Table Of ContentClaude Sabbah
Isomonodromic
Deformations and
Frobenius Manifolds
An Introduction
With10Figures
ProfessorClaudeSabbah
CNRS,CentredeMathématiquesLaurentSchwartz
ÉcolePolytechnique
F-91128PalaiseauCedex
France
Mathematics Subject Classification(2000): 14F05, 32A10, 32G20, 32G34, 32S40, 34M25, 34M35,
34M50,53D45,33E30,34E05
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LibraryofCongressControlNumber:2007939825
SpringerISBN-13:978-1-84800-053-7 e-ISBN-13:978-1-84800-054-4
EDPSciencesISBN978-2-7598-0047-6
TranslationfromtheFrenchlanguageedition:
DéformationsisomonodromiquesetvariétésdeFrobeniusbyClaudeSabbah
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Contents
Preface ........................................................ IX
Terminology and notation .....................................XIII
0 The language of fibre bundles............................ 1
1 Holomorphic functions on an open set of Cn .............. 1
2 Complex analytic manifolds............................. 2
3 Holomorphic vector bundle ............................. 5
4 Locally free sheaves of O -modules...................... 7
M
5 Nonabelian cohomology ................................ 10
6 Cˇech cohomology...................................... 14
7 Line bundles.......................................... 16
8 Meromorphic bundles, lattices........................... 17
9 Examples of holomorphic and meromorphic bundles ....... 19
10 Affine varieties, analytization, algebraic differential forms... 25
11 Holomorphic connections on a vector bundle .............. 27
12 Holomorphic integrable connections and Higgs fields ....... 32
13 Geometry of the tangent bundle......................... 37
14 Meromorphic connections............................... 44
15 Locally constant sheaves ............................... 48
16 Integrable deformations and isomonodromic deformations... 53
17 Appendix: the language of categories..................... 57
I Holomorphic vector bundles on the Riemann sphere..... 61
1 Cohomology of C, C∗ and P1............................ 61
2 Line bundles on P1 .................................... 63
3 A finiteness theorem and some consequences .............. 68
4 Structure of vector bundles on P1........................ 69
5 Families of vector bundles on P1......................... 76
VIII Contents
II The Riemann-Hilbert correspondence on a Riemann
surface .................................................. 83
1 Statement of the problems.............................. 83
2 Local study of regular singularities....................... 85
3 Applications .......................................... 97
4 Complements .........................................100
5 Irregular singularities: local study .......................102
6 The Riemann-Hilbert correspondence in the irregular case ..109
III Lattices..................................................121
1 Lattices of (k,∇)-vector spaces with regular singularity ....122
2 Lattices of (k,∇)-vector spaces with an irregular singularity 133
IV The Riemann-Hilbert problem and Birkhoff’s problem ..145
1 The Riemann-Hilbert problem ..........................146
2 Meromorphic bundles with irreducible connection..........152
3 Application to the Riemann-Hilbert problem..............155
4 Complements on irreducibility ..........................158
5 Birkhoff’s problem.....................................159
V Fourier-Laplace duality ..................................167
1 Modules over the Weyl algebra..........................168
2 Fourier transform......................................176
3 Fourier transform and microlocalization ..................183
VI Integrable deformations of bundles with connection on
the Riemann sphere .....................................191
1 The Riemann-Hilbert problem in a family ................192
2 Birkhoff’s problem in a family...........................200
3 Universal integrable deformation for Birkhoff’s problem ....208
VII Saito structures and Frobenius structures on a complex
analytic manifold ........................................223
1 Saito structure on a manifold ...........................224
2 Frobenius structure on a manifold .......................233
3 Infinitesimal period mapping............................237
4 Examples.............................................242
5 Frobenius-Saito structure associated to a singularity .......254
References.....................................................263
Index of Notation .............................................273
Index..........................................................275
Preface
Despite a somewhat esoteric title, this book deals with a classic subject,
namely that of linear differential equations in the complex domain. The pro-
totypes ofsuchequations arethelinear homogeneous equations (with respect
to the complex variable t and the unknown function u(t))
du α du 1
= u(t) (α∈C), = u(t).
dt t dt t2
The solutions of the first equation are the “multivalued” functions t (cid:4)→ ctα
(α ∈ C, c ∈ C) and those of the second equation are the functions t (cid:4)→
cexp(−1/t). On the other hand, the “multivalued” function log is a solution
of the inhomogeneous linear equation
du 1
= ,
dt t
or, if one wants to continue with homogeneous equations as we do in this
book, of the equation of order 2:
d2u du
t + =0.
dt2 dt
Thus, the solutions of a differential equation with respect to the variable t,
having polynomial or rational fractions as coefficients, are, in general, tran-
scendental functions. Needless to say, other families of equations, such as the
hypergeometric equations or the Bessel equations, are also celebrated.
