Preface

This book is based on courses of lectures which I delivered at University of

Tokyo, Nagoya University, Osaka University and Tokyo Institute of Technology

between 1996 and 1998.

The purpose of the lectures was to discuss various properties of the Hilbert

schemes of points on surfaces. This object is originally studied in algebraic ge-

ometry, but as it has been realized recently, it is related to many other branches

of mathematics, such as singularities, symplectic geometry, representation theory,

and even to theoretical physics. The book reflects this feature of Hilbert schemes.

The subjects are analyzed from various points of view. Thus this book tries to tell

the harmony between different fields, rather than focusing attention on a particular

one.

These lectures were intended for graduate students who have basic knowl-

edge on algebraic geometry (say chapter 1 of Hartshorne: “Algebraic Geometry”,

Springer) and homology groups of manifolds. Some chapters require more back-

ground, say spectral sequences, Riemannian geometry, Morse theory, intersection

cohomology (perverse sheaves), etc., but the readers who are not comfortable with

these theories can skip those chapters and proceed to other chapters. Or, those

readers could get some idea about these theories before learning them in other

books.

I have tried to make it possible to read each chapter independently. I believe

that my attempt is almost successful. The interdependence of chapters is outlined

in the next page. The broken arrows mean that we need only the statement of

results in the outgoing chapter, and do not need their proof.

Sections 9.1, 9.4 are based on A. Matsuo’s lectures at the University of Tokyo.

His lectures contained Monster and Macdonald polynomials. I regret omitting these

subjects. I hope to understand these by Hilbert schemes in the future.

The notes were prepared by T. Gocho and N. Nakamura. I would like to thank

them for their efforts. I am also grateful to A. Matsuo and H. Ochiai for their

comments throughout the lectures. A preliminary version of this book has been

circulating since 1996. Thanks are due to all those who read and reviewed it, in par-

ticular to V. Baranovsky, P. Deligne, G. Ellingsrud, A. Fujiki, K. Fukaya, M. Furuta,

V. Ginzburg, I. Grojnowski, K. Hasegawa, N. Hitchin, Y. Ito, A. King, G. Kuroki,

M. Lehn, S. Mukai, I. Nakamura, G. Segal, S. Strømme, K. Yoshioka, and M. Ver-

bitsky. Above all I would like to express my deep gratitute to M. A. de Cataldo for

his useful comments throughout this book.

February, 1999

Hiraku Nakajima

ix

This book is based on courses of lectures which I delivered at University of

Tokyo, Nagoya University, Osaka University and Tokyo Institute of Technology

between 1996 and 1998.

The purpose of the lectures was to discuss various properties of the Hilbert

schemes of points on surfaces. This object is originally studied in algebraic ge-

ometry, but as it has been realized recently, it is related to many other branches

of mathematics, such as singularities, symplectic geometry, representation theory,

and even to theoretical physics. The book reflects this feature of Hilbert schemes.

The subjects are analyzed from various points of view. Thus this book tries to tell

the harmony between different fields, rather than focusing attention on a particular

one.

These lectures were intended for graduate students who have basic knowl-

edge on algebraic geometry (say chapter 1 of Hartshorne: “Algebraic Geometry”,

Springer) and homology groups of manifolds. Some chapters require more back-

ground, say spectral sequences, Riemannian geometry, Morse theory, intersection

cohomology (perverse sheaves), etc., but the readers who are not comfortable with

these theories can skip those chapters and proceed to other chapters. Or, those

readers could get some idea about these theories before learning them in other

books.

I have tried to make it possible to read each chapter independently. I believe

that my attempt is almost successful. The interdependence of chapters is outlined

in the next page. The broken arrows mean that we need only the statement of

results in the outgoing chapter, and do not need their proof.

Sections 9.1, 9.4 are based on A. Matsuo’s lectures at the University of Tokyo.

His lectures contained Monster and Macdonald polynomials. I regret omitting these

subjects. I hope to understand these by Hilbert schemes in the future.

The notes were prepared by T. Gocho and N. Nakamura. I would like to thank

them for their efforts. I am also grateful to A. Matsuo and H. Ochiai for their

comments throughout the lectures. A preliminary version of this book has been

circulating since 1996. Thanks are due to all those who read and reviewed it, in par-

ticular to V. Baranovsky, P. Deligne, G. Ellingsrud, A. Fujiki, K. Fukaya, M. Furuta,

V. Ginzburg, I. Grojnowski, K. Hasegawa, N. Hitchin, Y. Ito, A. King, G. Kuroki,

M. Lehn, S. Mukai, I. Nakamura, G. Segal, S. Strømme, K. Yoshioka, and M. Ver-

bitsky. Above all I would like to express my deep gratitute to M. A. de Cataldo for

his useful comments throughout this book.

February, 1999

Hiraku Nakajima

ix