Selected titles in This Series
Volume
8 José Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters Introduction to Hodge theory (2002)
7 Jean-Pierre Otal
The hyperbolization theorem for fibered 3-manifolds (2001)
6 Laurent Manivel
Symmetric functions. Schubert polynomials and degeneracy loci (2001) 5 Daniel Alpay
The Schur algorithm, reproducing kernel spaces and system theory (2001) 4 Patrick le Calvez
Dynamical properties of diffeomorphisms of the annulus and of the torus (2000) 3 Bernadette Perrin-Riou
p-adic functions and p-adic representations (2000) 2 Michel Zinsmeister
Thermodynamic formalism and holomorphic dynamical systems (2000) 1 Claire Voisin
Mirror symmetry (1999)
Introduction to Hodge Theory
SMF/AMS TEXTS and MONOGRAPHS • Volume 8 Panoramas et Synthèses • Numéro 3 • 1996
Introduction to Hodge Theory
Jose Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters
Translated by
James Lewis Chris Peters
American Mathematical Society
Société Mathématique de France
(Introduction to Hodge Theory)
by José Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters Originally published in French by Société Mathématique de France.
Copyright @ 1996 Societe Mathematique de France
L 2 Hodge theory and vanishing theorems by Jean-Pierre Demailly and Frobenius and Hodge degeneration by Luc Illusie were translated from the French by James Lewis.
Variations of Hodge structure. Calabi- Yau manifolds and, mirror symmetry by José Bertin and Chris Peters was translated from the French by Chris Peters.
2000 Mathematics Subject Classification. Primary 14C30, 14D07, 14F17, 13A35, 58A14, 14-02, 32-02; Secondary 81-02.
A BSTRACT . Hodge theory is a powerful tool in analytic and algebraic geometry. This book consists of expositions of aspects of modern Hodge theory, with the purpose of providing the nonexpert reader with a clear idea of the current state of the subject. The three main topics are: L 2 Hodge theory and vanishing theorems; Hodge theory in characteristic p; and variations of Hodge structures and mirror symmetry. Each section has a detailed introduction and numerous references. Many open problems are also included. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry. This is the English translation of a volume previously published as volume 3 in the Panoramas et Synthèses series.
Library of Congress Cataloging-in-Publication Data [Introduction à la théorie de Hodge. English.]
Introduction to Hodge theory / Jose Bertin ... [et ah] ; translated by James Lewis, Chris Peters.
p. cm. — (SMF/AMS texts and monographs, ISSN 1525-2302 ; 8) (Panoramas et synthèses ; n. 3, 1996) Includes bibliographical references.
ISBN 0-8218-2040-0
1. Hodge theory. I. Bertin, Jose. II. Series. III. Panoramas et syntheses ; 3.
QA564.15913 2002
516.3'5 — dc21 2002019611
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10 9 8 7 6 5 4 3 2 1 0 7 0 6 0 5 0 4 0 3 0 2
vii CONTENTS
Foreword ix
L² Hodge Theory and Vanishing Theorems 1
J EAN -P IERRE D EMAILLY
0. Introduction 5
Part I. L² Hodge Theory 9
1. Vector bundles, connections and curvature 9
2. Differential operators on vector bundles 12
3. Fundamental results on elliptic operators 14
4. Hodge theory of compact Riemannian manifolds 19
5. Hermitian and Kähler manifolds 24
6. Fundamental identities of Kählerian geometry 27
7. The groups H p,q (X,E) and Serre duality 35
8. Cohomology of compact Kähler manifolds 36
9. The Hodge-Frölicher spectral sequence 42
10. deformations and the semi-continuity theorem 47
Part II. L² estimates and vanishing theorems 53
11. Concepts of pseudoconvexity and of positivity 53
12. Hodge theory of complete Kähler manifolds 60
13. Bochner techniques and vanishing theorems 70
14. L² estimates and existence theorems 73
15. Vanishing theorems of Nadel and Kawamata-Viehweg 75
16. On the conjecture of Fujita 82
17. An effective version of Matsusaka's big theorem 89
Bibliography 95
vii
Frobenius and Hodge Degeneration 99 Luc I LLUSIE
0. Introduction 101
1. Schemes: differentials, the de Rham complex 103
2. Smoothness and liftings 107
3. Frobenius and Cartier isomorphism 113
4. Derived categories and spectral sequences 119
5. Decomposition, degeneration and vanishing theorems in characteristic p > 0 124
6. From characteristic p > 0 to characteristic zero 130
7. Recent developments and open problems 137
8. Appendix: parallelizability and ordinarity 143
Bibliography 147
Variations of Hodge Structure, Calabi-Yau Manifolds and Mirror Symmetry 151 J OSÉ B ERTIN AND C HRIS P ETERS
0. Introduction 155
Part I. Variations of Hodge structures 161
1. Hodge bundles 161
2. Gauss-Manin connection 163
3. Variation of Hodge structures 172
4. Degenerations 179
5. Higgs bundles 187
6. Hodge modules 188
Part II. Mirror symmetry and Calabi-Yau manifolds 193
7. Introduction to mirror symmetry 193
8. Cohomology of hypersurfaces 199
9. Picard-Fuchs equations 205
10. Calabi-Yau threefolds and mirror symmetry 210
11. Relation with mixed Hodge theory 222
Bibliography 229
viii
Foreword
Each of the three chapters collected in this book is concerned with various aspects – important ones in several respects – of Hodge theory. The text is an expanded version, including substantial additions, of lectures presented on the occasion of the meeting “l'Etat de la Recherche” devoted to Hodge theory, that has been held at Université Joseph Fourier in Grenoble from Friday November 25, 1994 till Sunday November 27, under the auspices of the SMF (Société Mathématique de France). The authors wishes would be fulfilled if, in accordance with the general goals of sessions
“l'Etat de la Recherche”, this book could help the nonexpert reader to get a precise idea of the current status of Hodge theory.
