puvq{v “ E u
n
So π ˚ E x {y has good formal decomposition along on U0 . In the chart U1 where x “ u
et y “ uv, the pole locus of π ˚ E x{y is uv “ 0, and
pπ ˚ E x{y q|U1 » E u{puvq “ E 1{v
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
17
4. Application to periods of algebraic flat connections
4.1. Irregularity in any dimension. — Let X be a complex manifold and let
i : Z ÝÑ X be an analytic subspace of X. Set OX|Z :“ i´1 OX and let OX|Z
z be
the formalization of OX along Z. Mimicking the 1-dimensional case, we consider the
exact sequence of DX -modules
/ OX|Z
0
/ Oz
X|Z
/ QZ
/0
For an holonomic DX -module M, we deduce a distinguished triangle
/ RHompM, O z q
X|Z
i´1 SolpMq
/ RHompM, QZ q
`1
/
Following [Meb90], we define the irregularity sheaf of M along Z as Irr˚Z M :“
RHompM, QZ qr´1s viewed as a complex on X with support in Z.
Definition 4.1.1. — We say that M is regular if Irr˚Z M » 0 for every analytic
subspace Z of X.
Example 4.1.2. — If M is a meromorphic connection with poles along a divisor D,
and i : D ÝÑ X the inclusion. One can show
Irr˚D M “ i˚ i´1 SolpMq
where SolpMq :“ RHompM, OX q.
The following theorem generalizes theorem 1.5.2 to any dimension
Theorem 4.1.3. — Let M be a meromorphic connection with poles along a divisor
D, and i : D ÝÑ X the inclusion. The following conditions are equivalent
(1) M is regular.
(2) The irregularity sheaf of M along D vanishes on a dense open subset of D.
(3) For every A P D, let px1 , . . . , xn q be local coordinates with D given by x1 ¨ ¨ ¨ xm “ 0
xA is regular in the sense
in a neighbourhood of A. The Rn,m differential module M
xA admits a basis in which for every i “ 1, . . . , n, the matrix of
of 3.2. That is, M
xi ∇ B has no poles.
Bxi
Note that the equivalence between p1q and p2q does not require D to be a normal
crossing. See [Meb04].
4.2. Grothendieck-Deligne comparison theorem. — Let X be a smooth complex algebraic variety of dimension d. Let pE, ∇q be a flat connection on X. The De
Rham complex of E is the complex DR E
0
/E
∇
/ Ω1 b E
X
/ ¨¨¨
/ Ωd b E
X
/0
where E lies in degree 0 and where the C-linear morphism d : ΩkX b E ÝÑ Ωk`1
bE
X
is given by
dpω b mq “ dω b m ` ω ^ ∇m
18
J.-B. TEYSSIER
In particular, there is a comparison morphism for every k ě 0
H k pX, DR Eq ÝÑ H k pX an , DR E an q
(4.2.1)
By Poincaré lemma, DR E an is acyclic in degree ą 0, so
H k pX an , DR E an q » H k pX an , Ker ∇an q
By Cauchy theorem, Ker ∇an is a local system on X an of rank rk E and (4.2.1) reads
H k pX, DR Eq ÝÑ H k pX an , Ker ∇an q
(4.2.2)
The morphism (4.2.2) has no reason to be an isomorphism in general.
Example 4.2.3. — Take X “ A1C and
Ex
2
{2
“ pOA1C , f ÝÑ df ´ xf dxq
The only possibly non zero De Rham cohomology spaces are H 0 and H 1 , and we have
H 0 pA1C , DR E x
H
Since Ker ∇an » Cex
2
{2
1
2
{2
q»0
2
pA1C , DR E x {2 q
» Cdx
,
H 0 pC, Ker ∇an q » Cex
2
{2
H 1 pC, Ker ∇an q » 0
We have however the following positive result, proven by Grothendieck [Gro66]
for the trivial connection OX and in general by Deligne [Del70].
Theorem 4.2.4. — If pE, ∇q is regularp4q , the comparison morphism (4.2.2) is an
isomorphism.
