IRMA Lectures in Mathematics and Theoretical Physics 20 Edited by Christian Kassel and Vladimir G. Turaev Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René-Descartes 67084 Strasbourg Cedex France Chalk drawing by Tatsuo Suwa Singularities in Geometry and Topology Strasbourg 2009 Vincent Blanlœil Toru Ohmoto Editors Editors: Vincent Blanloeil Toru Ohmoto IRMA, UFR Mathématiques et Informatique Department of Mathematics Université de Strasbourg Faculty of Science 7, rue René Descartes Hokkaido University 67084 Strasbourg, France Sapporo 060-0810, Japan E-mail: [email protected] E-mail: [email protected] 2010 Mathematical Subject Classification: 13A35, 14A22, 14B05, 14B07, 14B15, 14C17, 14D06, 14E15, 14E18, 14F99, 14H25, 14J17, 14M25, 14N99, 18F30, 19K10, 32A27, 32C37, 32F75, 32G15, 32SXX, 35Q75, 52C35, 53C05, 55N35, 55R40, 57N05, 57R18, 57R20, 58A30, 58K10, 58K40, 58K60, 60D05, 83C57 Key words: singularity theory, singularities, characteristic classes, Milnor fiber, jet schemes, equisingularity, intersection homology, knot theory, Hodge theory, Fulton–MacPherson bivariant theory, mixed weighted homogeneous, nearby cycles, vanishing cycles, affine toric variety, (versal) deformation of surface singularities, noncommutative resolution, cyclic quotient surface singularity, splice quotient singularity, F-regular singulari- ties, semiquasihomogeneous isolated singularities, general relativity, statistical learning theory, singular distri- butions, localization of characteristic classes, Frobenius morphism, b-function, motivic Grothendieck group, motivic Hirzebruch class, monodromy covering, algebraic local cohomology, Riemann–Roch theorem for embeddings, birational invariant, Riemann surface, stable reduction, Teichmüller space, moduli space, orbifold ISBN 978-3-03719-118-7 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. ©2012 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: M. Zunino, Stuttgart Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface InAugust2009weorganizedthefifthFranco–JapaneseSymposiumonSingularitiesat theDepartmentofMathematicsofStrasbourgUniversity. Thissymposiumfollowed thefourthoneheldinToyama,Japan,twoyearsbefore. Thefirstdaywescheduleda JSPSForumonSingularitiesandApplications, andsomeapplicationsofsingularity theoryinphysics,medicineandstatisticswerepresented. Thefollowingdayswehad aconference;therewereadvancedtalksintopology,algebraicgeometryandcomplex geometry,andrecentresultsonsingularitieswerediscussed. In this volume we collected some research papers from participants of the con- ferenceandsurveysofsometalksintheJSPSForum. Moreoverweaddtwolecture notes of T. Suwa and S.Yokura. All papers in this volume have been refereed and areinfinalform. Wehopethatthisbookwillgiveanopportunitytoreaderstogeta deeperunderstandingofthemarvelousfieldofSingularities. Onbehalfoftheeditorsofthisproceedings,wewouldliketoexpressourthanks toStrasbourgUniversity,JSPS,CNRS,RegionAlsaceandCEEJA,fortheirsupport, andtoallcontributorsfortheproceedingsandtheparticipantsofthesymposium. VincentBlanlœil,Strasbourg ToruOhmoto,Sapporo TheparticipantsoftheConferenceinfrontoftheOpéradeStrasbourg Contents Preface....................................................................v AlainJoets Opticalcausticsandtheirmodellingassingularities (JSPSForum).............1 HelmutA.Hamm Onlocalequisingularity....................................................19 ShihokoIshii,AkiyoshiSannai,andKei-ichiWatanabe Jetschemesofhomogeneoushypersurfaces...................................39 TatsuhikoKoike Singularitiesinrelativity (JSPSForum).....................................51 YukioMatsumoto OntheuniversaldegeneratingfamilyofRiemannsurfaces.....................71 YayoiNakamuraandShinichiTajima Algebraiclocalcohomologiesandlocalb-functionsattached tosemiquasihomogeneoussingularitieswithL.f/D2.......................103 T.Ohmoto, AnoteontheChern–Schwartz–MacPhersonclass............................117 MutsuoOka Onmixedprojectivecurves................................................133 TomohiroOkuma Invariantsofsplicequotientsingularities....................................149 OswaldRiemenschneider AnoteonthetoricdualitybetweenA andA .........................161 n;q n;n(cid:2)q JörgSchürmann Nearbycyclesandcharacteristicclassesofsingularspaces....................181 TatsuoSuwa Residuesofsingularholomorphicdistributions (lecture) .................... 