The three main subjects developed here (L² Hodge theory and vanishing theorems, Frobenius and Hodge degeneration, Variations of Hodge structures and mirror symmetry) cover a wide range of techniques: elliptic PDE theory, complex differential geometry, algebraic geometry in characteristic p, cohomological and sheaf-theoretic methods, deformation theory of complex varieties, Calabi-Yau manifolds, a few aspects of singularity theory ... This accumulation of tools arising from various fields probably makes the access to the theory rather uneasy for newcomers.
We hope that the present book will greatly facilitate this access: a special effort has been made to approach various themes by their most natural starting point, each of the three chapters being supplemented with a detailed introduction and numerous references. The reader will find precise statements of quite a number of open problems which have been the subject of active research in the last years.
The authors are grateful to SMF and MESR (Ministère de l'Enseignement Supérieur et de la Recherche) for their decisive action – both psychological and financial – without which the Grenoble session “Hodge theory” would probably never have taken place. They address special thanks to the Scientific Committee of Sessions l'Etat de la Recherche, in behalf of its two successive directors Pierre Schapira and Colette Mœglin, as well as to Michèle Audin, Editor in Chief of the Journal “Panoramas et Synthèses”, for her strong encouragement to publish the present manuscript. Finally, they express their gratitude to the referee for his careful reading of the manuscript and a large number of invaluable suggestions.
November 27, 1995
José Bertin*, Jean-Pierre Demailly*. Luc Illusie**, Chris Peters*
* Université de Grenoble I, Institut Fourier, BP 74, 38402 Saint-Martin d'Heres. France
** Université de Paris-Sud, Département de Mathématiques, Bâtiment 425, 91405 Orsay. France
ix
L Hodge Theory and Vanishing Theorems
Jean-PierreDemailly
UniversitedeGrenobleI,
InstitutFourier,BP74,38402
Saint-Martind'Heres,Frane
0. Introdution
TheaimofthesenotesistodesribetwofundamentalappliationsofL 2
Hilbert
spaetehniquestoanalyti oralgebraigeometry: Hodgetheory,and thetheory
of L 2
estimatesfor the operator. Thepointof viewadopted here isessentially
analyti.
Therstpartis foussedonHodgetheoryanditis intendedto berather in-
trodutory. Thusthereaderwillndhereonlythemostelementarytopis,mostly
those dueto W.V.D. Hodgehimself [Hod41℄orto A. Weil[Wei57℄. Hodgethe-
ory, as rst oneived by its reator, onsists of the study of the ohomology of
Riemannian orKahlerianmanifolds,bymeansof adesriptionof harmoniforms
andtheirproperties. Wereferto thetreatmentof J.Bertin-Ch. Peters[BePe95℄
and L.Illusie [Ill95℄for apresentation of moreadvaned topis and appliations
(variationofHodgestruture,appliationofperiods,Hodgetheoryinharateristi
p> 0 :::). Weonsider aRiemannianmanifoldX andaEulideanorHermitian
bundleE overX. Weassumethat E isequippedwithaonnetionD ompatible
withthemetri: Aonnetionis bydenitionadierentialoperatoranalogous to
exteriordierentiation, ating onforms ofarbitrarydegree withvaluesin E, and
satises Leibniz rule for the exterior produt. The Laplae-Beltrami operator is
theself-adjointdierentialoperatorofseondorder
E
=D
E D
E +D
E D
E ,where
D
E
is the Hilbert spae adjoint of D
E
. One easily shows that
E
is an ellipti
operator. The niteness theorem forellipti operators showsthen that the spae
H q
(X;E)ofharmoni q-formswith valuesin E is nite dimensional ifX is om-
pat(wesaythataformuisharmoniif
E
u=0). Ifweassumeinadditionthat
the onnetion satises D 2
E
=0, the operator D
E
ating on forms of alldegrees
denes aomplex alled thede Rham omplex with values in theloal systemof
oeÆientsdenedbyE. Theorrespondingohomologygroupswillbedenotedby
H q
DR
(X;E). ThefundamentalobservationofHodgetheoryisthatanyohomology
lassontainsauniqueharmonirepresentative,sineX isompat. It leadsthen
toanisomorphism,alledtheHodgeisomorphism
(0.1) H
q
DR
(X;E)'H q
DR (X;E):
When the manifold X and the bundle E are holomorphi, there exists aunique
onnetionD
E
alledtheChernonnetion,ompatiblewiththeHermitianmetri
on E and has the following properties: D
E
splits into a sum D
E
= D 0
E +D
00
E
of a onnetion D 0
E
of type (1;0) and a onnetion D 0 0
E
of type (0;1), suh that
D 02
E
= D 00 2
E
= 0 and D 0
E D
00
E +D
00
E D
0
E
= (E) (Chern urvature tensor of the
bundle). TheoperatorD 0 0
E
atingon theforms of bidegree (p;q)denes then for
xedp,aomplexalledtheDolbeaultomplex. WhenX isompat,theDolbeault
ohomology groupsH p;q
(X;E) satisfy a Hodge isomorphism analogous to (0:1),
namely
(0.2) H
p;q
(X;E)'H p;q
(X;E);
where H p;q
(X;E) denotes the spae of harmoni (p;q)-forms with values in E,
relativeto the anti-holomorphiLaplaian 0 0
E
=D 0 0
E D
00
E +D
00
E D
0 0
E
. Byutilizing
thislatterresult,oneeasilyprovestheSerre dualitytheorem
(0.3) H
p;q
(X;E)
'H n p;n q
(X;E
); n=dim X;
whihistheomplexversionofthePoinaredualitytheorem. Theentraltheorem
ofHodgetheoryonernsompatKahlermanifolds: AHermitianmanifold(X;!)
is alled Kahlerian if the Hermitian (1;1)-form ! = i P
j;k
!
