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Progress in Mathematics Volume261 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein Edgar Lee Stout Polynomial Convexity Birkhaüser Boston • Basel • Berlin EdgarLeeStout DepartmentofMathematics UniversityofWashington Seattle,WA98195 U.S.A. MathematicsSubjectClassification(2000):32A40,32D20,32E10,32E20,32E30 LibraryofCongressControlNumber:2007923422 ISBN-10:0-8176-4537-3 e-ISBN-10:0-8176-4538-1 ISBN-13:978-0-8176-4537-3 e-ISBN-13:978-0-8176-4538-0 Printedonacid-freepaper. (cid:1)c2007BirkhaüserBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(BirkhaüserBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 987654321 www.birkhauser.com (JLS/SB) Preface Thisbookisdevotedtoanexpositionofthetheoryofpolynomiallyconvexsets.Acompact subset of CN is polynomially convex if it is defined by a family, finite or infinite, of polynomial inequalities. These sets play an important role in the theory of functions of severalcomplexvariables,especiallyinquestionsconcerningapproximation.Ontheone hand,thepresentvolumeisastudyofpolynomialconvexityperse,ontheother,itstudies theapplicationofpolynomialconvexitytootherpartsofcomplexanalysis,especiallyto approximationtheoryandthetheoryofvarieties. Not every compact subset of CN is polynomially convex, but associated with an (cid:1) arbitrarycompactset,sayX,isitspolynomiallyconvexhull,X,whichistheintersection of all polynomially convex sets that contain X. Of paramount importance in the study ofpolynomialconvexityisthestudyofthecomplementarysetX(cid:1)\X.Theonlyobvious reason for this set to be nonempty is for it to have some kind of analytic structure, and initiallyonewonderswhetherthissetalwayshascomplexstructureinsomesense.Itis notlongbeforeoneisdisabusedofthisnaivehope;anaturalproblemthenisthatofgiving conditionsunderwhichthecomplementarysetdoeshavecomplexstructure.Inanatural classofone-dimensionalexamples,suchanalyticstructureisfound.Thestudyofthisclass ofexamplesisoneofthemajordirectionsoftheworkathand. This book is not self-contained. Certainly it is assumed that the reader has some previousexposuretothetheoryoffunctionsofseveralcomplexvariables.Hereandthere wedrawonsomemajorresultsfromthetheoryofSteinmanifolds.Thisseemsreasonable in the context: Stein manifolds are the natural habitat of the complex analyst. We draw freelyontheelementsofrealvariables,functionalanalysis,andclassicalfunctiontheory. Atcertainpointsinthetext,partsofalgebraictopologyandMorsetheoryareinvoked.For resultsinalgebraictopologythatgobeyondwhatonecouldreasonablyexpecttomeetin anintroductorycourseinthesubject,precisereferencestothetextbookliteraturearegiven, asarereferencesforMorsetheory.Inaddition,itisnecessarytoinvokecertainresultsfrom geometric measure theory, particularly some of the seminal work of Besicovitch on the structureofone-dimensionalsets,workthatisquitetechnical.Again,precisereferences aregivenasrequired. Chapterbychapter,thecontentsofthebookcanbesummarizedasfollows.Chap- ter1isintroductoryandcontainstheinitialdefinitionsofthesubject,developssomeofthe toolsthatwillbeusedinsubsequentchapters,andgivesillustrativeexamples.Chapter2 is concerned mainly with general properties of polynomially convex sets, for the most vi Preface part properties that are independent of particular structural requirements. Chapter 3 is a systematicstudyofthepolynomialhullofaone-dimensionalsetthatisconnectedandhas finitelengthor,moregenerally,thatiscontainedinaconnectedsetoffinitelength.Forex- ample,inthischapter,itisfoundthatarectifiablearcispolynomiallyconvex,aresultthat, despitethesimplicityofitsformulation,isnotatallsimpletoprove.Alsointhischapter thetheoryofpolynomiallyconvexsetsisappliedtothestudyofone-dimensionalvarieties, especiallytoquestionsofanalyticcontinuation.Chapter4continuesthestudyofthepoly- nomiallyconvexhullofone-dimensionalsets,thistimeadmittingsetsmoregeneralthan