Introduction_to_Algebraic_Topology_and_Algebraic_Geometry-U_Bruzzo.pdf

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INTERNATIONALSCHOOL
FORADVANCEDSTUDIES
Trieste
U.Bruzzo
INTRODUCTIONTO
ALGEBRAICTOPOLOGYAND
ALGEBRAICGEOMETRY
Notesofacoursedeliveredduringtheacademicyear2002/2003
Lafilosofia`escrittainquestograndissimolibrochecontinuamente
cistaapertoinnanziagliocchi(iodicol’universo),manonsipu`o
intendereseprimanonsiimparaaintenderlalingua,econosceri
caratteri,ne’quali`escritto.Egli`escrittoinlinguamatematica,e
icaratterisontriangoli,cerchi,edaltrefiguregeometriche,senza
iqualimezi`eimpossibileaintenderneumanamenteparola;senza
questi`eunaggirarsivanamenteperunoscurolaberinto.
GalileoGalilei(from“IlSaggiatore”)
i
Preface
ThesenotesassemblethecontentsoftheintroductorycoursesIhavebeengivingat
SISSAsince1995/96.Originallythecoursewasintendedasintroductionto(complex)
algebraicgeometryforstudentswithaneducationintheoreticalphysics,tohelpthemto
masterthebasicalgebraicgeometrictoolsnecessaryfordoingresearchinalgebraically
integrablesystemsandinthegeometryofquantumfieldtheoryandstringtheory.This
motivationstilltranspiresfromthechaptersinthesecondpartofthesenotes.
Thefirstpartonthecontraryisabriefbutrathersystematicintroductiontotwo
topics,singularhomology(Chapter2)andsheaftheory,includingtheircohomology
(Chapter3).Chapter1assemblessomebasicsfactinhomologicalalgebraanddevelops
thefirstrudimentsofdeRhamcohomology,withtheaimofprovidinganexampleto
thevariousabstractconstructions.
Chapter4isanintroductiontospectralsequences,aratherintricatebutverypower-
fulcomputationtool.Theexamplesprovidedherearefromsheaftheorybutthiscom-
putationaltechniquesisalsoveryusefulinalgebraictopology.
Ithankallmycolleaguesandstudents,inTriesteandGenovaandotherlocations,
whohavehelpedmetoclarifysomeissuesrelatedtothesenotes,orhavepointedout
mistakes.InthisconnectionspecialthanksareduetoFabioPioli.MostofChapter3is
anadaptationofmaterialtakenfrom[2].IthankmyfriendsandcollaboratorsClaudio
BartocciandDanielHern´andezRuip´erezforgrantingpermissiontousethatmaterial.
IthankLotharG¨ottscheforusefulsuggestionsandforpointingoutanerrorandthe
studentsofthe2002/2003coursefortheirinterestandconstantfeedback.
Genova,4December2004
Contents
Part1.AlgebraicTopology 1
Chapter1. Introductorymaterial 3
1.Elementsofhomologicalalgebra 3
2.DeRhamcohomology 7
3.Mayer-VietorissequenceindeRhamcohomology 10
4.Elementaryhomotopytheory 11
Chapter2.Singularhomologytheory 17
1.Singularhomology 17
2.Relativehomology 25
3.TheMayer-Vietorissequence 28
4.Excision
32
Chapter3. Introductiontosheavesandtheircohomology 37
1.Presheavesandsheaves 37
2.Cohomologyofsheaves 43
Chapter4.Spectralsequences 53
1.Filteredcomplexes 53
2.Thespectralsequenceofafilteredcomplex 54
3.Thebidegreeandthefive-termsequence 58
4.Thespectralsequencesassociatedwithadoublecomplex 59
5.Someapplications
62
Part2.Introductiontoalgebraicgeometry 67
Chapter5.Complexmanifoldsandvectorbundles 69
1.Basicdefinitionsandexamples 69
2.Somepropertiesofcomplexmanifolds 72
3.Dolbeaultcohomology 73
4.Holomorphicvectorbundles 73
5.Chernclassoflinebundles 77
6.Chernclassesofvectorbundles 79
7.Kodaira-Serreduality 81
8.Connections
82
iii

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