jk dz
j
^dz
k
satises
d!=0. Afundamental exampleofaompatKahlerianmanifold isgivenbythe
projetivealgebraimanifolds. IfX isompatKahlerianandifEisaloalsystem
ofoeÆientsonX,theHodgedeompositiontheoremassertsthat
H k
DR
(X;E)= M
p+q=k H
p;q
(X;E) (Hodgedeomposition) (0.4)
H p;q
(X;E)'H q;p
(X;E
): (Hodgesymmetry) (0.5)
Theintrinsiharaterofthedeompositionwillbeshownhereinasomewhatorig-
inalway,viatheutilizationoftheBott-Chernohomologygroups(-ohomology
groups). It follows from these results that the Hodge numbers h p;q
=
dim
C H
p;q
(X;C)satisfythesymmetrypropertyh p;q
=h q;p
=h n p;n q
=h n q;n p
,
and that they are onneted to the Betti numbers b
k
= dim
C H
k
DR
(X;C) by the
relation b
k
= P
p+q=k h
p;q
. A ertain number of other remarkable ohomologial
properties of ompat Kahler manifolds are obtained by means of the primitive
deompositionandthehardLefshetztheorems(whihinturnisaresultoftheex-
isteneofansl(2;C) ationonharmoniforms). Theseresultsallowustodesribe
in a preise way the struture of the Piard group Pi(X) = H 1
(X;O
) in the
Kahlerianase. Inamoregeneralsetting,wedisusstheHodge-Froliherspetral
sequene(thespetralsequeneonnetingDolbeaulttodeRhamohomology),and
weshowhowoneanutilizethisspetralsequeneto obtainsomegeneralresults
on the Hodgenumbersh p;q
of ompat omplex manifolds. Finally, weestablish
thesemi-ontinuityofthedimensionoftheohomologygroupsH q
(X
t
;E
t
)ofbun-
dles arisingfrom a proper and smooth holomorphibration X ! S (result due
toKodaira-Spener),andwededuefrom itthattheHodgenumbersh p;q
(X
t )are
onstantifthebersX
t
areKahlerian(invarianeoftheh p;q
underdeformations);
theholomorphinatureoftheHodgeltrationF p
H k
(X
t
;C)=
rp H
r;k r
(X
t
;C)
relativeto theGauss-Maninonnetionisprovenbymeansofthetheoremonthe
ohereneofdiretimages,appliedtotherelativedeRhamomplex
X=S
ofX!S.
Intheseondpart,afterreallingsomeoftherelevantoneptsofpositivityand
pseudoonvexity,weestablishtheBohner-Kodaira-Nakanoidentityonnetingthe
Laplaians 0
E and
0 0
E
. Theidentity in questionfurnishesan expliitexpression
of thedierene 00
E
0
E
in termsof theurvature (E) of thebundle. Under
adequatehypothesis(weakpseudoonvexityofX,positivityoftheurvatureofE),
onearrivesaprioriattheestimate
jjD 00
E ujj
2
+jjD 0 0
E ujj
2
Z
X (z)juj
2
dV(z)
where is a positive funtion depending on the eigenvalues of urvature. The
inequalityisvalid hereforanyform uofbidegree (n;q); n=dimX; q1,with
values in E, u belonging to the Hilbert spae domains of D 00
E
and D 00
E
. By an
argumentofHilbertspaedualityonededuesfromthisthefollowingfundamental
theorem,essentiallyduetoHormander[Hor65℄and Andreotti-Vesentini [AV65℄:
0.6. Theorem. Let (X;!) be a Kahler manifold, dimX = n. Assume that
eigenvaluesof the urvatureformi(E)withrespettothe metri! ateahpoint
x2X,satisfy
1
(x)
n (x):
Further,suppose thatthe urvatureissemi-positive, i.e.
1
0everywhere. Then
for any formg2L 2
(X;
n;q
T
X
E)suhthat
D 00
E
g=0 and Z
X (
1
++
q )
1
jgj 2
dV
!
<+1;
thereexistsf 2L 2
(X;
n;q 1
T
X
E)suhthat
D 00
E
f =g and Z
X jfj
2
dV
!
Z
X (
1
++
q )
1
jgj 2
dV
! :
An importantobservation isthat the abovetheorem still remainsvalid when
themetrihof E aquiressingularities. Themetrihisthen givenin eahhart
by a weight e 2'
assoiated to a plurisubharmoni funtion ' (by denition '
ispsh ifthematrixof seondderivatives( 2
'=z
j z
k
), alulatedin thesense of
distributions,issemi-positiveateahpoint). TakingintoaountTheorem(0.6),it
isnaturaltointroduethemultiplieridealsheafJ(h)=J('),madeupofthegerms
ofholomorphifuntionsf 2O
X ;x
suhthat R
V jfj
2
e 2'
onvergesinasuÆiently
smallneighbourhoodV ofx. AreentresultofA.Nadel[Nad89℄guaranteesthat
J(') is alwaysaoherentanalyti sheaf, whatever thesingularities of '. In this
ontext, onededues from (0.6) the following qualitative version, onerning the
ohomology with valuesin theoherentsheaf O(K
X
E)J(h) (K
X
=
n
T
X
beingtheanonial bundleofX).
0.7. Nadel Vanishing Theorem ([Nad89℄, [Dem93b℄). Let (X;!) be a
weakly pseudoonvexKahlermanifold, andletE beaholomorphilinebundleover
X equipped with a singular Hermitian metri h of weight '. Suppose that there
exists a ontinuous positive funtion on X suh that the urvature satises the
inequality i
h
(E)! inthe sense ofurrents. Then
H q
(X;O(K
X
E)J(h))=0 for allq1:
Inspiteoftherelativesimpliityofthetehniquesinvolved,itisanextremely
powerfultheorem,whihbyitselfontainsmanyofthemostfundamentalresultsof
analytioralgebraigeometry. Theorem(0.7)alsoontainsthesolutionoftheLevi
problem (equivalene ofholomorphionvexityandpseudoonvexity),thevanish-
ing theorems of Kodaira-Serre, Kodaira-Akizuki-Nakano and Kawamata-Viehweg
forprojetivealgebraimanifolds,aswellastheKodairaembeddingtheoremhar-
aterizingthese manifoldsamong theompatomplexmanifolds. Byitsintrinsi
harater, the \analyti" statement of Nadel's theorem appears useful even for
purelyalgebraiappliations. (Thealgebraiversionofthetheorem,knownasthe
Kawamata-Viehwegvanishing theorem, utilizesthe resolutionof singularities and
doesnotgivesuhaleardesriptionofthemultipliersheafJ(h).) Inareentwork
[Siu96℄,Y.T.Siuhasshown thefollowingremarkableresult,byutilizingonlythe
Riemann-Roh formula and an indutive Noetherian argument forthe multiplier
sheaves. Thetehniqueisdesribedinx16(withsomeimprovementsdevelopedin
0.8. Theorem [Siu96℄,[Dem96℄). LetX be aprojetive manifold andL an
amplelinebundle(i.e. haspositiveurvature)onX. Then thebundleK 2
X L
m
isvery amplefor mm
0
(n)=2+ 3n+1
n
,where n=dimX.
