Schubert Varieties, Equivariant Cohomology and Characteristic Classes

356 Pages·2018·2.407 MB·English

by  Michałek, Mateusz

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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
Introductionto“Schubertvarieties,equivariantcohomologyandcharacteristic
classes,IMPANGA15volume”
byJarosławBuczyn´ski,MateuszMichałekandElisaPostinghel: : : : : : : : : : 1
FriedrichHirzebruch–ahandfulofreminiscences
byPiotrPragacz : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9
PieriruleforthefactorialSchurP-functions
bySoojinChoandTakeshiIkeda : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25
Restrictionvarietiesandtherigidityproblem
byIzzetCoskun : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 49
OnPlückerequationscharacterizingGrassmanncones
byLetterioGattoandParhamSalehyan : : : : : : : : : : : : : : : : : : : : : : : : : 97
Kempf–LaksovSchubertclassesforeveninfinitesimalcohomologytheories
byThomasHudsonandTomooMatsumura : : : : : : : : : : : : : : : : : : : : : : : 127
Onthemulticanonicalsystemsofquasi-ellipticsurfacesincharacteristic3
byToshiyukiKatsura: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 153
CharacteristicclassesofmixedHodgemodulesandapplications
byLauren¸tiuMaximandJörgSchürmann : : : : : : : : : : : : : : : : : : : : : : : : 159
OnacertainfamilyofU.b/-modules
byPiotrPragacz : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 203
EquivariantChern–Schwartz–MacPhersonclasses in partial flag varieties:
interpolationandformulae
byRichárdRimányiandAlexanderVarchenko : : : : : : : : : : : : : : : : : : : : : 225
Thom polynomials in A-classification I: counting singular projections
ofasurface
byTakahisaSasajimaandToruOhmoto : : : : : : : : : : : : : : : : : : : : : : : : : 237
Schubertpolynomialsanddegeneracylocusformulas
byHarryTamvakis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 261
vi Contents
Hirzebruch(cid:2) -generaofcomplexalgebraicfiberbundles–themultiplicativity
y
ofthesignaturemodulo4
byShojiYokura : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 315
Pushing-forwardSchurclassesusingiteratedresiduesatinfinity
byMagdalenaZielenkiewicz: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 331
Listofcontributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Introduction to “Schubert varieties, equivariant
cohomology and characteristic classes,
IMPANGA15 volume”
JarosławBuczyn´ski1,MateuszMichałek2 and ElisaPostinghel
The volume
This volumeis a conclusionofthe activities ofIMPANGA in the years2010–2015,
whichcelebrated15yearsofitsexistencein2015.Itisafollowuptopreviousbooks
[1,2,3]anditcontainscontributionsoftheparticipantsoftheanniversaryconference
IMPANGA15.
InthisintroductionwebrieflyreviewwhatIMPANGAis,describetheconference
andsummarisethecontentofthevolume.
FriedrichHirzebruchpassedawayin2012,duringtheaforementionedperiod.We
dedicatethisbooktohismemory.
1 The seminarIMPANGA
IMPANGA is the name of the activities of Algebraic Geometers at the Institute of
Mathematics,PolishAcademyofSciences,includingoneofthemostimportantsem-
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-
turesusuallyfitwithintheframeworkofcomplexalgebraicgeometry,althoughtalks
about positive characteristics, real algebraic geometry, symplectic geometry, com-
plex analysis, singularity theory and other neighbouring areas of mathematics are
alsowelcomed.
The acronym IMPANGA stands for the Polish names of the Institute of Mathe-
maticsofthePolishAcademyofSciences(IMPAN,InstytutMatematycznyPolskiej
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
departmentofAlgebraandAlgebraicGeometryoftheinstituteisonethemostactive
