J Schubert Series of Congress Reports arosła Series of Congress Reports w Buczyn V arieties, ´ ski, Schubert Varieties, and Elisa Michałek Mateusz Equivariant Cohomology and Schubert Varieties, Equi Characteristic Classes Equivariant Jarosław Buczyn´ ski, Mateusz Michałek and va Elisa Postinghel, Editors Classes and Characteristic riant Cohomology Cohomology and Characteristic Classes IMPANGA stands for the activities of algebraic geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The topics of the lectures usually fit within the framework of complex algebraic geometry and neighbouring areas of mathematics. Pos This volume is a collection of contributions by the attendees of the 15 conference IMPANGA15, organised by participants of the seminar, as well tinghel, as notes from the major lecture series of the seminar in the period 2010–2015. Both original research papers and self-contained expository surveys can be found here. The articles circulate around a broad range of topics within algebraic geometry such as vector bundles, Schubert Editors Jarosław Buczyn´ ski varieties, degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and Mateusz Michałek polynomials, and algebraic geometry in positive characteristic. Elisa Postinghel Editors A tribute to Friedrich Hirzebruch ISBN 978-3-03719-182-8 www.ems-ph.org SCR_Impanga 15 | Fonts: Nuri, Helvetica Neue | Farben: Pantone 116, pantone 287 | RB ??? mm EMS Series of Congress Reports EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowron´ski (eds.) Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.) Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.) Functional Analysis and Operator Theory for Quantum Physics, Jaroslav Dittrich, Hynek Kovarˇík and Ari Laptev (eds.) Schubert Varieties, Equivariant Cohomology and Characteristic Classes IMPANGA 15 Jarosław Buczyn´ski Mateusz Michałek Elisa Postinghel Editors Editors: Jarosław Buczyn´ski Mateusz Michałek Institute of Mathematics Institute of Mathematics Polish Academy of Sciences Polish Academy of Sciences ul. S´ niadeckich 8 ul. S´ niadeckich 8 00-656 Warszawa, Poland 00-656 Warszawa, Poland and and Faculty of Mathematics, Computer Science and Mechanics Max Planck Institute for of University of Warsaw Mathematics in the Sciences ul. Banacha 2 Inselstr. 22 02-097 Warszawa, Poland 04103 Leipzig, Germany Email: [email protected] Email: [email protected] Elisa Postinghel Department of Mathematical Sciences Loughborough University Loughborough LE11 3TU, United Kingdom Email: [email protected] 2010 Mathematics Subject Classification: Primary 14-06; secondary 32L10, 14M15, 55N91, 14C17, 14G17. Key words: IMPANGA, vector bundles, Schubert varieties and degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, quasi-elliptic surfaces ISBN 978-3-03719-182-8 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2018 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich, Switzerland Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Contents Introduction to “Schubert varieties, equivariant cohomology and characteristic classes, IMPANGA15 volume” by Jarosław Buczy´nski, Mateusz Michałek and Elisa Postinghel :::::::::: 1 Friedrich Hirzebruch – a handful of reminiscences by Piotr Pragacz :::::::::::::::::::::::::::::::::::::::: 9 Pieri rule for the factorial Schur P -functions by Soojin Cho and Takeshi Ikeda :::::::::::::::::::::::::::::: 25 Restriction varieties and the rigidity problem by Izzet Coskun ::::::::::::::::::::::::::::::::::::::::: 49 On Plücker equations characterizing Grassmann cones by Letterio Gatto and Parham Salehyan ::::::::::::::::::::::::: 97 Kempf–Laksov Schubert classes for even infinitesimal cohomology theories by Thomas Hudson and Tomoo Matsumura ::::::::::::::::::::::: 127 On the multicanonical systems of quasi-elliptic surfaces in characteristic 3 by Toshiyuki Katsura :::::::::::::::::::::::::::::::::::::: 153 Characteristic classes of mixed Hodge modules and applications by Lauren¸tiu Maxim and Jörg Schürmann :::::::::::::::::::::::: 159 On a certain family of U.b/-modules by Piotr Pragacz :::::::::::::::::::::::::::::::::::::::: 203 Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae by Richárd Rimányi and Alexander Varchenko ::::::::::::::::::::: 225 Thom polynomials in A-classification I: counting singular projections of a surface by Takahisa Sasajima and Toru Ohmoto ::::::::::::::::::::::::: 237 Schubert polynomials and degeneracy locus formulas by Harry Tamvakis ::::::::::::::::::::::::::::::::::::::: 261 vi Contents Hirzebruch y-genera of complex algebraic fiber bundles – the multiplicativity of the signature modulo 4 by Shoji Yokura ::::::::::::::::::::::::::::::::::::::::: 315 Pushing-forward Schur classes using iterated residues at infinity by Magdalena Zielenkiewicz ::::::::::::::::::::::::::::::::: 331 List of contributors ........................................ 347 Introduction to “Schubert varieties, equivariant cohomology and characteristic classes, IMPANGA15 volume” Jarosław Buczynski´ 1, Mateusz Michałek2 and Elisa Postinghel The volume This volume is a conclusion of the activities of IMPANGA in the years 2010–2015, which celebrated 15 years of its existence in 2015. It is a follow up to previous books [1, 2, 3] and it contains contributions of the participants of the anniversary conference IMPANGA15. In this introduction we briefly review what IMPANGA is, describe the conference and summarise the content of the volume. Friedrich Hirzebruch passed away in 2012, during the aforementioned period. We dedicate this book to his memory. 1 The seminar IMPANGA IMPANGA is the name of the activities of Algebraic Geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important sem- inars in algebraic geometry in Poland. The head of the seminar is Piotr Pragacz, and the first seminar was held on 30 October 2000. The topics of the seminar lec- tures usually fit within the framework of complex algebraic geometry, although talks about positive characteristics, real algebraic geometry, symplectic geometry, com- plex analysis, singularity theory and other neighbouring areas of mathematics are also welcomed. The acronym IMPANGA stands for the Polish names of the Institute of Mathe- matics of the Polish Academy of Sciences (IMPAN, Instytut Matematyczny Polskiej Akademii Nauk) and Algebraic Geometry (GA, Geometria Algebraiczna). IMPAN https://www.impan.pl/ is a Polish institute designated to mathematical research. Its headquarters are in Warsaw and it has branches in 6 other major Polish cities. The department of Algebra and Algebraic Geometry of the institute is one the most active research groups and it is chaired by Piotr Pragacz. Its members collaborate closely with other research groups of algebraic geometers in Poland, particularly those at the 1Buczynski´ is supported by the research grant from Polish National Science Center, number 2013/11/D/ST1/02580, and by a scholarship of Polish Ministry of Science. 2Michałek is supported by the research grant from Polish National Science Center, number UMO- 2016/22/E/ST1/00574 and by the Foundation for Polish Science (FNP). 2 J. Buczynski,´ M. Michałek,