The importane of having an eetive bound for the integer m
0
(n) is that
oneanalsoobtainembeddingsofmanifoldsX in projetivespae,withapreise
ontrolofthedegreeoftheembedding. Asaonsequeneofthis,onehasarather
simpleproofofasigniantnitenesstheorem,namely\Matsusaka's big theorem"
(f. [Mat72℄,[KoM83℄,[Siu93℄,[Dem96℄):
0.9. Matsusaka's BigTheorem. LetX beaprojetive manifold andL an
amplelinebundleoverX. Thereexistsanexpliitboundm
1
=m
1 (n;L
n
;K
X L
n 1
)
dependingonlyonthe dimensionn=dimX andonthersttwooeÆientsof the
Hilbertpolynomialof L,suhthat mLisvery amplefor mm
1 .
Fromthistheorem,oneeasilydeduesnumerousnitenessresults,inpartiular
thefatthatthereexistonlyanitenumberoffamiliesofdeformationsofpolarized
projetivemanifolds(X;L),whereLisanamplelinebundlewithgivenintersetion
numbersL n
andK
X L
n 1
.
1. Vetorbundles, onnetions and urvature
Thegoalofthis setionis to reallsomebasidenitionsof Hermitiandier-
entialgeometry withregardtotheoneptsof onnetion,urvatureandtherst
Chernlassoflinebundles.
1.A. Dolbeault ohomologyand the ohomology of sheaves. Assume
given X aC-analyti manifold of dimensionn. Wedenote by p;q
T
X
thebundle
of dierential forms of bidegree (p;q) on X, i.e. dierential forms whih an be
written
u=
X
jIj=p;jJj=q u
I;J dz^z
J
; dz
I :=dz
i1
^^dz
ip
; dz
J :=dz
j1
^^dz
jq
;
where (z
1
;:::;z
n
) are loal holomorphi oordinates, and where I = (i
1
;::: ;i
p )
and J =(j
1
;:::;j
q
)are multi-indies (inreasing sequenes ofintegers in the in-
terval[1;:::;n℄,with lengthsjIj=p; jJj=q). LetA p;q
bethesheafofgermsof
dierentialformsofbidegree(p;q)withomplexvaluedC 1
oeÆients. Wereall
thattheexteriorderivativeddeomposesintod=d 0
+d 00
where
d 0
u =
X
jIj=p;jJj=q;1k n u
I;J
z
k dz
k
^dz
I
^dz
J
;
d 00
u =
X
jIj=p;jJj=q;1k n u
I;J
z
k dz
k
^dz
I
^dz
J
areoftype(p+1;q),(p;q+1)respetively. ThewellknownDolbeault-Grothendiek
Lemmaassertsthatalld 00
-losedformsoftype(p;q)withq>0areloallyd 00
-exat
(this is the analoguefor d 00
of the usual PoinareLemma for d, see for example
[Hor66℄). In other words, the omplex of sheaves (A p;
;d 00
) is exat in degree
q>0: and indegreeq=0,Kerd 00
isthesheaf p
X
ofgermsofholomorphiforms
ofdegreeponX.
More generally, if E is aholomorphi vetorbundle of rankr over X, there
existsanaturaloperatord 00
atingonthespaeC 1
(X;
p;q
T
X
E)ofC 1
(p;q)-
forms with valuesin E. Indeed, ifs = P
1r s
e
isa (p;q)-form expressed in
termsofaloalholomorphiframeofE,weandened 00
s:=
P
(d 0 0
s
)e
;byrst
observing that thetransition matries orresponding to a hange of holomorphi
frame are holomorphi, and whih ommute with the operation of d 00
. It then
followsthattheDolbeault-GrothendiekLemmastillholdsforformswithvaluesin
E. Foreveryintegerp=0; 1;:::;n,theDolbeaultohomologygroupsH p;q
(X;E)
are dened asbeingthe ohomologyof theomplexof globalforms of type(p;q)
(indexedbyq):
(1.1) H
p;q
(X;E)=H q
(C 1
(X;
p;
T
X E)):
There is thefollowingfundamental resultof sheaf theory(de Rham-Weil Isomor-
phismTheorem): Let(L
;Æ)bearesolutionofasheafF byaylisheaves,i.e. a
omplex(L
;Æ)givenbyanexatsequeneofsheaves
0!F j
!L 0
Æ 0
!L 1
!!L q
Æ q
!L q+1
!;
whereH s
(X;L q
)=0forallq0ands1. (Toarriveatthis latteronditionof
ayliity, itis enough forexample that theL q
are asqueorsoft, forexamplea
sheafofmodulesoverthesheafofringsC 1
.) Thenthereisafuntorialisomorphism
(1.2) H
q
( (X;L
))!H q
(X;F):
Weapplythisinthefollowingsituation. LetA p;q
(E)bethesheafofgermsofC 1
setionsof p;q
T
X
E. Then (A p;
(E);d 00
)isaresolutionoftheloally freeO
X -
module p
X
O(E)(Dolbeault-GrothendiekLemma),andthesheavesA p;q
(E)are
ayliasC 1
-modules. Aordingto(1.2),weobtain
1.3. DolbeaultIsomorphismTheorem(1953). Forallholomorphivetor
bundlesE on X,thereexistsaanonial isomorphism
H p;q
(X;E)'H p
(X;
p
X
O(E)):
IfX isprojetivealgebraiandifEisanalgebraivetorbundle,thetheorem
ofSerre(GAGA)[Ser56℄showsthatthealgebraiohomologygroupsH q
(X;
p
X
O(E))omputedviatheorrespondingalgebrai sheafin theZariskitopologyare
isomorphi to the orresponding analyti ohomology groups. Sine our point of
viewhereisexlusivelyanalyti,wewillnolongerneedtorefertothisomparison
theorem.
