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London 2nd July

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Niveau: Supérieur, Doctorat, Bac+8
Extremal Laurent Polynomials? Alessio Corti London, 2nd July 2011 1 Introduction The course is an elementary introduction to my experimental work in progress with Tom Coates, Sergei Galkin, Vasily Golyshev and Alexander Kasprzyk ( funded by EPSRC. I owe a special intellectual debt to Vasily, who already several years ago insisted that we should pursue these themes. I am an algebraic geometer. In the Fall 1988, my first “quarter” in grad- uate school at the University of Utah, I attended lectures by Mori on the classification of Fano 3-folds. (The Minimal Model Program classifies pro- jective manifolds in three classes of manifolds with negative, zero and positive “curvature:” Fano manifolds are those of positive curvature.) I learned from Mori that there are 105 (algebraic) deformation families of Fano 3-folds. I hope that what I tell you in this course is interesting from several perspectives, but one source of motivation for me is to get a picture of the classification of Fano 4-folds: how many families are there? Is it 100,000 families; is it 1,000,000 or perhaps 10,000,000 families? If we are seriously to do algebraic geometry in ≥ 4 dimensions, there is a dearth of examples: what does a “typical” Fano 4-fold look like? ?These notes reproduce almost exactly my 4 lectures at the Summer School on “Moduli of curves and Gromov–Witten theory” held at the Institut Fourier in Grenoble during 20th June–8th July 2011.

  • polynomial differential operator1

  • c?n ?

  • motivic sheaves

  • polarised pure

  • local system

  • extremal local

  • classify extremal


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ExtremalLaurentPolynomialsAlessioCortiLondon,2ndJuly20111IntroductionThecourseisanelementaryintroductiontomyexperimentalworkinprogresswithTomCoates,SergeiGalkin,VasilyGolyshevandAlexanderKasprzyk(http://coates.ma.ic.ac.uk/fanosearch/),fundedbyEPSRC.IoweaspecialintellectualdebttoVasily,whoalreadyseveralyearsagoinsistedthatweshouldpursuethesethemes.Iamanalgebraicgeometer.IntheFall1988,myfirst“quarter”ingrad-uateschoolattheUniversityofUtah,IattendedlecturesbyMoriontheclassificationofFano3-folds.(TheMinimalModelProgramclassifiespro-jectivemanifoldsinthreeclassesofmanifoldswithnegative,zeroandpositive“curvature:”Fanomanifoldsarethoseofpositivecurvature.)IlearnedfromMorithatthereare105(algebraic)deformationfamiliesofFano3-folds.IhopethatwhatItellyouinthiscourseisinterestingfromseveralperspectives,butonesourceofmotivationformeistogetapictureoftheclassificationofFano4-folds:howmanyfamiliesarethere?Isit100,000families;isit1,000,000orperhaps10,000,000families?Ifweareseriouslytodoalgebraicgeometryin4dimensions,thereisadearthofexamples:whatdoesa“typical”Fano4-foldlooklike?Thesenotesreproducealmostexactlymy4lecturesattheSummerSchoolon“ModuliofcurvesandGromov–Wittentheory”heldattheInstitutFourierinGrenobleduring20thJune–8thJuly2011.Iwanttothanktheorganisersforatrulyoutstandingschoolandfortheextremelylovelyatmosphere.Thesenoteswerewritteninhaste:pleaseletmeknowifyoufoundmistakes.Youwillfindanupdatedversiononmyteachingpagehttp://www2.imperial.ac.uk/acorti/teaching.html1
Topics1IgiveashortdiscussionoflocalsystemsonP1\S(whereSP1isafiniteset)andintroducethenotion(duetoVasilyGolyshev)ofextremallocalsystem:thatis,alocalsystemthatisnontrivial,irreducible,andofsmallestpossibleramification.2GivenaLaurentpolynomialf:C×nC,IexplainhowtoconstructthePicard–FuchsdifferentialoperatorLfanditsnaturalsolution,theprincipalperiod.Bydefinition,fisextremalifthelocalsystemofsolutionsofLfisextremal.Iexplainthegeneraltheoryandgivesomeexamples.Inparticular,wediscoveredaninterestingclassofLaurentpolynomialswherethePicard–Fuchslocalsystemshas(conjecturallyandexperimentally)lowramification,calledMinkowskipolynomials.3IbrieflysummarizequantumcohomologyofaFanomanifoldXandquick-and-dirtymethodsofcalculation.MuchofthestructureisencodedinadifferentialoperatorQbXandpowerseriessolutionIbX.ImotivatewithexamplestheconjecturethatQbXisofsmall(oftenminimal)ramification.4AFanomanifoldXismirror-dualtoaLaurentpolynomialfifQbX=Lf.Thisisaveryweaknotionofmirrorsymmetry:toaFanomanifoldXtherecorrespond(infinitely)manyf.IdemostratehowtoderivetheclassificationofFano3-folds(Iskovskikh,Mori–Mukai)fromtheclassificationof3-variableMinkowskipolynomials.Ioutlineaprogramtousetheseideasin4dimensions.ReferencesThisdocumentcontainsnoreferences.ThisisdueinparttothefactthatIampresentingafreshnewscience,andpartlytothefactthatthenotesarewritteninhasteandIwaslazywhenitcomestohistory,attribution,anddetail.Letmeatleasthereacknowledgemygreaterintellectualdebts.Thedef-initionofextremallocalsystems(withthename“low-ramifiedlocalsys-tems”)andextremalLaurentpolynomials(withthename“specialLau-rentpolynomials”)appearedfirstin[VasilyGolyshev,SpectraandStrain,arXiv:0801.0432(hep-th)].Theviewofmirrorsymmetryadvocatedhere2