Complex Abelian Varieties

635 Pages·2010·4.213 MB·English

by  Christina Birkenhake, Herbert Lange

Grundlehren der
mathematischen Wissenschaften 302
ASeriesofComprehensiveStudiesinMathematics
Serieseditors
A.Chenciner S.S.Chern B.Eckmann
P.delaHarpe F.Hirzebruch N.Hitchin
L.Ho¨rmander M.-A.Knus A.Kupiainen
G.Lebeau M.Ratner D.Serre
Ya.G.Sinai N.J.A.Sloane B.Totaro
A.Vershik M.Waldschmidt
Editor-in-Chief
M.Berger J.Coates S.R.S.Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Christina Birkenhake
Herbert Lange
Complex
Abelian Varieties
Second, Augmented Edition
1 3
ChristinaBirkenhake
HerbertLange
MathematischesInstitut
UniversitätErlangen-Nürnberg
Bismarckstraße11/2
91054Erlangen
Germany
e-mail:
[email protected]
[email protected]
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Mathematics Subject Classification (2000): 14-02, 14KXX, 32G20, 14H37,
14H40,14H42,14F05,14G25
ISSN0072-7830
ISBN 978-3-642-05807-3 ISBN 978-3-662-06307-1 (eBook)
DOI 10.1007/978-3-662-06307-1
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Preface to the Second Edition
Duringthelast15yearsthetheoryofabelianvarietieshasseenprogressinseveral
directions.Weincludesomeofitinthissecondedition.Infact,therearefivenew
chapters,onautomorphisms,onvectorbundlesonabelianvarieties,somenewresults
onlinebundlesandthethetadivisor,andoncyclesonabelianvarieties.Finallywe
giveanintroductiontotheHodgeconjectureforabelianvarieties.
Whereasthefirsteditionwasmoreorlessself-contained,thiscannotbesaidanymore
ofthenewchapters.Weapplyseveralresults,theproofsofwhichwouldbebeyond
the scope of the book. To give an example, the theory of abelian varieties relies
heavilyonMukai’sFourierfunctorandthisisexpressedinthelanguageofderived
categories.Forreadersnotfamiliarwiththislanguage,wegiveashortintroduction
inAppendixD.Fortheconvenienceofthereaderwecompilemoreoverinthreeother
appendicessomemoreadvancedresultsonAlgebraicGeometry,whichareneeded
inthenewchapters.
EachchapterendswithasectioncalledExercisesandFurtherResults.Apartfroma
fewexercisesitcontainsmainlysomerecentresultsforwhichwewouldhaveliked
toincludefullproofs,butfoundneitherthetimenorthespacetodoso.
WewouldliketoexpressourgratitudetoO.Debarre,whopointedoutseveralerrors
inthefirstedition,toM.S.NarasimhanforsomevaluablehintsandtoW.-D.Geyer
foransweringmanyofourquestions.Finally,wethankS.Hu¨bnerforsomeadvice
concerningLatex.
