Table Of ContentAnalytic D-Modules and Applications
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
V.olume 247
Analytic I)-Modules
and Applications
by
Jan-Erik Bjork
Department of Mathematics,
Stockholm University,
Stockholm, Sweden
....
"
SPRINGER-SCIENCE+BUSINESS MEDIA, B. V.
Library of Congress Cataloging-in-Publication Data
BJărk, Jan-Erlk.
Analyt1c 0-modules and appl icat1ons·/ by Jan-Erik Bjork.
p. cm. -- <MathematlCS and its applicatlons; v. 247)
Inc 1u des 1n dex.
ISBN 978-90-481-4238-5 ISBN 978-94-017-0717-6 (eBook)
DOI 10.1007/978-94-017-0717-6
1. Oifferentiable dynamical systems. • 2. 0-modules. 3. Complex
manifolds. 4. Oifferent ial topology. I. T1tle. II. Series:
Mathematlcs and its applications <Kiuwer Academ1c Publlshers>; v.
247.
OA614.8.B58 1993
514' .74--dc20 92-44826
ISBN 978-90-481-4238-5
Typeset by AMS-TEX
Printed on acid-free paper
Ali Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner_
SERIES EDITOR'S PREFACE
'Et moi, ... , si favait su comment eo reveoir. je One service mathematics has rendered the
n'y serais point all6.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next to
the dusty canister labelled 'discarded nonsense'.
Tbe series is divergent; therefore we may be EricT. Bell
ajle to do something with it
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari
tLes abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci
ences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser
vice topology has rendered mathematical physics .. .'; 'One service logic has rendered computer science
. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements
etre
obtainable this way form part of the raison d' of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have
appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and textbooks
on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and
related fields does not grow only by putting forth new branches. It also happens, quite often in
fact, that branches which were thought to be completely disparate are suddenly seen to be
related. Further, the kind and level of sophistication of mathematics applied in various sci
ences has changed drastically in recent years: measure theory is used (non-trivially) in
regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky
lemma, coding theory and the structure of water meet one another in packing and covering
theory; quantum fields, crystal defects and mathematical programming profit from homotopy
theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use
Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experi
mental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large
scale order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics. "
By and large, all this still applies today. It is still true that at first sight mathematics seems rather frag
mented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so
He books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make
mch books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of interaction
iIeas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots,
quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to
the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination
of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to
~tatistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections
of VLSI chips (via difference sets). There seems to be no part of (S(H;alled pure) mathematics that is not
in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of
much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combina
torics, algebra, probability, and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra
vi
mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest
and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal
inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics
were linear we would have no fun with transistors and computers; we would have no TV; in fact you
would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and
anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical
engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the
shortest path between 'real' results. Similarly, the first two topics named have already provided a number
of 'wormhole' paths. There is no telling where all this is leading -fortunately.
Thus the series still aims at books dealing with:
a central concept which plays an important role in several different mathematical and/or scientific
specialization areas;
new applications of the results and ideas from one area of scientific endeavour into another;
influences which the results, problems and concepts of one field of enquiry have, and have had, on the
development of another.
Analytic D-module theory studies holomorphic differential systems of equations on complex manifolds. It
is a quite new area in mathematics that has drawn much attention in recent years and that has, correspond
ingly, enjoyed vigorous development. Together with its tightly interlined algebraic counterpart it has
brought new insights and methods in many areas such as representation theory, hypergeometric functions,
intersection cohomology, and residues for several complex variables, and it has been instrumental in the
solution of some outstanding conjectures.
The present book is the first monograph on analytic D-modules and it offers a complete and systematic
treatment of the foundations together with a thorough discussion of such modem topics as the Riemann
Hilbert correspondence, Bemstein-Sata polynomials and a large variety of results concerning micro
differential analysis.
The book has been long in the making. Preliminary versions in various stages of completeness have cir
culated for many years among the specialists, and many improvements resulted from their reactions and
comments. It is a real pleasure to welcome the finished product, impressive as it is.