Grothendieck and Deligne’s proofs are global. A local proof of 4.2.4 is given in
[Meb89]. It lies in the existence of a triangle
/ RΓpX an , Irr˚Dan pj˚ Eqan qr1s
/ RΓpX an , Ker ∇an q
RΓpX, DR Eq
If pE ˚ , ∇˚ q denotes the connection dual to pE, ∇q, theorem 4.2.4 can be equivalently
formulated by saying that the pairing
(4.2.5)
H k pX, DR Eq ˆ Hk pX an , Ker ∇˚ an q ÝÑ C
induced by
ż
(4.2.6)
ppω b eq, pγ b ϕqq ÝÑ
ϕpeqω
γ
is perfect if pE, ∇q is regular.
p4q Regularity
in the algebraic context means the following: for a (and actually, any) smooth compactification j : X ÝÑ X of X with D :“ XzX a normal crossing divisor, the meromorphic connection
pj˚ Eqan with poles along D is regular in the sense of 4.1.3.
19
A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
4.3. Rapid decay homology. — Let us raise the following
Question. — How to generalize the perfect pairing (4.3.2) to possibly irregular flat
connections?
In example 4.2.3, the pairing (4.3.2) fails to be perfect because the algebraic H 1 is
non zero whereas the analytic H 1 is 0. This means that there are not enough cycles on
X an to be integrated on. An easy way to remedy this is to look for cycles drawn in a
smooth compactification X of X with D :“ XzX a normal crossing divisor. However,
a flat section ϕ P Ker ∇˚ an may diverge at infinity. So integrating ϕpeq along γ does
not make sense. To solve this issue, Deligne [Del07] proposed to consider only pairs
γ b ϕ for which ϕ has rapid decay along γ. For a precise sheaf theoretic definition
using the language of rapid decay functions on oriented blow-up introduced in section
2, we refer to [Hie09, 5.1].
The idea of Deligne leads to rapid decay homology Hkrd pX, E ˚ , ∇˚ q of pE ˚ , ∇˚ q. This
homology restores the perfectness of the period pairing (4.3.2), due to the following
Theorem 4.3.1. — The canonical pairing
H k pX, DR Eq ˆ Hkrd pX, E ˚ , ∇˚ q ÝÑ C
(4.3.2)
induced by (4.2.6) is perfect.
This theorem was proved by Bloch-Esnault [BE04] in dimension 1 and by Hien
[Hie09] in dimension ą 1. Hien’s proof reduces essentially to the case of exponential
modules using Kedlaya-Mochizuki theorem 3.2.7.
Example 4.3.3. — Let us compute the H1rd of the dual
2
E ´x
2
{2
“ pOA1C , f ÝÑ df ` xf dxq
2
of the connexion E x {2 introduced in example 4.2.3. We have Ker ∇˚ an » Ce´x {2 .
Taking P1C as compactification, we have to look for segments γ drawned on P1,an
C
2
such that e´x {2 has rapid decay at 8 along γ. In the local coordinate τ “ 1{x of 8,
2
2
the function e´x {2 takes the form e´1{2τ . The rapid decay condition is achieved for
5π
π π
cycles approaching 8 along directions in p´ 3π
4 , 4 q or p´ 4 , 4 q.
20
J.-B. TEYSSIER
2
In particular, e´x {2 has rapid decay along the real line r´8, `8s. Thus, r´8, `8sb
´x2 {2
e
is a rapid decay cycle. The corresponding integral is
ż `8
?
2
e´x {2 dx “ 2π
´8
1
2
pA1C , DR E x {2 q
It is non zero. Since H
is one dimensional, theorem 4.3.1 implies that
2
rd
´x2 {2
H1 pX, E
q is the line generated by the class of r´8, `8s b e´x {2 .
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A COURSE ON IRREGULAR SINGULARITIES OF MEROMORPHIC CONNECTIONS
[Sab00]
[Sv00]
21
C. Sabbah, Equations différentielles à points singuliers irréguliers et phénomène
de Stokes en dimension 2, Astérisque, vol. 263, SMF, 2000.
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J.-B. Teyssier, The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Givat
Ram. 9190401 Jerusalem, Israel ‚ E-mail : teyssier@zedat.fu-berlin.de