Vector Bundles on Algebraic Varieties: Papers Presented at the Bombay Colloquium 1984

429 Pages·1987·2.338 MB·English

by  Michael Francis Atiyah

Vector Bundles On
Algebraic Varieties

VECTOR BUNDLES ON
ALGEBRAIC VARIETIES
Paperspresented attheBombayColloquium 1984, by
ATIYAHAYOUBBARTHBHATWADEKAR
COLLIOT-THE´LE`NEDREZETHIRSCHOWITZ
HORROCKSHULEKKEMPFKNUSLEPOTIER
LINDELMARUYAMAMOOREMUKAI
NARASIMHANOJANGURENPARIMALA
RAMANANRAMANATHANSESHADRI
SRIDHARANSRINIVASTRAUTMANN
Publishedforthe
TATA INSTITUTE OF FUNDAMENTALRESEARCH,
BOMBAY
OXFORDUNIVERSITY PRESS
BOMBAYDELHICALCUTTAMADRAS
1987
OxfordUniversity Press,WaltonStreet,OxfordOX26DP
NEWYORKTORONTO
DELHIBOMBAYCALCUTTAMADRASKARACHI
PETALINGJAYASINGAPOREHONGKONGTOKYO
NAIROBIDARESSALAAM
MELBOURNEAUCKLAND
andassociates in
BEIRUTBERLINIBADANNICOSIA
(cid:13)c TataInstitute ofFundamental Research,1987
PrintedinIndiabyK.M.Aarif,
PaperPrintIndia, Worli,Bombay400018
andpublished byR.Dayal,OxfordUniversityPress
OxfordHouse,ApolloBunder, Bombay400039
Contents
1. M.F.Atiyah: Magneticmonopoles inhyperbolic spaces 1
2. W.Barth,K.HulekandR.Moore: Shioda’smodular 29
surface S(5)andtheHorrocks-Mumford bundle
3. S.M.Bhatwadekar: Someresultsonaquestion ofQuillen 79
4. J.-L.ColliotThe´le`ne: Aproblem ofZariski 93
5. J.M.DrezetandJ.LePotier: Conditionsd’existence des 97
fibresstablesderange´leve´ surP2
6. A.Hirschowitz: Ranktechniques andjumpstratifications 115
7. G.Horrocks: Vectorbundles onthepunctured spectrum 155
ofalocalringII
8. G.R.Kempf: SomemetricsonPicardbundles 165
9. M.-A.Kuns: BundlesonP2 withaquaternionic structure 173
10. H.Lindel: Onprojectivemodulesoverpositively graded 195
rings
11. M.Maruyama: VectorbundlesonP2 andtorsionsheaves 213
onthedualplane
12. S.Mukai: Onthemodulispaceofbundles onK3 263
surfaces, I
13. M.S.NarasimhanandS.Ramanan: 2θ-linear systemson 315
abelian varieties
14. M.S.NarasimhanandG.Trautmann: Compactification of 325
M(0,2)
15. M.OjangurenandG.Ayoub: TheWittgroupofareal 337
surface
16. M.Ojanguren, R.ParimalaandR.Sridharan: Anisotropic 353
quadratic spacesovertheplane
17. A.Ramanathan : Unitarybundlesandrestrictions of 373
stablesheaves
18. C.S.Seshadri: Linebundles onSchubertvarieties 379
19. V.Srinivas: Zerocyclesonasingular surface: an 401
introduction
International Colloquium on Vector Bundles
on Algebraic Varieties
Bombay, 9-16 January 1984
REPORT
ANINTERNATIONALCOLLOQUIUMon‘VectorBundlesonAl-
gebraic Varieties’ was held at the Tata Institute of Fundamental Re-
search, Bombay from January 9 to January 16, 1984. The purpose of
theColloquiumwastohighlightrecentdevelopmentsinthegeneralarea
of Vector Bundles as well as principal bundles on both affine and pro-
jective varieties. Projective modules and quadratic spaces over general
rings werealsoamongthetopics covered bytheColloquium. TheCol-
loquiumwasjointlysponsoredbytheInternationalMathematicalUnion
andtheTataInstituteofFundamentalResearch,andwasfinanciallysup-
portedbythemandtheSirDorabjiTataTrust.
The Organizing Committee for the Colloquium consisted of Pro-
fessors Sir M.F. Atiyah, M.P. Murthy, M.S. Narasimhan, M.S. Raghu-
nathan, S. Ramanan and R. Sridharan. The International Mathematical
Unionwasrepresented byProfessorsAtiyahandNarasimhan
The following mathematicians gave one-hour addresses at the Col-
loquium :
M.F. Atiyah, W. Barth, S.M. Bhatwadekar, J.L. Colliot-The´le`ne, A.
Hirschowitz, G. Horrocks, G.R. Kempf, M.A. Knus, J. Le Potier, H.
Lindel, M. Maruyama, N. Mohan Kumar, S. Mukai, M. Ojanguren, R.
Parimala, S. Ramanan, A. Ramanathan, C.S. Seshadri, V. Srinivas and
G.Trautmann.
REPORT
BesidesthemembersoftheSchoolofMathematicsoftheTataInstitute,
mathematicians from universities and educational institutions in India
andFrancewerealsoinvited toattendtheColloquium.
Thesocial programmefortheColloquium included aTeaPartyonJan-
uary9,aClassicalIndianDance(BharataNatyam)PerformanceonJan-
uary 10, a Concert (Classical Indian Music) on January 12, a Dinner