207 viii Contents SumioWatanabe Twobirationalinvariantsinstatisticallearningtheory (JSPSForum)..........249 TakehikoYasuda Frobeniusmorphismsofnoncommutativeblowups...........................269 ShojiYokura BivariantmotivicHirzebruchclass andazetafunctionofmotivicHirzebruchclass (lecture)....................285 MasahikoYoshinaga Minimalityofhyperplanearrangementsandbasisoflocalsystemcohomology..345 Optical caustics and their modelling as singularities Alain Joets LaboratoiredePhysiquedesSolides,Bât.510 UniversitéParis-Sud,91405Orsaycedex,France e-mail: [email protected] Abstract. Opticalcausticsarebrightpatterns,formedbythelocalfocalizationoflightrays.They arecaused,forinstance,bythereflectionortherefractionofthesunraysthroughawavywater surface. Intheabsenceofanappropriatemathematicalframe,theirmaincharacteristicshave remainedunrecognizedforalongtimeandthecausticsappearedintheliteratureunderdifferent names: evolutes, envelopes, focals, etc. The creation of the singularity theory in the middle of the XX century radically changed the situation. Caustics are now understood as physical realizationsofLagrangiansingularities. Inthismodelling,onepredictstheirlocalclassification into five stable types (R. Thom,V.Arnold): folds, cusps, swallowtails, elliptic umbilics and hyperbolicumbilics. Thislocalclassificationisindeedobservedinexperiments. Howeverthe globalpropertiesofthecausticsareonlypartiallytakenintoaccountbytheLagrangianmodel.In fact,ithasbeenprovedbyYu.Chekanovthatthespecialformoftheeikonalequationgoverning thepropagationoftheopticalwavefrontsimposestheexistenceofatopologicalconstrainton thesingularset(representingthecausticinthephasespace)andrestrictsthenumberofpossible bifurcations. Ourexperimentsoncausticsproducedbybi-periodicstructuresinliquidcrystals confirmtheexistenceofthetopologicalconstraint,andvalidatethemodellingofthecausticsas specialtypesofLagrangiansingularities. Causticsconstituteaphenomenonoflightfocalization,usuallystudiedintheframeof geometricalopticsorofwaveoptics. Itisremarkablethattheynowconstitutealsoa purelymathematicalobject,expressedintermsofsingularities. Thesetwonotionsare notuncorrelated. Themathematicalnotionisthefinaloutcome ofalongprocessof successivemodellingsofthephysicalphenomenon,thatwewillcallhereafteroptical caustics. The aim of this paper is to show how the singularity theory drastically changedourviewpointabouttheopticalcaustics. Wewillshowthatsomeproblems, forinstancethelocalclassificationofcausticpoints,maybesolvedonlywiththehelp ofthesingularitytheory,andthat,conversely,thesingularitytheoryisattheoriginof newproblemsandnewexperimentsonopticalcaustics. 2 AlainJoets 1 Physical aspects 1.1 Thephysicalphenomenonofopticalcaustics Therearedifferentwaystopresentcaustics,accordingtowhetheroneconsiderslight ascomposedofrays,orofscalarwavesorofelectromagneticwaves. However,inview of our purpose, we shall mainly consider the geometrical description in which light iscomposedofrays, orequivalentlyofwavefronts. Inotherwords, thewavelength of the lightwillbe supposed tobe 0, orverysmall withrespect tothe characteristic dimensionsofthesystem. Givenasetofrays(acongruenceofrays),acausticpointisapoint(ofourphysical space)wheretheraysarelocallyfocusing,i.e. wheretwofinitelycloseraysintersect (seeFigure1).Atanoncausticpoint,thatistosayataregularpointofthecongruence, theraysformalocalbeam(orthesuperpositionofafinitenumberoflocalbeams). In contrast, the light beam is shrunk at a caustic point and the energy density becomes infinite(atleastintheframeconsideredhere). Thisisthereasonforthename“caustic”, thatcomesfromtheGreekroot“kausticos”meaning“burning”. Fromthegeometrical viewpoint,thecausticistheenvelopeoftheraycongruence. Thismeansthattherays are tangent to the caustic (at the corresponding caustic point). In the usual case of straightraysinourphysical3D-space,eachraycontributesto2causticpointsandthe causticiscomposedof2sheets. Averysimpleexampleofcausticsisprovidedbythebrightmovinglinesonesees onthebottomofaswimmingpool. Anotherexampleisprovidedbyaperfectfocus. However, this example is a somewhat misleading, since a focus is a fully unstable causticpoint disappearingunder anysmallperturbation ofthe congruence. Suchan unstablesituationmustbeexcludedfromthegeneralstudyofcaustics. In the plane, the caustic points constitute curves (Figure 1). In the physical 3D-space,theyconstitutesurfaces. Thesegeometricalobjectsaregenerallynotreg- ular. They may possess special points: regression points for the caustic curves, and regressionedgesforthecausticsurfaces. Theregressionedgesthemselvesmaypos- sessmoreparticularpoints. Inbrief,causticsarestructuredobjectsandanimportant problemistounderstandtheirstructureintodifferenttypesofpoints. Thereisnospecialconditionforproducingopticalcaustics. Everycongruenceof raysgeneratesacaustic,moreorlessintricate. Eveninthecaseofabeamofparallel rays,onemayconsiderthatacausticpointisgeneratedatinfinity. Thecausticsthen constituteanopticalphenomenonofgreatgenerality. 1.2 Observationofcaustics As (singular) surfaces in our physical space R3, the caustics cannot be directly ob- served, since they are not material surfaces. However they are easily visualized by interposing some screen transversely to the rays. In other words, one sees only 2D-sections of a caustic surface and the whole caustic itself necessitates a (tedious)