1.B. Connetions on dierentiable manifolds. Assume given a real or
omplexC 1
vetorbundleEofrankronadierentiablemanifoldM oflassC 1
.
AonnetionD onE isalineardierentialoperatoroforder1
D:C 1
(M;
q
T
M
E)!C 1
(M;
q+1
T
M
E)
suhthatD satisesLeibnitzrule:
(1.4) D(f^u)=df^u+( 1)
degf
f^Du
forallformsf 2C 1
(M;
p
T
M
); u2C 1
(X;
q
T
M
E). Onanopenset M
whereE admitsatrivialization :E
j
'
!C
r
, aonnetionDanbewritten
Du'
du+ ^u
where 2C 1
(;
1
T
M
Hom(C r
;C r
))is agivenmatrixof1-formsandwhered
atsomponentwiseonu'
(u
)
1r
. Itistheneasyto verifythat
D 2
u'
(d + ^ )^uon:
SineD 2
isagloballydened operator,thereexistsaglobal2-form
(1.5) (D)2C
1
(M;
2
T
M
Hom(E;E))
suhthatD 2
u=(D)^uforanyformuwithvaluesinE. This2-formwithvalues
inHom(E;E)isalledtheurvaturetensoroftheonnetionD.
NowsupposethatE isequippedwithaEulideanmetri(resp. Hermitian)of
lass C 1
and that the isomorphismE
j
'C
r
isgivenby aC 1
frame(e
).
Wethenhaveaanonialbilinearpairing,(resp. sesquilinear).
C 1
(M;
p
T
M
E)C 1
(M;
q
T
M
E)!C 1
(M;
p+q
T
M C) (1.6)
(u;v)7!fu;vg
givenby
fu;vg= X
; u
^v
he
;e
i; u= X
u
e
; v= X
v
e
:
TheonnetionD isalled Hermitianifitsatisestheadditionalproperty
dfu;vg=fDu;vg+( 1) degu
fu;Dvg:
Byassumingthat(e
)isorthonormal,oneeasilyveriesthatDisHermitianifand
onlyif
= . Inthisase(D)
= (D),therefore
i(D)2C 1
(M;
2
T
M
Herm(E;E)):
1.7. A partiular ase. For a omplex line bundle L (a omplex vetor
bundleofrank1),theonnetionform ofaHermitianonnetionDanbetaken
to bea1-formwith purely imaginaryoeÆients =iA (A real). Wethen have
(D)=d =idA. Inpartiulari(L)isalosed2-form. TherstChernlass of
Lisdened tobetheohomologylass
1 (L)
R
=
i
2 (D)
2H 2
DR (M;R):
This ohomologylassis independentofthehoieof onnetion,sineanyother
onnetion D
1
diers by a global 1-form, D
1
u = Du+B^u, so that (D
1 ) =
(D)+dB. It iswell-knownthat
1 (L)
R
is theimagein H 2
(M;R) ofan integral
lass
1
(L) 2 H 2
(M;Z). Indeedif A = C 1
is thesheaf of C 1
funtions on M,
thenviatheexponentialexatsequene
0!Z!A e
2 i
!A
!0;
1
(L) an be dened in
Ceh ohomology as the image of the oyle fg
jk g 2
H 1
(M;A
)deningLbytheoedgemapH 1
(M;A
)!H 2
(M;Z).Seeforexample
[GH78℄formoredetails.
1.C.Connetionson omplexmanifolds. Wenowstudythoseproperties
ofonnetionsgovernedbytheexisteneofaomplexstrutureonthebase mani-
fold. IfM=X isaomplexmanifold,anyonnetionD onaomplexC 1
vetor
bundle E anbe split in aunique manner as asum of a(1;0)-onnetion and a
(0;1)-onnetion, D =D 0
+D 00
. Inaloal trivialization givenby aC 1
frame,
oneanwrite
D 0
u '
d
0
u+ 0
^u;
(1.8 0
)
D 00
u '
d
00
u+ 00
^u;
(1.8 00
)
with = 0
+ 00
. TheonnetionisHermitianifandonlyif 0
= (
00
)
relative
to any orthonormal frame. As a onsequene, there exists a unique Hermitian
00
NowsupposethatthebundleEisendowedwithaholomorphistruture. The
unique Hermitian onnetion whose omponentD 00
is the operator d 00
dened in
x1.A is alled the Chern onnetion of E. With respet to a loal holomorphi
frame (e
) ofE
j
,the metriis givenby theHermitian matrixH =(h
) where
h
=he
;e
i. Wehave
fu;vg= X
; h
u
^v
= u y
^Hv;
whereu y
isthetransposematrixofu,andaneasyalulationgives
dfu;vg=(du) y
^Hv+( 1) degu
u y
^(dH^v+Hdv)
=(du+H 1
d 0
H^u) y
^Hv+( 1) degu
u y
^(dv+H 1
d 0
H^v);
by using the fat that dH =d 0
H +d 0
H and H y
= H. Consequently the Chern
onnetionD oinideswiththeHermitianonnetiondened by
(
Du '
du+H 1
d 0
H^u;
D 0
'
d
0
+H 1
d 0
H^=H 1
d 0
(H); D
00
=d 0 0
: (1.9)
TheserelationsshowthatD 02
=D 002
=0. ConsequentlyD 2
=D 0
D 00
+D 00
D 0
,and
theurvaturetensor(D)isoftype(1;1). Sined 0
d 00
+d 00
d 0
=0,weobtain
(D 0
D 00
+D 00
D 0
)u'
H
1
d 0
H^d 0 0
u+d 00
(H 1
d 0
H^u)=d 00
(H 1
d 0
H)^u:
1.10. Proposition. The Chernurvaturetensor (E):=(D) satises
i (E)2C 1
(X;
1;1
T
X
Herm(E;E)):
If : E
! C
r
is a holomorphi trivialization and if H is the Hermitian
matrixrepresentativeof themetri alongthe bersof E
,then
i(E)'
id
0 0
(H 1
d 0
H) on :
If(z
1
;::: ;z
n
)areholomorphioordinatesonXandif(e
)
1r
isanorthog-
onalframeofE,oneanwrite
(1.11) i(E)=
X
1j;k n;1;r
jk dz
j
^dz
k e
e
;
where(
jk
(x))aretheoeÆientsoftheurvaturetensorofEatanypointx2X.
