London 2nd July

Laurent Polynomials?

Description

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

Sujets

• Grenoble
• Galkin
• Coates
• Polynomial
• Local system

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 26
• Langue: English

Extrait

ExtremalLaurentPolynomials ∗ AlessioCorti London,2 nd July2011 1Introduction Thecourseisanelementaryintroductiontomyexperimentalworkinprogress withTomCoates,SergeiGalkin,VasilyGolyshevandAlexanderKasprzyk (http://coates.ma.ic.ac.uk/fanosearch/),fundedbyEPSRC.Ioweaspecial intellectualdebttoVasily,whoalreadyseveralyearsagoinsistedthatwe shouldpursuethesethemes. Iamanalgebraicgeometer.IntheFall1988,myfirst“quarter”ingrad- uateschoolattheUniversityofUtah,IattendedlecturesbyMorionthe classificationofFano3-folds.(TheMinimalModelProgramclassifiespro- jectivemanifoldsinthreeclassesofmanifoldswithnegative,zeroandpositive “curvature:”Fanomanifoldsarethoseofpositivecurvature.) IlearnedfromMorithatthereare105(algebraic)deformationfamilies ofFano3-folds.IhopethatwhatItellyouinthiscourseisinterestingfrom severalperspectives,butonesourceofmotivationformeistogetapictureof theclassificationofFano4-folds:howmanyfamiliesarethere?Isit100,000 families;isit1,000,000orperhaps10,000,000families?Ifweareseriously todoalgebraicgeometryin ≥ 4dimensions,thereisadearthofexamples: whatdoesa“typical”Fano4-foldlooklike? ∗ Thesenotesreproducealmostexactlymy4lecturesattheSummerSchoolon“Moduli ofcurvesandGromov–Wittentheory”heldattheInstitutFourierinGrenobleduring 20 th June–8 th July2011.Iwanttothanktheorganisersforatrulyoutstandingschool andfortheextremelylovelyatmosphere.Thesenoteswerewritteninhaste:pleaselet meknowifyoufoundmistakes.Youwillfindanupdatedversiononmyteachingpage http://www2.imperial.ac.uk/ ∼ acorti/teaching.html 1

Topics 1Igiveashortdiscussionoflocalsystemson P 1 \ S(whereS ⊂ P 1 isa finiteset)andintroducethenotion(duetoVasilyGolyshev)ofextremallocal system:thatis,alocalsystemthatisnontrivial,irreducible,andofsmallest possibleramification. 2GivenaLaurentpolynomialf: C × n → C ,Iexplainhowtoconstructthe Picard–FuchsdifferentialoperatorL f anditsnaturalsolution,theprincipal period.Bydefinition,fisextremalifthelocalsystemofsolutionsofL f is extremal.Iexplainthegeneraltheoryandgivesomeexamples.Inparticular, wediscoveredaninterestingclassofLaurentpolynomialswherethePicard– Fuchslocalsystemshas(conjecturallyandexperimentally)lowramification ,calledMinkowskipolynomials. 3IbrieflysummarizequantumcohomologyofaFanomanifoldXand quick-and-dirtymethodsofcalculation.Muchofthestructureisencoded inadifferentialoperatorQ bX andpowerseriessolutionI bX .Imotivatewith examplestheconjecturethatQ b X isofsmall(oftenminimal)ramification. 4AFanomanifoldXismirror-dualtoaLaurentpolynomialfifQ bX = L f .Thisisaveryweaknotionofmirrorsymmetry:toaFanomanifold Xtherecorrespond(infinitely)manyf.Idemostratehowtoderivethe classificationofFano3-folds(Iskovskikh,Mori–Mukai)fromtheclassification of3-variableMinkowskipolynomials.Ioutlineaprogramtousetheseideas in4dimensions. References Thisdocumentcontainsnoreferences.ThisisdueinparttothefactthatI ampresentingafreshnewscience,andpartlytothefactthatthenotesare writteninhasteandIwaslazywhenitcomestohistory,attribution,and detail. Letmeatleasthereacknowledgemygreaterintellectualdebts.Thedef- initionofextremallocalsystems(withthename“low-ramifiedlocalsys- tems”)andextremalLaurentpolynomials(withthename“specialLau- rentpolynomials”)appearedfirstin[VasilyGolyshev,SpectraandStrain, arXiv:0801.0432(hep-th)].Theviewofmirrorsymmetryadvocatedhere 2