research groups and it is chaired by Piotr Pragacz. Its members collaborate closely
withotherresearchgroupsofalgebraicgeometersinPoland,particularlythoseatthe
1Buczyn´ski is supported by the research grant from Polish National Science Center, number
2013/11/D/ST1/02580,andbyascholarshipofPolishMinistryofScience.
2Michałek is supported by the research grant from Polish National Science Center, number UMO-
2016/22/E/ST1/00574andbytheFoundationforPolishScience(FNP).
2 J.Buczyn´ski,M.Michałek,E.Postinghel
Figure1.ProfessorFriedrichHirzebruchatIMPAN,2009.
UniversityofWarsaw,theJagiellonianUniversityandthePedagogicalUniversityin
Kraków.
TheseminarIMPANGAusuallyrunstwomeetingspermonthonFridays. During
eachmeeting(typicallyinWarsaw)eithertwotalksaredeliveredbytwospeakers,or
aseriesoftwo lecturesis presentedbyasinglespeaker. Themeetingisattendedby
participantsfromWarsawandKrakówandoftenalsobyvisitorsfromothercities.
Theseminar’sparticipantsalsoorganisemeetings(schools,workshops,conferen-
ces)ofvarioussize, topics andimpact, seehttps://www.impan.pl/~pragacz/impanga
_miniszkoly.htm. The IMPANGA School 2010 https://www.impan.pl/~impanga/
school/andtheIMPANGA15conferencehttps://www.impan.pl/~impanga15aretwo
ofthelargestevents. Wehopetoorganisesimilareventsevery5years.
2 The conference IMPANGA15
The IMPANGA15 conference took place in Be˛dlewo during the week 12–18 April
2015. Inadditionto hostingan excellentscientific program, wecelebrated15years
ofIMPANGAandthe60thbirthdayofProfessorPiotrPragacz.
Theconferenceexposedarichvarietyoftopicsinalgebraicgeometry,especially:
symmetric functionsandpolynomials,Schubertvarieties anddegeneracyloci, char-
acteristicclasses(particularlyofsingularvarieties),Thompolynomials,characteristic
p problems,arithmeticalgebraicgeometry,moduliproblems,tropicalgeometry.
IMPANGA15 gathered about 80 participants from Europe, United States, and
Asia. Besides17plenarylectures,therewerealso9shortertalksdeliveredbythepar-
ticipants. Duringtheconferencetherewasapostersessionwitharangeofinteresting
presentations.Numerouspeoplecontributedtothesuccessoftheconference.Theor-
ganisingcommitteeconsistedofJ.Buczyn´ski(PolishacademyofSciences;Univer-
sity of Warsaw), M. Donten-Bury(University of Warsaw; Freie UniversitätBerlin),
Introduction 3
Figure2.ParticipantsoftheIMPANGA15conferenceinBe˛dlewo,2015.
G. Kapustka (Polish Academy of Sciences; Jagiellonian University), O. Ke˛dzierski
(UniversityofWarsaw;PolishAcademyofSciences),M.Michałek(PolishAcademy
of Sciences; University of California, Berkeley), E. Postinghel (University of Leu-
ven) and J. Szpond (PedagogicalUniversity of Kraków). The scientific committee,
thathelpedmaketheeventexceptional,included:PaoloAluffi(FlordiaStateUniver-
sity), Bernard Leclerc (Université de Caen), Richárd Rimányi (University of North
Carolina at Chapel Hill), Matthias Schütt (Leibniz Universität Hannover) and Ravi
Vakil(StanfordUniversity).
Theinvitedspeakerswere:DavidAnderson(OhioStateUniversity),AndersBuch
(RutgersUniversity),HélèneEsnault(FreieUniversitätBerlin),GerardvanderGeer
(UniversiteitvanAmsterdam),JuneHuh(PrincetonUniversityandInstitute forAd-
vancedStudy), ToshiyukiKatsura(HoseiUniversity),Maxim Kazarian(SteklovIn-
stitute of Mathematics; Moscow Independent University), JongHae Keum (Korea
Institute for Advanced Study), Allen Knutson (Cornell University), Adrian Langer
(Polish Academy of Sciences; University of Warsaw), Lauren˛tiu Maxim (Univer-
sityofWisconsin,Madison),ToruOhmoto(HokkaidoUniversity),SamPayne(Yale
University),PiotrPragacz(PolishAcademyofSciences),StevenSam(Universityof
California,Berkeley),HarryTamvakis(UniversityofMaryland)andOrsolaTommasi
(LeibnizUniversitätHannover).