2. Dierentialoperators onvetor bundles
Werstdesribesomebasioneptsonerningdierentialoperators(symbol,
omposition,elliptiity,adjoint),in thegeneralontextofvetorbundles. Assume
given M amanifoldof dierentiable lassC 1
; dim
R
M =m, and E; F givenK
vetorbundles onM, overtheeld K =R orK =C suh thatrankE =r,rank
0
2.1. Definition. A(linear)dierentialoperatorofdegreeÆfromEtoF isa
K-linearoperatorP :C 1
(M;E)!C 1
(M;F); u7!Puoftheform
Pu(x)= X
jjÆ a
(x)D
u(x);
where E
' K
r
; F
' K
r 0
are loal trivializations on an open hart
M with loal oordinates (x
1
;:::;x
m
), and the oeÆients a
(x) are r 0
r matries (a
(x))
1r 0
;1r
with C 1
oeÆients on . One writes here
D
=(=x
1 )
1
(=x
m )
m
asusual,andthematriesu=(u
)
1r ,D
u=
(D
u
)
1r
areviewedasolumn vetors.
Ift2K isaparameterandf 2C 1
(M;K); u2C 1
(M;E),aneasyalulation
showsthate tf(x)
P(e tf(x)
u(x))isapolynomialofdegreeÆint,oftheform
e tf(x)
P(e tf(x)
u(x))=t Æ
P
(x;df(x))u(x) + terms
j (x)t
j
ofdegreej<Æ;
where
P
isahomogeneouspolynomialmapT
M
!Hom(E;F)dened by
(2.2) T
M;x
37!
P
(x;)2Hom(E
x
;F
x
);
P
(x;)= X
jj=Æ a
(x)
:
Then
P
(x;) is a C 1
funtion of the variables (x;) 2 T
M
, and this funtion
is independent of thehoieof oordinatesortrivialization usedfor E; F.
P is
alled theprinipalsymbolofP. Theprinipalsymbolofaomposition QÆP of
dierentialoperatorsissimplytheprodut.
(2.3)
QÆP
(x;)=
Q (x;)
P (x;);
alulatedasaprodutofmatries. Thedierentialoperatorsforwhihthesymbols
areinjetiveplayaveryimportantrole:
2.4. Definition. A dierentialoperatorP is saidto beelliptiif
P
(x;)2
Hom(E
x
;F
x
)isinjetiveforallx2M and2T
M;x nf0g.
Let us now assumethat M is orientedand assume given aC 1
volume form
dV(x)=(x)dx
1
^^dx
m
,where(x)>0isaC 1
density. IfEisaEulideanor
Hermitianvetorbundle,weandeneaHilbertspaeL 2
(M;E)ofglobalsetions
withvaluesinE,beingthespaeofformsuwithmeasurableoeÆientswhihare
squaresummablesetionswithrespettothesalarprodut
jjujj 2
= Z
M ju(x)j
2
dV(x);
(2.5)
hhu;vii= Z
M
hu(x);v(x)idV(x); u;v2L 2
(M;E):
(2.5 0
)
2.6. Definition. IfP:C 1
(M;E)!C 1
(M;F)isadierentialoperatorand
ifthebundles E; F areEulideanorHermitian, there existsauniquedierential
operator
P
:C 1
(M;F)!C 1
(M;E);
alled the formal adjointof P, suh that for allsetions u2 C 1
(M;E) and v 2
C 1
(M;F)onehasanidentity
hhPu;vii=hhu;P
vii; wheneverSuppu \Supp v M:
Proof. Theuniqueness is easy to verify, beingaonsequeneof thedensity
of C 1
forms with ompat support in L 2
(M;E). By a partition of unity argu-
ment, we redue the veriation of the existene of P
to the proof of its loal
existene. NowletPu(x)=
jjÆ a
(x)D
u(x)bethedesriptionofP relativeto
thetrivializationsofE; F assoiatedto anorthonormalframeand tothesystem
ofloal oordinatesonanopenset M. ByassumingSuppu\Suppv,
integrationbypartsgives
hhPu;vii= Z
X
jjÆ;;
a
D
u
(x)v
(x)(x)dx
1
;:::;dx
m
= Z
X
jjÆ;;
( 1) jj
u
(x)D
((x)a
v
(x) dx
1
;:::;dx
m
= Z
hu;
X
jjÆ ( 1)
jj
(x) 1
D
(x)a y
v(x)
idV(x):
Wethus seethatP
exists,andisdened inauniquewayby
(2.7) P
v(x)= X
jjÆ ( 1)
jj
(x) 1
D
(x)a y
v(x)
:
Formula(2.7)showsimmediatelythat theprinipal symbolofP
isgivenby
(2.8)
P (x;
)=( 1) Æ
X
jj=Æ a
y
=( 1) Æ
P (x;)
:
Ifrank E =rankF, theoperator P is ellipti ifandonly if
P
(x;)is invertible
for6=0,thereforetheelliptiityofP isequivalenttothat ofP
.