The shortest path between two bUlbs in the real Never lend books, for no one ever returns them;
domain passes through the complex domain. the only books I have in my library are books
1. Hadamard that other folk have lent me.
Anatole France
La physique ne nons donne pas seulement
I'occasion de resoudre des probl~es ... elle The function of an expert is not to be more right
nons fait pressentir Ia solution. than other people, but to be wrong for more
H. Poincar~ sophisticated reasons.
David Butler
Amsterdam, October 1992 Michiel Hazewinkel
Table of contents
Series editor's preface ...................................................... v
Preface ..................................................................... xiii
Introduction ................................................................ 1
Chapter I. The sheaf Vx and its modules
Summary .................................................................. 7
1.1. The ring Vn ............................................................ 7
1.2 The sheaf Vx .......................................................... 17
1.3 The category of Vx-modules ........................................... 21
1.4 Coherent V-modules ................................................... 24
1.5 Spencer's resolution .................................................... 29
1.6 Characteristic varieties ................................................. 32
1.7. V-modules and homological algebra ................................... 38
1.8. Miscellaneous results .................................................. 43
1.9. Twisted rings of differential operators ................................. 48
Notes ..................................................................... 52
Chapter II. Operations on V-modules
Summary ................................................................. 53
ILl. The derived category of V-modules ................................... 53
II.2. V-modules and derived functors ...................................... 57
II.3. Direct and inverse images of V-modules .............................. 61
II.4. Further results on images of V-modules ............................... 67
11.5. Temperate localisations ............................................... 73
11.6. Images of V-modules under closed imbeddings ........................ 81
II.7. Non-characteristic inverse images ..................................... 90
11.8. Images of coherent V-modules ........................................ 97
II.9. Direct images and Edge maps ....................................... 104
viii
n.10. Fuchsian filtrations ................................................. 110
II.11. Duality on V-modules .............................................. 119
Notes .................................................................... 128
Chapter III. Holonomic V-modules
Summary .................. .............................................. 129
IIL1. Basic facts about holonomic modules ............................... 130
IIL2. Holonomic modules and b-functions ................................. 133
nL3. Holonomic solution complexes ...................................... 140
I1I.4. The sheaf V'X ...................................................... 148
III.5. Further results on holonomic modules .............................. 151
III.6. Connections and holonomic modules ................................ 156
III.7. V-filtrations on holonomic modules ................................. 160
Notes .............. ...................................................... 166
Chapter IV. Deligne modules
Summary ................................................................ 167
IV.1. Deligne sheaves ......... ........................................... 167
IV.2. The normal crossing case ........................................... 173
IV.3. Fundamental solutions .............................................. 177
IV.4. A study of the L2-lattice ........................................... 181
IV.5. Deligne modules .................................................... 184
IV.6. Further results on Deligne modules ................................. 187
Notes ....................... ............ ................................. 192
Chapter V. Regular holonomic V-modules
Summary ................................................................ 193
V.l. V-modules in dimension one ......................................... 194
V.2. Regular holonomic V1-modules ...................................... 198
V.3. Regular holonomic Vx-modules ................. .................... 207
V.4. Preservation of regular holonomicity ................................. 212
V.5. The Riemann-Hilbert correspondence .... ........... ................. 215
V.6. Further results on regular holonomic modules ........................ 224
V.7. The L2-lattice .............. ...... ................................... 235
V.8. Algebraic V- modules ..... . . ....................... ............... 244
Kotes .................. . . ....................................... 249
ix
Chapter VI. b-functions.
Summary ................................................................ 251
VI.l. Regular holonomic 'V-modules and b-functions ...................... 252
VI.2. On characteristic varieties .......................................... 259
VI.3. The Bernstein-Sato polynomial ..................................... 262
VI.4. Vanishing cycles .................................................... 270
VI.5. The poles of s -> J Ifl2s ..........•...............•..•........••.... 276
Notes .................................................................... 280
Chapter VII. Distributions and regular holonomic systems
Summary ................................................................ 281
VII. 1. 'V-modules on real manifolds ....................................... 281
VII.2. Extendible distributions ........................................... 286
VII.3. The conjugation functor ........................................... 289
VII.4. Regular holonomic distributions ................................... 293
VII.5. Distributions on real manifolds .................................... 297
VII.6. Meromorphic continuations of distributions ........................ 300
VII.7. Examples of regular holonomic distributions ....................... 304
VII.8. The temperate Hom-functor ....................................... 312
VII.9. The Ill-functor ..................................................... 320
VILlO. The decomposition of direct images ............................... 328
Notes .................................................................... 332
Chapter VIII. Microdifferential operators
Summary ................................................................ 333
VIII.l. The sheaf of microdifferential operators ........................... 334
VIII.2. Coherent [-modules .............................................. 339
VIII.3. Operations on [-modules ......................................... 346
VIII.4. [-modules with regular singularities ............................... 350
VIII. 5. Monodromy operators ............................................ 355
VIII. 6. Holonomic [-modules ............................................. 361
VIII. 7. Regular holonomic [-modules ..................................... 373
VIII.8. Regular holonomic [-modules and b-functions ..................... 378