Erlangen,December2003 Ch.Birkenhake
H.Lange
Contents
PrefacetotheSecondEdition ..................................... V
Introduction.................................................... 1
Notation ....................................................... 5
1. ComplexTori............................................... 7
1.1 ComplexTori ............................................ 7
1.2 Homomorphisms ......................................... 9
1.3 CohomologyofComplexTori .............................. 13
1.4 TheHodgeDecomposition................................. 15
1.5 ExercisesandFurtherResults............................... 21
2. LineBundlesonComplexTori ................................ 23
2.1 LineBundlesonComplexTori.............................. 24
2.2 TheAppell-HumbertTheorem.............................. 29
2.3 CanonicalFactors ........................................ 32
2.4 TheDualComplexTorus .................................. 34
2.5 ThePoincare´ Bundle ...................................... 37
2.6 ExercisesandFurtherResults............................... 41
3. CohomologyofLineBundles.................................. 45
3.1 Characteristics ........................................... 46
3.2 ThetaFunctions .......................................... 49
3.3 ThePositiveSemidefiniteCase ............................. 54
3.4 TheVanishingTheorem ................................... 56
3.5 CohomologyofLineBundles............................... 61
3.6 TheRiemann-RochTheorem ............................... 64
3.7 ExercisesandFurtherResults............................... 66
4. AbelianVarieties............................................ 69
4.1 PolarizedAbelianVarieties................................. 70
4.2 TheRiemannRelations.................................... 73
4.3 TheDecompositionTheorem............................... 74
4.4 TheGaussMap .......................................... 81
4.5 ProjectiveEmbeddings .................................... 84
VIII Contents
4.6 SymmetricLineBundles................................... 88
4.7 SymmetricDivisors....................................... 92
4.8 KummerVarieties ........................................ 97
4.9 MorphismsintoAbelianVarieties ...........................100
4.10 ThePontryaginProduct....................................102
4.11 HomologicalVersusNumericalEquivalence ..................105
4.12 ExercisesandFurtherResults...............................109
5. EndomorphismsofAbelianVarieties ........................... 113
5.1 TheRosatiInvolution .....................................114
5.2 Polarizations.............................................119
5.3 Norm-EndomorphismsandSymmetricIdempotents............122
5.4 EndomorphismsAssociatedtoCycles........................128
5.5 TheEndomorphismAlgebraofaSimpleAbelianVariety .......131
5.6 ExercisesandFurtherResults...............................140
6. ThetaandHeisenbergGroups................................. 145
6.1 ThetaGroups ............................................146
6.2 ThetaGroupsunderHomomorphisms........................149
6.3 TheCommutatorMap.....................................151
6.4 TheCanonicalRepresentationoftheThetaGroup .............153
6.5 TheIsogenyTheorem .....................................156
6.6 HeisenbergGroupsandThetaStructures .....................159
6.7 TheSchro¨dingerRepresentation ............................164
6.8 TheIsogenyTheoremforFiniteThetaFunctions ..............166
6.9 SymmetricThetaStructures ................................169
6.10 ExercisesandFurtherResults...............................174
7. EquationsforAbelianVarieties................................ 179
7.1 TheMultiplicationFormula ................................180
7.2 SurjectivityoftheMultiplicationMap........................184
7.3 ProjectiveNormality ......................................187
7.4 TheIdealofanAbelianVarietyinP ........................190
N
7.5 Riemann’sThetaRelations.................................196
7.6 CubicThetaRelations.....................................198
7.7 ExercisesandFurtherResults...............................203
8. Moduli .................................................... 209
8.1 TheSiegelUpperHalfSpace ...............................210
8.2 TheAnalyticModuliSpace ................................213
8.3 LevelStructures ..........................................216
8.3.1 LevelD-Structure..................................217
8.3.2 GeneralizedLeveln-Structure........................218
8.3.3 DecompositionoftheLattice.........................219
8.4 TheThetaTransformationFormula,PreliminaryVersion........220
8.5 ClassicalThetaFunctions ..................................222
Contents IX
8.6 TheThetaTransformationFormula,FinalVersion .............227
8.7 TheUniversalFamily .....................................229
8.8 TheActionoftheSymplecticGroup.........................232
8.9 OrthogonalLevelD-Structures .............................234
8.10 TheEmbeddingofA (D) intoProjectiveSpace..............235
D 0