3. Fundamentalresultson ellipti operators
WeassumethroughoutthissetionthatMisaompatorientedC 1
manifold
of dimension m, with volume form dV. Let E ! M be aC 1
Hermitian vetor
bundleofrankr onM.
3.A. Sobolev spaes. For anyreal numbers, we dene the Sobolev spae
W s
(R m
)to be theHilbert spaeof tempereddistributions u2S 0
(R m
)suh that
theFouriertransformu^isaL 2
lo
funtion satisfyingtheestimate
(3.1) jjujj
2
s
= Z
R m
(1+jj 2
) s
j^u()j 2
d() < +1:
Ifs2N, wehave
jjuj 2
s
Z
R m
X
jjs jD
u(x)j 2
d(x);
thereforeW s
(R m
)istheHilbert spaeof funtions usuh thatallthe derivatives
D
uoforderjjsareinL 2
(R m
).
Moregenerally,wedenotebyW s
(M;E)theSobolevspaeofsetionsu:M!
E whose omponents are loally in W s
(R m
) on all open harts. More preisely,
hoose a nite subovering (
j
) of M by open oordinate harts
j ' R
m
on
Consideranorthonormalframe(e
j;
)
1r ofE
j
andwriteuintermsofits
omponents,i.e. u= P
u
j;
e
j;
. Wethenset
jjujj 2
s
= X
j;
jj
j u
j;
jj 2
s
where (
j
) is a \partition of unity" subordinate to (
j
), suh that P
2
j
= 1.
Theequivaleneofnormsjj jj
s
isindependentofhoiesmade. Wewillneedthe
following fundamental fats, that the reader will be able to nd in many of the
speializedworksdevotedto thetheoryofpartial dierentialequations.
3.2. Sobolevlemma. Foranintegerk2N andanyrealnumberssk+ m
2 ,
wehave W s
(M;E)C k
(M;E)andthe inlusion isontinuous.
Itfollowsimmediatelyfrom theSololevlemma that
\
s0 W
s
(M;E)=C 1
(M;E);
[
s0 W
s
(M;E)=D 0
(M;E):
3.3. Rellih lemma. For allt>s,the inlusion
W t
(M;E),!W s
(M;E)
isaompat linearoperator.
3.B.Pseudodierentialoperators. IfP = P
jjÆ a
(x)D
isadierential
operatoronR m
,theFourierinversionformulagives
Pu(x)= Z
R m
X
jjÆ a
(x)(2i )
^ u()e
2ix
d(); 8u2D(R m
);
whereu()^ = R
R m
u(x)e 2ix
d(x) istheFouriertransformofu. Weall
(x;)= X
jjÆ a
(x)(2i)
;
thesymbol(ortotalsymbol) ofP.
A pseudodierential operator is an operator Op
dened by a formulaof the
type
(3.4) Op
(u)(x)= Z
R m
(x;)^u ()e 2ix
d(); u2D(R m
);
where belongs to a suitable lass of funtions on T
R
m. The standard lass of
symbolsS Æ
(R m
)isdened asfollows: Assume givenÆ2R; S Æ
(R m
)isthelassof
C 1
funtions(x;)onT
R m
suhthatforany; 2N m
andanyompatsubset
KR m
onehasanestimate
(3.5) jD
x D
(x;)jC
;
(1+jj) Æ jj
; 8(x;)2KR m
;
where Æ2R isregardedas the\degree"of. ThenOp
(u) isawell dened C 1
funtiononR m
,sineu^belongstothelassS(R m
)offuntionshavingrapiddeay.
in abundle F overa ompat manifold M, weintrodue the analogous spae of
symbolsS Æ
(M;E;F). TheelementsofS Æ
(M;E;F)arethefuntions
T
M
3(x;)7!(x;)2Hom(E
x
;F
x )
satisfyingondition(3.5)inalloordinatesystems. Finally,wetakeanitetrivial-
izingover(
j
)ofM anda\partitionofunity" (
j
)subordinateto
j
suh that
P
2
j
=1,andwedene
Op
(u)=
X
j Op
(
j
u); u2C 1
(M;E);
in away whih redues thealulationstothe situationof R m
. The basiresults
pertainingtothetheoryofpseudodierentialoperatorsaresummarizedbelow.
3.6. Existeneofextensionsto thespaesW s
. If2S Æ
(M;E;F),then
Op
extendsuniquelytoaontinuouslinearoperator
Op
:W
s
(M;E)!W s Æ
(M;F):
Inpartiularif2S 1
(M;E;F):=
T
S Æ
(M;E;F),thenOp
isaontinuous
operator sending an arbitrary distributional setionof D 0
(M;E) into C 1
(M;F).
Suhanoperator isalled aregularoperator. It isastandardresultin thetheory
of distributions that the lass R of regular operators oinides with the lass of
operators dened bymeans of aC 1
kernelK(x;y)2 Hom(E
y
;F
x
). That is, the
operatorsoftheform
R:D 0
(M;E)!C 1
(M;F); u7!R u; R u(x)= Z
M
K(x;y)u(y)dV(y):
Conversely,ifdV(y)=(y)dy
1 dy
m on
j
andifwewriteR u= P
R (
j
u),where
(
j
) isapartition of unity, theoperatorR (
j
) is thepseudodierentialoperator
assoiatedtothesymbol dened bythepartial Fouriertransform
(x;)= (y)
j
(y)K(x;y)
^
y
(x;); 2S 1
(M;E;F):
Whenoneworkswithpseudodierentialoperators,itisustomarytoworkmodulo
theregularoperatorsand toallowoperatorsmoregenerallyof theform Op
+R
whereR2Risanarbitraryregularoperator.