Once these facts are understood, the question of knowing if it is neces-
sary to explicitly solve the equations to obtain interesting properties of their
solutions can be stated. In other words, one wants to know which properties
of the solutions only depend in an algebraic way on (hence are in principle
computable from) the coefficients of the equation, and which are those which
need transcendental manipulations.
Following this reasoning to its end leads one to develop the theory of
differential equations in the complex domain with the tools of algebraic or
X Preface
complex analytic geometry (i.e., the theory of complex algebraic equations).
Oneisthusledtotreatsystems oflineardifferentialequations, whichdepend
on many variables. The algebraic geometry also invites us to consider the
global properties of such systems, that is, to consider systems defined on
algebraic or complex analytic manifolds.
Thedifferentialequationsthatwewillconsiderinthisbookwillbenamed
integrable connections on a vector bundle.OurDrosophila melanogaster (fruit
fly) will be the complex projective line, more commonly called the “Riemann
sphere” and denoted by P1(C) or P1, and will be the subject of some experi-
ments concerning connections: analysis of singularities and deformations.
Thetheoryofisomonodromicdeformationsservesasamachinetoproduce
systems of nonlinear (partial) differential equations in the complex domain,
starting from an equation or from a system of linear differential equations
of one complex variable. It provides at the same time a procedure (far from
being explicit in general) to solve them, as well as remarkable properties of
the solutions of these systems (among others, the property usually called the
“Painlev´e property”). If, at the beginning, the main object of interest was
the deformation of linear differential equations of a complex variable with
polynomial coefficients, it has now been realized that the deformation theory
oflinearsystemsofmanydifferentialequationscanshedlightonthisquestion,
thankstotheuseoftoolscomingfromalgebraicordifferentialgeometry,such
as vector bundles, connections, and the like.
For a long time (and such remains the case), this method serve specialists
indynamicalsystemsandphysicistswhoanalyzethenonlinearequationspro-
ducedbyintegrabledynamicalsystems;toexhibittheseequationsasisomon-
odromyequationsis,inaway,alinearizationoftheinitialproblem.Fromthis
pointofview,thePainlev´eequationshaveplayedaprototypicrole,beginning
with the article by R.Fuchs [Fuc07] (followed by those of R.Garnier) who
showed how the sixth one can be written as an isomonodromy equation, thus
avoiding the strict framework of the search for new transcendental functions.
A nice application of this theory is the introduction of the notion of a
Frobenius structure on a manifold. If this notion had clearly emerged from
the articles of Kyoji Saito on the unfoldings of singularities of holomorphic
functions,ithasbeenextensivelydevelopedbyBorisDubrovin,whousedmo-
tivations coming from physics, opening new perspectives on, and establishing
a new link between, mathematical domains which are apparently not related
(singularities, quantum cohomology, mirror symmetry).
Myaimtokeepthistextamoderatelengthandlevelofcomplexity,aswell
asmylackofknowledgeonmorerecentadvances,ledmetolimitthenumber
of themes, and to refer to the foundational article of B.Dubrovin [Dub96], or
to the book of Y.Manin [Man99a], for further investigation of other topics.
Chapter 0, although slightly long, can be skipped by any reader having
a basic knowledge of complex algebraic geometry; it can serve as a reference
fornotation.Itpresentstheconceptsreferredtointhebookconcerningsheaf
Preface XI
theory,vectorbundles,holomorphicandmeromorphicconnections,andlocally
constant sheaves. The results are classic and exist, although scattered, in the
literature.
The same considerations apply to Chapter I, although it can be more
difficulttofindareferencefortherigiditytheoremoftrivialvectorbundlesin
elementary books on algebraic geometry. We restrict ourselves to bundles on
theRiemannsphere,minimizingtheknowledgeneededofalgebraicgeometry.
In this chapter, we do not give the proof of the finiteness theorem for the
cohomology of a vector bundle on a compact Riemann surface, for which
good references exist; we only need it for the Riemann sphere.
