Algebraic Geometry over the Complex Numbers

326 Pages·2012·2.109 MB·English

by  Donu Arapura (auth.)

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Donu Arapura
Algebraic Geometry
over the Complex Numbers
Donu Arapura
Department of Mathematics
Purdue University
150 N. University Street
West Lafayette, IN 47907
USA
[email protected]
ISSN 0172-5939 e-ISSN 2191-6675
ISBN 978-1-4614-1808-5 e-IS BN 978-1 -4614-1809-2
DOI 10.1007/978-1-4614-1809-2
Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2012930383
Mathematics Subject Classification (2010): 14-XX, 14C30
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Preface
Algebraicgeometryisthegeometricstudyofsetsofsolutionstopolynomialequa-
tions over a field (or ring). These objects, called algebraic varieties (or schemes
or ...), can be studied using tools from commutative and homological algebra.
Whenthefieldisthefieldofcomplexnumbers,thesemethodscanbesupplemented
with transcendental ones, that is, by methods from complex analysis, differential
geometry, and topology. Much of the beauty of the subject stems from the rich
interplayofthesevarioustechniquesandviewpoints.Unfortunately,thisalsomakes
it a hard subject to learn. This book evolved from various courses in algebraic
geometry that I taught at Purdue. In these courses, I felt my job was to act as a
guide to the vast terrain. I did not feel obligated to cover everything or to prove
everything,becausethestandardaccountsofthealgebraicandtranscendentalsides
of the subject by Hartshorne [60] and Griffiths and Harris [49] are remarkably
complete, and perhaps a little daunting as a consequence. In this book I have
tried to maintain a reasonable balance between rigor, intuition,and completeness.
As for prerequisites, I have tried not to assume too much more than a mastery of
standardgraduatecoursesinalgebra,analysis,andtopology.Consequently,Ihave
included discussions of a number of topics that are technically not part of alge-
braicgeometry.Ontheotherhand,sincethebasicsarecoveredquickly,someprior
exposureto elementaryalgebraicgeometry(atthe levelof say Fulton[40],Harris
[58,Chapters1–5]orReid[97])andcalculuswithmanifolds(asinGuilleminand
Pollack[56,Chapters1&4]orSpivak[109])wouldcertainlybedesirable,although
notabsolutelyessential.
Thisbookisdividedintoanumberofsomewhatindependentpartswithslightly
differentgoals.Thestarredsectionscanbeskippedwithoutlosingmuchcontinuity.
The first part, consisting of a single chapter, is an extended informalintroduction
illustrated with concrete examples. It is really meant to build intuition without a
lot of technical baggage. Things really get going only in the second part. This is
wheresheavesareintroducedandusedto definemanifoldsandalgebraicvarieties
in a unified way. A watered-down notion of scheme—sufficient for our needs—
is also presented shortly thereafter. Sheaf cohomology is developed quickly from
scratch in Chapter 4, and appliedto de Rham theoryand Riemann surfacesin the
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viii Preface
next few chapters. By Part III, we move into Hodge theory, which is really the
heartoftranscendentalalgebraicgeometry.Thisiswherealgebraicgeometrymeets
differentialgeometryontheonehand,andsomeserioushomologicalalgebraonthe
other.AlthoughI haveskirtedaroundsomeof the analysis,I didnotwantto treat
thisentirelyasa blackbox.Ihaveincludeda sketchofthe heatequationproofof
the Hodge theorem, which I think is reasonably accessible and quite pretty. This
theoremalong with the weak and hardLefschetz theoremshave some remarkable
consequencesfor the geometryand topologyof algebraic varietes. I discuss some
of these applications in the remaining chapters. From Hodge theory, one extracts
a set of useful invariants called Hodge numbers, which refine the Betti numbers.
Inthe fourthpart, I considersome methodsforactually computingthese numbers
forvariousexamples,suchashypersurfaces.ThetaskofcomputingHodgenumbers
canbeconvertedtoanessentiallyalgebraicproblem,thankstotheGAGAtheorem,
whichisexplainedhereaswell.Thistheoremgivesanequivalencebetweencertain
algebraic and analytic objects called coherent sheaves. In the fifth part, I end the
bookbytouchingonsomeofthedeepermysteriesofthesubject,forexample,that
theseeminglyseparateworldsofcomplexgeometryandcharacteristic pgeometry
arerelated.IwillalsoexplainsomeoftheconjecturesofGrothendieck,Hodge,and
othersalongwithacontexttoputthemin.
I would like to thank Bill Butske, Harold Donnelly, Ed Dunne, Georges
Elencwajg, Anton Fonarev, Fan Honglu, Su-Jeong Kang, Mohan Ramachandran,
Peter Scheiblechner, Darren Tapp, and Razvan Veliche for their suggestions and
clarifications.MythanksalsototheNSFfortheirsupportovertheyears.
DonuArapura
PurdueUniversity
November,2011
Contents
Preface............................................................ vii
PartI IntroductionthroughExamples
1 PlaneCurves .................................................. 3
1.1 Conics.................................................... 3
1.2 Singularities............................................... 5
1.3 Be´zout’sTheorem.......................................... 7
1.4 Cubics.................................................... 9
1.5 Genus2and3 ............................................. 11
1.6 HyperellipticCurves........................................ 14
PartII SheavesandGeometry
2 ManifoldsandVarietiesviaSheaves.............................. 21
2.1 SheavesofFunctions ....................................... 22
2.2 Manifolds................................................. 24
2.3 AffineVarieties ............................................ 28
2.4 AlgebraicVarieties ......................................... 32
2.5 StalksandTangentSpaces ................................... 35
2.6 1-Forms,VectorFields,andBundles .......................... 41
2.7 CompactComplexManifoldsandVarieties..................... 45
3 MoreSheafTheory............................................. 49
3.1 TheCategoryofSheaves .................................... 49
3.2 ExactSequences ........................................... 53
3.3 AffineSchemes ............................................ 58
3.4 SchemesandGluing ........................................ 62
3.5 SheavesofModules ........................................ 66
3.6 LineBundlesonProjectiveSpace............................. 70
3.7 DirectandInverseImages ................................... 72
3.8 Differentials............................................... 76
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