Contemporary trends in algebraic geometry and algebraic topology

270 Pages·2002·9.99 MB·English

by  Shiing-Shen Chern

CONTEMPORARY TRENDS IN
ALGEBRAIC GEOMETRY
AND
ALGEBRAIC TOPOLOGY
NANKAI TRACTS IN MATHEMATICS
Series Editors: Shiing-shen Chern, Yiming Long, and Weiping Zhang
Nankai institute of Mathematics
Published
Vol. 1 Scissors Congruences, Group Homology and
Characteristic Classes
by J. L Dupont
Vol. 2 The Index Theorem and the Heat Equation Method
byY.LYu
Vol. 3 Least Action Principle of Crystal Formation of Dense Packing Type
and Kepler's Conjecture
by W. Y. Hsiang
Vol. 4 Lectures on Chern-Weil Theory and Witten Deformations
by W. P. Zhang
Nankai Tracts in Mathematics - Vol. 5
CONTEMPORARY TRENDS IN
ALGEBRAIC GEOMETRY
AND
ALGEBRAIC TOPOLOGY
Editors
Shiing-Shen Chern
Lei Fu
Nankai Institute of Mathematics
P. R. China
Richard Hain
Duke University, USA
fe World Scientific
III New Jersey • London • Singapore • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Fairer Road, Singapore 912805
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
CONTEMPORARY TRENDS IN ALGEBRAIC GEOMETRY AND
ALGEBRAIC TOPOLOGY
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 981-02-4954-3
Printed in Singapore by Mainland Press
PREFACE
Wei-Liang Chow and Kuo-Tsai Chen are two of the most distinguished
Chinese mathematicians of the 20th century. Chen's collected papers have
been published in 2001 by Birkhauser, while Chow's collected papers have
been edited by V. Shokurov and myself and are awaiting publication by
World Scientific. A memorial conference organized by the Nankai Institute
of Mathematics, took place in Tianjin in Oct. 9-Oct. 13, 2000. This volume
contains only a small sample of the papers presented at the conference. We
are glad that it covers broad areas.
S. S. Chern
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CONTENTS
Preface v
Mathematics in the 20th Century
Sir M. Atiyah 1
The $4 of Minimal Gorenstein 3-Folds of General Type
M. Chen 23
Morphisms of Curves and the Fundamental Group
M. Cushman 39
Iterated Integrals and Algebraic Cycles: Examples
and Prospects
R. Hain 55
Chen's Iterated Integrals and Algebraic Cycles
B. Harris 119
On Algebraic Fiber Spaces
Y. Kawamata 135
Local Holomorphic Isometric Embeddings Arising From
Correspondences in The Rank-1 Case
N. Mok 155
Multiple Polylogarithms: Analytic Continuation, Monodromy,
and Variations of Mixed Hodge Structures
J. Zhao 167
Deformation Types of Real and Complex Manifolds
F. Catanese 195
vii
viii Contents
Wei-Liang Chow, 1911-1995
S. S. Chern 239
Comments on Chow's Work
S. Lang 243
The Life and Work of Kuo-Tsai Chen
R. Hain and Ph. Tondeur 251
MATHEMATICS IN THE 20TH CENTURY
Sir Michael Atiyah
Thank you for inviting me here to take part in this program. Of course,
if you talk about the end of one century and the beginning of the next
you have two choices, both of them equally difficult. One is to survey the
mathematics over the past hundred years; the other is to predict the math
ematics of the next hundred years. I have chosen the more difficult task.
Everybody can predict and we will not be around to find out whether we
were wrong. But giving an impression of the past is something that every
body can disagree with.
All I can do is give you a personal view. It is impossible to cover every
thing, and in particular I will leave out significant parts of the story, partly
because I am not an expert, partly because it is covered elsewhere. I will
say nothing, for example, about the great events in the area between logic
and computing associated with the names of people like Hilbert, Godel,
and Turing. Nor will I say much about the applications of mathematics, ex
cept in fundamental physics, because they are so numerous and they need
such special treatment. Each would require a lecture to itself. But perhaps
you will hear more about those in some of the other lectures taking place
during this meeting. Moreover there is no point in trying to give just a list
of theorems or even a list of famous mathematicians over the last hundred
years. That would be rather a dull exercise. So instead I am going to try
and pick out some themes that I think run across the board in many ways
and underline what has happened.
Let me first make a general remark. Centuries are crude numbers. We
do not really believe that, after a hundred years something suddenly stops
and starts again. So when I describe the mathematics of the 20th century,
1
2 Sir M. Atiyah
I am going to be rather cavalier about dates. If something started in the
1890's and moved into the 1900's — I shall ignore such detail. I will behave
like an astronomer and work in rather approximate numbers. In fact many
things started in the 19th century and only came to fruition in the 20th
century.
One of the difficulties of this exercise is that it is very hard to put oneself
back in the position of what it was like in 1900 to be a mathematician,
because so much of the mathematics of the last century has been absorbed
by our culture, by us. It is very hard to imagine a time when people did not
think in those terms. In fact, if you make a really important discovery in
mathematics you will then get omitted altogether! You simply get absorbed
into the background. So going back, you have got to try to imagine what
it was like in a different era when people did not think the same way.
1. Local to Global
I am going to start off by listing some themes and talking around them.
My first theme is broadly under what you might call the passage from
the local to the global. In the classical period, people on the whole would
have studied things on a small scale, in local co-ordinates and so on. In
this century, the emphasis has shifted to try and understand the global,
large-scale behaviour. And because global behaviour is more difficult to
understand, much of it is done qualitatively, and topological ideas become
very important. It was Poincare who both made the pioneering steps in
topology and who forecast that topology would be an important ingredient
in 20th century mathematics. Incidentally, Hilbert, who made his famous
list of problems, did not. Topology hardly figured in his list of problems.
But for Poincare it was quite clear it would be an important factor.
Let me try to list a few of the areas and you can see what I have in
mind. Consider, for example, complex analysis ("function theory", as it was
called), which was at the centre of mathematics in the 19th century, the
work of great figures like Weierstrass. For them, a function was a function
of one complex variable and for Weierstrass a function was a power series.
Something you could lay your hands on, write down, and describe explicitly;
or a formula. Functions were formulas: they were explicit things. But then
the work of Abel, Riemann, and subsequent people, moves us away, so that
functions became denned not just by explicit formulae but more by their
global properties: by where their singularities were, where their domains of