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
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