With Chapters II and III begins the study of linear systems of differential
equations of a complex variable and their deformations. The type of singular
points is analyzed there. Here also we do not give the proof of two theorems
of analysis, inasmuch as the techniques needed, although very accessible, go
too much beyond the scope of this book.
One of the fundamental objects attached to a differential equation or,
moregenerally,toanintegrableconnectiononavectorbundle,isthegroup of
monodromytransformations initsnaturalrepresentation,reflectingthe“mul-
tivaluedness” of the solutions of this equation or connection. The Riemann-
Hilbert correspondence—at least when the singularities of the equation are
regular—expresses that this group contains the complete information on the
differential equation. Thus, one of the classic problems of the theory consists
of, given a differential equation, computing its monodromy group. Letus also
mention another object, the differential Galois group—which we will not use
in this book—that has the advantage of being defined algebraically from the
equation.
We will not deal with this problem in this book, and one will not find
explicit computations of such groups. As indicated above, we rather try to
express the properties of the solutions of the equation in terms of algebraic
objects,herethe(meromorphic)vectorbundlewithconnection.Inthismero-
morphic bundle exist lattices (i.e., holomorphic bundles), which correspond
to the various equivalent ways to write the differential system.
Tofindthesimplestwaytopresentadifferentialsystemuptomeromorphic
equivalence is the subject of the Riemann-Hilbert problem (in the case of
regular singularities) or of Birkhoff’s problem. In all cases, it is a matter of
writingthesystemasaconnectiononthetrivialbundle.ChapterIVexpounds
onsometechniquesusedintheresolutionoftheRiemann-HilbertorBirkhoff’s
problem. One will find in the works of A.Bolibrukh [AB94] and [Bol95] many
more results.
Chapter V introduces the Fourier transform (which should possibly
more accurately be called the Laplace transform) for systems of differential
equations of one variable. It helps one in understanding the link between
Schlesinger equations and the deformation equations for Birkhoff’s prob-
lem, analyzed in Chapter VI. In the latter, the notion of isomonodromic
deformation is explained in detail.
XII Preface
ChapterVIIgivesanaxiomaticpresentationofthenotionofaSaitostruc-
ture (as introduced by K.Saito) as well as that of a Frobenius structure (as
introduced by B.Dubrovin, with its terminology). We show the equivalence
between these notions, using the term “Frobenius-Saito structure”. Many ex-
amples are given in order to exhibit various aspects of these structures. This
chaptercanserveasanintroductiontothetheoryofK.SaitoontheFrobenius-
Saito structure associated with unfoldings of holomorphic functions with iso-
latedsingularities.Theproofsofmanyresultsofthistheoryrequiretechniques
ofalgebraicgeometryindimension(cid:2)1,techniqueswhichgobeyondthescope
ofthisbookandwouldneedanotherbook(HodgetheoryfortheGauss-Manin
system).
This text, a much expanded version of my article [Sab98] on the same
topic, stemmed from a series of graduate lectures that I gave at the univer-
sities of ParisVI, BordeauxI and Strasbourg, and during a summer school
on Frobenius manifolds at the CIRM (Luminy). Mich`ele Audin, Alexandru
Dimca, Claudine Mitschi and Pierre Schapira gave me the opportunity to
lecture on various parts of this text.
Many ideas, as well as their presentation, come directly from the articles
of Bernard Malgrange, as well as from numerous conversations that we had.
Many aspects of Frobenius manifolds would have remained obscure to me
without the multiple discussions with Mich`ele Audin. I also had the pleasure
oflongdiscussionswithAndreyBolibrukh,whoexplainedtomehiswork,par-
ticularlyconcerningtheRiemann-Hilbertproblem.JosephLePotieranswered
my electronic questions on bundles with good grace.
Various people helped me to improve the text, or pointed out a few mis-
takes:GillesBailly-Maitre,AlexandruDimca,AntoineDouai,ClausHertling,
Adelino Paiva, Mathias Schulze and the anonymous referees.
I thank all of them.
Theoriginal(French)versionofthisbookhasbeenwrittenwithinINTAS
program no.97-1644.
TheEnglishtranslationdiffersfromtheoriginalFrenchversiononlyinthe
correction of various mistakes or inaccuracies, a list of which can be found on
the author’s web page math.polytechnique.fr/~sabbah.
