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Hyperplane Arrangements an Introduction Universitext Universitext Universitext Alexandru Dimca Hyperplane Arrangements An Introduction Universitext Universitext Series editors Sheldon Axler San Francisco State University, San Francisco, CA, USA Carles Casacuberta Universitat de Barcelona, Barcelona, Spain Angus MacIntyre Queen Mary University of London, London, UK Kenneth Ribet University of California, Berkeley, CA, USA Claude Sabbah École polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France Endre Süli University of Oxford, Oxford, UK Wojbor A. Woyczyński Case Western Reserve University, Cleveland, OH, USA Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. More information about this series at http://www.springer.com/series/223 Alexandru Dimca Hyperplane Arrangements An Introduction 123 Alexandru Dimca Université Côte d’Azur Nice France ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-319-56220-9 ISBN 978-3-319-56221-6 (eBook) DOI 10.1007/978-3-319-56221-6 Library of Congress Control Number: 2017935563 Mathematics Subject Classification (2010): 32S22, 32S55, 32S35, 14F35, 14F40, 14F45, 52C35 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Hyperplane arrangement theory is a very active area of research, combining ideas from combinatorics, algebraic topology and algebraic geometry in a blend that is both tasty and useful. The classical textbook was written by Orlik and Terao [180] about 25 years ago, when the subject was already vast and mature. A continuation by the same authors was produced 10 years later, to discuss important progress involving local systems and twisted cohomology, see [183]. On the combinatorial side, the book by Stanley [213] is an excellent introduc- tion, see also the monograph [29] by Björner, Las Vergnas, Sturmfels, White, Ziegler, as well as Cartier’s report [40]. The recent and detailed monograph by De Concini and Procesi [58] adds new important directions to the subject, by looking at polytopes and polyhedra, matroids and root systems, splines and num- bers of integral points in polytopes. The results continued to accumulate over recent years, and a number of leading experts in the area joined efforts to produce a rather complete survey of the current situation, but their book is still a project, see [50] for a preliminary version. Our aim in writing these notes was more modest: We intended to write an introduction to hyperplane arrangement theory which is both accessible and motivating. On the accessibility side, we have recalled in the first few chapters many of the basic results from the book by Orlik and Terao [180], choosing to avoid the proofs that are long and technical. As for the motivating side, our choice of topics in the latter chapters was inspired by the current frontiers of research and includes both new results and open problems. As a result, some important topics are not treated at all, e.g., the Lie algebras attached to arrangements, which have a whole chapter devoted to them in [50]. Other subjects, such as free arrangements, are treated from a very personal viewpoint. One of our main concerns was to keep the book at a reasonable size, so that the resulting text is more an invitation to explore a beautiful area of mathematics and/or to embark on a related research project, rather than a reference monograph. The exercises at the end of each chapter are a good test of the understanding of the material and should make the book (or parts of it) readily usable in a graduate course. v vi Preface We describe the contents of this book in more detail. Chapter 1 is a brief introduction, in which we point out the interplay between combinatorics, topology, geometry and arithmetic in the realm of line arrangements in the plane, where the reader’s intuition can be strongly supported by drawings. Each of the themes introduced here is developed fully in a later chapter. We include a discussion of the Sylvester–Gallai property for real line arrangements, both the classical projective version in Theorem 1.5 and an affine version in Theorem 1.4. The proof of both results is inspired by Hirzebruch’s approach in [135]. In Chap. 2, we collect the basic definitions and results involving the intersection lattice LðAÞ of a hyperplane arrangement A; we explain the key induction tech- nique based on triples of hyperplane arrangements (A,A0,A00) and apply it to deduce the main properties of the characteristic polynomial v(A; t) and of the Poincaré polynomial p(A; t). The characteristic polynomial enters into Zaslavsky’s Theorem 2.8, expressing the number of regions (resp. bounded regions) of the complement of a real arrangement A in terms of v(A, Æ1). These numbers are also related to the number of singularities of the polynomial Q on the complement MðAÞ, for any essential affine arrangement A : QðxÞ¼0, see Theorem 2.9.We also introduce the supersolvable arrangements and state the factorization property of their Poincaré polynomials in Theorem 2.4. In the last section, we define the graphic arrangements and state the fact that the chromatic polynomial of a simple graph C coincides with the characteristic polynomial of the associated hyperplane arrangement AC, see Theorem 2.10. Finally, we discuss the reflection arrangements and introduce the main arrangements in this class, namely the monomial arrange- ments Aðr; r; nÞ in Example 2.24 and the full monomial arrangements Aðr; 1; nÞ in Example 2.23. The purely combinatorial definition of the Orlik–Solomon algebra AÃðAÞ of a hyperplane arrangement A is given in Chap. 3, and the fundamental result stating that this algebra is isomorphic to the cohomology algebra of the complex hyper- plane arrangement complement MðAÞ is proved in Theorem 3.5. To do this, we assume a technical result on the behavior of the Orlik–Solomon algebras with respect to triples, see Theorem 3.1. In this chapter, we also mention a tensor product decomposition of the Orlik–Solomon algebra of a supersolvable arrangement, see Theorem 3.3, as well as an alternative definition of the Orlik–Solomon algebra of a projective hyperplane arrangement, see Theorem 3.4. In Chap. 4, we discuss the minimality of the complement MðAÞ and its relation to the degree of the gradient mapping of the defining equation for A, see Theorem 4.4. In Remark 4.2, we collect some basic results on the topology of the n union of the hyperplanes in A, namely on the hypersurface NðAÞ¼fx 2 C : QðxÞ¼0g. Then, we mention two beautiful results of June Huh, the first on the log-concavity of the coefficients of the Poincaré polynomial pðA; tÞ, see Theorem 4.6, and the second on the relation between the degree of the gradient mapping of a projective hypersurface V and the multiplicities of its singularities, see Theorem 4.7. When V is a line arrangement, we give an elementary new proof of a more precise version of the latter result in Theorem 4.8. In this chapter, we also discuss the fundamental group of the complement MðAÞ and the arrangements Preface vii whose complements are Kðp; 1Þ-spaces. We state Deligne’s result which says that real simplicial hyperplane arrangements give rise to such Kðp; 1Þ-spaces, see Theorem 4.12, as well as Bessis’ result which says that the same holds for the complex reflection arrangements, see Theorem 4.15. We introduce the fiber type arrangements, show their relation to the Kðp; 1Þ-spaces in Theorem 4.15, and then state the surprising fact that a central arrangement is supersolvable if and only if it is fiber type, see Theorem 5.3. This is another deep connection between combinatorics and topology in this subject. Some very interesting groups occur as fundamental group of complements MðAÞ, for instance the Stallings group and the Bestvina– Brady groups are discussed in Remark 4.11. In Chap. 5, we start our discussion of the Milnor fiber F associated to a central hyperplane arrangement A, the monodromy action on the cohomology HÃðFÞ and the relation between monodromy eigenspaces and the twisted cohomology of the complement MðA0Þ, where A0 is the projective arrangement associated to A, see Proposition 5.4.
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