thoseconsideredinChapter3,setsthataresometimestermedgeometrically1-rectifiable. Chapter5studiesthreedistinctsubjectsthatdo,though,havesomeconnectionswithone another.Thefirstconcernscertainisoperimetricpropertiesofhulls.Next,wepresentsome resultsonremovablesingularities.Finally,thehullsofsurfacesinstrictlypseudoconvex boundaries are considered. Chapter 6 is devoted to approximation questions, mostly on compact sets, but with some consideration of approximation on unbounded sets. Chap- ter7appliesideasofpolynomialconvexitytothestudyofone-dimensionalsubvarieties of strictly pseudoconvex domains, for example the ball. In part, the motivation for this work comes from the well-developed theory of the boundary behavior of holomorphic functions.Chapter8isdevotedtosomeadditionaltopicsthateitherfurtherthesubjectof polynomialconvexityitselforareapplicationsofthistheory. As it stands, the book is not short, but it has been necessary to omit certain topics that might naturally have been considered. For example, it is with real reluctance that I omitalldiscussionofthehullsoftwo-spheresorofthehullsofsetsfiberedovertheunit circle.Theformeromissionisexplainedbythehighlytechnicalnatureofthesubject,the latterbyaperceptionthatthesubjecthasnotyetachieveditsdefinitiveform. Acknowledgmentsareinorder.Formanyyearsthemathematicsdepartmentofthe UniversityofWashingtonhasprovedtobeanexcellentplaceformywork;toitIamtruly thankful. Mary Sheetz of that department has been unflaggingly good-humored as she helpedwiththemanuscriptofthisbook,ofteninthefaceofveryfrustratingdifficulties. TheworkonthisbookwassupportedinpartbytheRoyaltyResearchFundattheUniversity ofWashington.NormanLevenbergreadmuchofthetextinmanuscriptandmademany helpfulsuggestions,allofwhichIappreciatedbutnotallofwhichIfollowed.Iamindebted toV.M.Gichev,MarkLawrence,andJean-PierreRosayforpermissiontoincludeasyet unpublished results of theirs. Other friends and colleagues have made useful comments andsuggestions;toallIexpressmythanks. The reader will note the great influence of the work of HerbertAlexander on our subject.Overthecourseofhiscareer,Alexandermademanypenetratingcontributionsto the theory of polynomial convexity. His friends and colleagues, who looked forward to hisfurtherdevelopmentofthesubject,wereappalledtolearnofhisuntimelydeathatthe ageof58,tolearnthatadistinguishedcolleagueandgoodfriendhadbeensoprematurely takenaway.Themanisgonebutnotforgotten;hisworkwillendure. EdgarLeeStout Seattle MayDay,2006 Contents Preface v IndexofFrequentlyUsedNotation ix 1 INTRODUCTION 1 1.1 PolynomialConvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 UniformAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 PlurisubharmonicFunctions . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 TheCauchy–FantappièIntegral . . . . . . . . . . . . . . . . . . . . . . . 28 1.5 TheOka–WeilTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6 SomeExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.7 HullswithNoAnalyticStructure . . . . . . . . . . . . . . . . . . . . . . 68 2 SOMEGENERALPROPERTIESOFPOLYNOMIALLYCONVEXSETS 71 2.1 ApplicationsoftheCousinProblems . . . . . . . . . . . . . . . . . . . . 71 2.2 TwoCharacterizationsofPolynomiallyConvexSets . . . . . . . . . . . 83 2.3 ApplicationsofMorseTheoryandAlgebraicTopology . . . . . . . . . . 93 2.4 ConvexityinSteinManifolds . . . . . . . . . . . . . . . . . . . . . . . . 106 3 SETSOFFINITELENGTH 121 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.2 One-DimensionalVarieties . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.3 GeometricPreliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4 Function-TheoreticPreliminaries . . . . . . . . . . . . . . . . . . . . . . 132 3.5 SubharmonicityResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.6 AnalyticStructureinHulls . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.7 FiniteArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.8 TheContinuationofVarieties . . . . . . . . . . . . . . . . . . . . . . . . 156 4 SETSOFCLASSA 169 1 4.1 IntroductoryRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.2 Measure-TheoreticPreliminaries . . . . . . . . . . . . . . . . . . . . . . 170 4.3 SetsofClassA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 1 viii Contents 4.4 FiniteArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.5 Stokes’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.6 TheMultiplicityFunction . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.7 CountingtheBranches . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5 FURTHERRESULTS 217 5.1 Isoperimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.2 RemovableSingularities . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5.3 SurfacesinStrictlyPseudoconvexBoundaries . . . . . . . . . . . . . . . 257 6 APPROXIMATION 277 6.1 TotallyRealManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6.2 HolomorphicallyConvexSets . . . . . . . . . . . . . . . . . . . . . . . 289 6.3 ApproximationonTotallyRealManifolds . . . . . . . . . . . . . . . . . 300 6.4 SomeToolsfromRationalApproximation . . . . . . . . . . . . . . . . . 310 6.5 AlgebrasonSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 6.6 TangentialApproximation . . . . . . . . . . . . . . . . . . . . . . . . . 341 7 VARIETIESINSTRICTLYPSEUDOCONVEXDOMAINS 351 7.1 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 7.2 BoundaryRegularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 7.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 8 EXAMPLESANDCOUNTEREXAMPLES 377 8.1 UnionsofPlanesandBalls . . . . . . . . . . . . . . . . . . . . . . . . . 377 8.2 PluripolarGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 8.3 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 8.4 SetswithSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References 415 Index 431 Index of Frequently Used Notation ⊥ A thespaceofmeasuresorthogonaltothealgebraA. bE theboundaryofthesetE. B theopenunitballinCN. N B (z,r) theopenballofradiusr centeredatthepointz∈CN. N B (r) theopenballofradiusr centeredattheorigininCN. N C(X) thespaceofcontinuousC-valuedfunctionsonthespaceX. D(M) the space of compactly supported functions of class C∞ onM. Dp(M) the space of compactly supported forms of degree p and classC∞onM. Dp,q(M) thespaceofcompactlysupportedformsofbidegree(p,q) andclassC∞onM. D (M)=Dp(cid:4)(M) thespaceofcontinuouslinearfunctionalsonDp(M). p D (M)=D(p,q)(cid:4)(M) thespaceofcontinuouslinearfunctionalsonD(p,q)(M). p,q G (C) theGrassmannianofallk-dimensionalcomplex-linearsub- N,k spacesofCN. G (R) theGrassmannianofallk-dimensionalreal-linearsubspaces N,k ofRN. Hˇ∗ Cˇechcohomology. Hp (M) thepthdeRhamcohomologygroupofthemanifoldM. deR (cid:6)z theimaginarypartofthecomplexnumberz. k (z,w) theBochner–Martinellikernel. BM L Lebesguemeasure. O(M) thealgebraoffunctionsholomorphiconthecomplexman- ifoldM. PN(C) N-dimensionalcomplexprojectivespace. x IndexofFrequentlyUsedNotation P(X) the algebra of functions on the set X uniformly approx- imablebypolynomials. Psh(M) the space of plurisubharmonic functions on the complex manifoldM. (cid:7)z therealpartofthecomplexnumberz. R-hullX therationallyconvexhullofX. R(X) the algebra of functions on X uniformly approximable by rationalfunctionswithoutpolesonX. Sn theunitsphereinRn+1. TN thetorus{(z ,...,z )∈CN :|z |=···=|z |=1}. 1 N 1 N UN theopenunitpolydiskinCN. U(N) theunitarygroup. (cid:1)p p-dimensionalHausdorffmeasure. ω(z) thedifferentialformdz ∧···∧dz . 1 N ω[k](z) thedifferentialformdz1∧···∧d(cid:2)zk ∧···∧dzN. (cid:9)f(cid:9) =sup{|f(x)|:x ∈X}. X (cid:1) E thepolynomiallyconvexhullofthecompactsetE. X (cid:1)Y XisarelativelycompactsubsetofY. ∅ theemptyset.