8.11 ExercisesandFurtherResults...............................239
9. ModuliSpacesofAbelianVarietieswithEndomorphismStructure.. 243
9.1 AbelianVarietieswithEndomorphismStructure ...............245
9.2 AbelianVarietieswithRealMultiplication....................246
9.3 SomeNotation ...........................................251
9.4 TotallyIndefiniteQuaternionMultiplication...................254
9.5 TotallyDefiniteQuaternionMultiplication....................257
9.6 FamiliesofAbelianVarietieswithComplexMultiplication......262
9.7 GroupActionsonH andH .............................267
r,s m
9.8 ShimuraVarieties.........................................270
9.9 TheEndomorphismAlgebraofaGeneralMember.............274
9.10 ExercisesandFurtherResults...............................279
10. AbelianSurfaces ............................................ 281
10.1 Preliminaries.............................................282
10.2 The16 -ConfigurationoftheKummerSurface ................285
6
10.3 AnEquationfortheKummerSurface ........................290
10.4 Reider’sTheorem.........................................293
10.5 PolarizationsofType(1,4) ................................300
10.6 ProductsofEllipticCurves.................................304
10.7 TheCobleHypersurface
ofaPrincipallyPolarizedAbelianSurface ....................308
10.8 ExercisesandFurtherResults...............................310
11. JacobianVarieties........................................... 315
11.1 DefinitionoftheJacobianVariety ...........................316
11.2 TheThetaDivisor ........................................322
11.2.1 ThetaCharacteristics ...............................325
11.2.2 TheSingularityLocusof(cid:2)..........................325
11.3 ThePoincare´ BundlesforaCurveC .........................327
11.4 TheUniversalProperty ....................................330
11.5 CorrespondencesofCurves ................................333
11.6 EndomorphismsAssociatedtoCurvesandDivisors ............335
11.7 ExamplesofJacobians ....................................337
11.8 TheCriterionofMatsusaka-Ran ............................341
11.9 TrisecantsoftheKummerVariety ...........................344
11.10 Fay’sTrisecantIdentity....................................347
11.11 AlbaneseandPicardVarieties ..............................353
11.12 ExercisesandFurtherResults...............................359
X Contents
12. PrymVarieties.............................................. 363
12.1 AbelianSubvarietiesofaPrincipallyPolarizedAbelianVariety ..364
12.2 Prym-TyurinVarieties .....................................368
12.3 PrymVarieties ...........................................372
12.4 TopologicalConstructionofPrymVarieties ...................374
12.5 TheAbel-PrymMap ......................................378
12.6 TheThetaDivisorofaPrymVariety.........................381
12.7 Recillas’Theorem ........................................385
12.8 Donagi’sTetragonalConstruction ...........................388
12.9 Kanev’sCriterion.........................................394
12.10 TheSchottky-JungRelations ...............................399
12.11 ExercisesandFurtherResults...............................406
13. Automorphisms............................................. 411
13.1 Fixed–PointFormulas .....................................412
13.2 TheFixed–PointSetofaFiniteAutomorphismGroup..........413
13.3 AbelianVarietiesofCM-Type..............................417
13.4 AbelianSurfaceswithFiniteAutomorphismGroup ............421
13.5 Poincare´’sReducibilityTheoremwithAutomorphisms .........428
13.6 TheGroupAlgebraDecompositionofanAbelianVariety .......431
13.7 ExercisesandFurtherResults...............................436
14. VectorbundlesonAbelianVarieties............................ 439
14.1 SomePropertiesofthePoincare´ Bundle......................440
14.2 TheFourierTransformforWIT–Sheaves .....................444
14.3 SomePropertiesoftheFourierTransform ....................448
14.4 TheDualPolarization .....................................453
14.5 Application:GlobalGenerationofVectorBundles .............455
14.6 PicardSheaves ...........................................459
14.7 TheFourierTransformofaComplex ........................464
14.8 VectorBundlesonAbelianSurfaces .........................469
14.9 ExercisesandFurtherResults...............................476
15. FurtherResultsonLineBundlesantheThetaDivisor ............ 479
15.1 VeryAmpleLineBundlesonGeneralAbelianVarieties.........480
15.2 SyzygiesofLineBundlesonAbelianVarieties ................484
15.3 SeshadriConstants........................................488
15.4 BoundsforSeshadriConstants..............................491
15.5 TheMinimalLengthofaPeriod ............................496
15.6 SeshadriConstantsofLineBundlesonAbelianSurfaces........503
15.7 SubvarietiesofAbelianVarieties ............................507
15.8 SingularitiesoftheThetaDivisor............................512
15.9 ExercisesandFurtherResults...............................517