PartyattheInstituteonJanuary13,anExcursiontoElephantaCaveson
January14,aFilmShowonJanuary15,andaFarewellDinnerPartyon
January 16,1984.
Magnetic Monopoles in Hyperbolic Spaces
By M.F. Atiyah
1 Introduction
1
In recent years, the Penrose twistor transform has been extensively and
successfully used to convert certain problems arising in physics into
problems of algebraic geometry [1]. More precisely, solutions of the
self-dual Yang-Mills equations on R4 (describing ‘instantons’) convert
into holomorphic bundles on the complex projective 3-space P . Simi-
3
larly, solutions oftheBogomolny equation inR3 (describing ‘magnetic
monopoles’)convertintoholomorphicbundlesonTP (thetangentbun-
1
dle of P ) [7]. In this talk, I shall consider the analogous problem, for
1
magneticmonopoles, whentheEuclidean3-spaceR3 isreplacedbythe
hyperbolic 3-space H3. Twistormethods stillapply andsothen‘hyper-
bolicmonopoles’ canalsobedescribed byholomorphic bundles.
Themotivationforstudyingthehyperboliccaseisthat,surprisingly,
itturnsouttobesimplerthantheEuclideancase,whileatthesametime
preserving all its essential features. Moreover, by varying the curva-
ture of hyperbolic space and letting it tend to zero, the Euclidean case
appears as a natural limit of the hyperbolic case. While the details of
this limiting procedure are a little delicate, and need much more care-
fulexaminationthanIshallgivehere,itseemsreasonable toconjecture
that the moduli (or parameter space) of monopoles remains unaltered
by passing to the limit. This conjecture (for SU(2)) receives substan-
tial confirmation from therecent result ofDonaldson [5]onthe moduli
spaceofEuclidean monopoles.
In§2,Iexplainhowhyperbolicmonopolessatisfyingsuitabledecay
conditions at infinity, can essentially be viewed as instantons invariant
1
2 M.F.Atiyah
under a circular rotation. Although this works quite generally for any
compact Lie group G, and for any asymptotic value for Higgs field, I
2 willforsimplicityconcentrateontheSU(2)case. Thisisthecasewhich
(for Euclidean 3-space) has been studied in the greatest detail [7], [8],
[9],[10]anditisthereforethemostusefulforcomparativepurposes. In
particular, Iwillshowhowthetwosplittings(reductiontothetriangular
group) usedbyWard[10]andHitchin[7]havesimple analogues inthe
hyperbolic case.
The twistor picture for S1-invariant instantons is developed in § 3
and then the ‘mini-twistor’ picture is derived in § 4. The analogue of
Hitchin’sspaceTP (aquadriccone)isnowanon-singular quadricsur-
1
face Q, and the analogue of Hitchin’s spectral curve is now a curve on
Q. However, in the hyperbolic space, this spectral curve also has an
additional interpretation in terms of the jumping lines of the instanton
bundle onthe3-dimensional twistorspace.
In § 5, I discuss the limiting process of letting the curvature tend
to zero. In the ‘mini-twistor’ picture, this means that we have a family
Q(t) of non-singular quadrics which degenerate to a quadric cone as
t → 0. Whilethegeneralsituationisfairlyclear,therearemanydetailed
technical questions aboutthislimitingprocesswhichIdonotenterinto
andwhichneedthorough investigation.
Finally in§6, using arecent result ofDonaldson [4]on themoduli
spaceofinstantons,IwillshowthatthemodulispaceofS1-invariantin-
stantons (i.e.ofhyperbolic monopoles) canbeidentified withthespace
ofrationalfunctions(ofonecomplexvariable). Thisresultwasalsode-
rived in [2] in a more general context, but the treatment given here is
somewhat more elementary. Moreover, it enables us to identify the ra-
tional functions assigned toamonopole aspart of thescattering matrix
forHitchin’sdifferentialoperator. Itseemslikelythatasimilarinterpre-
tationholdsformonopolesinEuclideanspace.
3 It will be clear from this introduction that many of the key points
restonresultsofS.K.Donaldson. IamalsoindebtedtohimandtoN.J.
Hitchinformuchvaluable discussion onthesetopics.