Donaldson Type Invariants for Algebraic Surfaces Transition of Modul
We are defining and studying an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface.We are interested in relations among the invariants, which are natural g
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Lecture Notes in Mathematics
1
Takuro Mochizuki
Donaldson Type Invariants for Algebraic Surfaces
1972 Transition of Moduli Stacks
123
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1972
Takuro Mochizuki
Donaldson Type Invariants for Algebraic Surfaces Transition of Moduli Stacks
ABC
Takuro Mochizuki Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan [email protected]
ISBN: 978-3-540-93912-2 e-ISBN: 978-3-540-93913-9 DOI: 10.1007/978-3-540-93913-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008943979 Mathematics Subject Classification (2000): 14D20, 14J60, 14J80 c 2009 Springer-Verlag Berlin Heidelberg ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper 987654321 springer.com
To my friends Yoshinobu Akahori and Shu Kawaguchi for memory of our informal student seminar
Preface
In this monograph, we define and investigate an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface. We may expect the existence of interesting “universal relations among invariants”, which would be a natural generalization of the “wall-crossing formula” and the “Witten conjecture” for classical Donaldson invariants. Our goal is to obtain a weaker version of such relations, in other brief words, to describe a relation as the sum of integrals over the products of moduli spaces of objects with lower ranks. Fortunately, according to a recent excellent work of L. G¨ottsche, H. Nakajima and K. Yoshioka, [53], a wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case. We hope that our work in this monograph would, at least tentatively, provides a part of foundation for the further study on such universal relations. In the rest of this preface, we would like to explain our motivation and some of important ingredients of this study. See Introduction for our actual problems and results.
Donaldson Invariants Let us briefly recall Donalds
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