3.7. Composition. If 2 S Æ
(M;E;F) and 0
2 S Æ
0
(M;F;G); Æ; Æ 0
2 R,
thereexistsasymbol 0
}2S Æ+Æ
0
(M;E;G) suhthat Op
0ÆOp
=Op
0
} mod
R. Moreover
0
}
0
2S Æ+Æ
0
1
(M;E;G):
3.8. Definition. A pseudodierentialoperator Op
of degreeÆ isalled el-
liptiifitanbedenedbyasymbol 2S Æ
(M;E;F)suhthat
j(x;)ujjj Æ
juj; 8(x;)2T
M
; 8u2E
x
forjjlargeenough,theestimatebeinguniform forx2M.
IfE andF havethesamerank,theelliptiityonditionimpliesthat(x;)is
invertiblefor large. Bytakingasuitabletrunatingfuntion ()equalto1for
large,oneseesthat thefuntion 0
(x;)=()(x;) 1
denes asymbolinthe
spaeS Æ
(M;F;E), andaordingto (3.8)wehaveOp
0 ÆOp
=Id+Op
; 2
1
expansionId + }2
++( 1) j
}j
+. Itislearthenthatoneobtainsan
inverseOp
} 0
of Op
modulo R. An easy onsequeneof thisobservation isthe
following:
3.9. Ga rding inequality. Assume given P : C 1
(M;E) ! C 1
(M;F) an
ellipti dierential operator of degree Æ, where rankE = rankF = r, and let
~
P
beanextensionofP withdistributionaloeÆientsetions. Forallu2W 0
(M;E)
suhthat
~
Pu2W s
(M;F), onethenhasu2W s+Æ
(M;E)and
jjujj
s+Æ
C
s (jj
~
Pujj
s +jjujj
0 );
whereC
s
isapositiveonstantdependingonlyons.
Proof. SineP isellipti,thereexists asymbol2S Æ
(M;F;E)suh that
Op
Æ
~
P = Id+R ; R 2 R. Then jjOp
(v)jj
s+Æ
Cjjvjj
s
by applying (3.6).
Consequently, in setting v =
~
Pu, we see that u = Op
(
~
Pu) R u satises the
desiredestimate.
3.C. Finiteness theorem. We onlude this setion with the proof of the
followingfundamental nitenesstheorem, whih isthestartingpointofL 2
Hodge
theory.
3.10. Finiteness Theorem. Assume given E; F Hermitian vetor bun-
dles on a ompat manifold M, suh that rank E = rank F = r; and given
P :C 1
(M;E)!C 1
(M;F)an ellipti dierential operator of degreeÆ. Then:
i)KerP isnitedimensional.
ii) P(C 1
(M;E)) is losed and of nite odimension in C 1
(M;F); moreover, if
P
isthe formaladjoint of P,there existsadeomposition.
C 1
(M;F)=P(C 1
(M;E))KerP
asan orthogonal diret sumin W 0
(M;F)=L 2
(M;F).
Proof. (i)The Garding inequality showsthat jjujj
s+Æ
C
s jjujj
0
for allu2
KerP. By the Sobolev Lemma, this implies that KerP is losed in W 0
(M;E).
Moreover,thejj jj
0
-losedunitballofKerPisontainedinthejjjj
Æ
-ballofradius
C
0
,thereforeitisompataordingtotheRellihLemma. RieszTheoremimplies
thatdimKerP <+1.
(ii)Werstshowthat theextension
~
P :W s+Æ
(M;E)!W s
(M;F)
haslosedimage foralls. Forany>0,there existsanite numberof elements
v
1
;:::;v
N 2W
s+Æ
(M;F); N =N(),suh that
(3.11) jjujj
0
jjujj
s+Æ +
N
X
j=1 jhhu;v
j ii
0 j:
Indeedtheset:
K
(vj)
=
u2W s+Æ
(M;F); jjujj
s+Æ +
N
X
jhhu;v
j ii
0 j1
;
is relatively ompat in W 0
(M;F) and T
(vj) K
(vj)
= f0g. It follows that there
are elements (v
j
) suh that K
(vj)
are ontained in the unit ball of W 0
(M;E),
as required. Substituting the main term jjujj
0
given by (3.11) in the Garding
inequality;weobtain
(1 C
s )jjujj
s+Æ
C
s
jj
~
Pujj
s +
N
X
j=1 jhhu;v
j ii
0 j
:
DeneT =fu2W s+Æ
(M;E); u?v
j
; 1jngandput=1=2C
s
. Itfollows
that
jjujj
s+Æ
2C
s jj
~
Pujj
s
; 8u2T:
Thisimpliesthat
~
P(T)islosed. Asaonsequene
~
P(W s+Æ
(M;E))=
~
P(T) + Vet
~
P(v
1 );:::;
~
P(v
N )
islosed inW s
(M;E). Considernowtheases=0. SineC 1
(M;E)is densein
W Æ
(M;E),weseethat inW 0
(M;E)=L 2
(M;E),onehas
~
P W
Æ
(M;E)
?
=
P C 1
(M;E)
?
=Ker
~
P
:
Wehavethusproventhat
(3.12) W
0
(M;E)=
~
P W
Æ
(M;E)
Ker
~
P
:
Sine P
is also ellipti, it follows that Ker
~
P
is nite dimensional and that
Ker
~
P
= KerP
is ontained in C 1
(M;F). Byapplying the Garding inequal-
ity,thedeomposition formula(3.12) gives
W s
(M;E)=
~
P W
s+Æ
(M;E)
KerP
; (3.13)
C 1
(M;E)=P C 1
(M;E)
KerP
: (3.14)
WenishthissetionbytheonstrutionoftheGreen'soperatorassoiatedto
aself-adjointelliptioperator.
3.15. Theorem. Assumegiven E a Hermitianvetor bundle of rankr ona
ompat manifold M,and P :C 1
(M;E)!C 1
(M;E)a self-adjoint ellipti dif-
ferentialoperatorofdegreeÆ. Then ifH denotestheorthogonalprojetion operator
H :C 1
(M;E)!KerP,there existsaunique operator Gon C 1
(M;E)suhthat
PG + H =GP + H = Id;
moreoverGisapseudo-dierential operator ofdegree Æ,alledthe Green'soper-
ator assoiatedtoP.
Proof. AordingtoTheorem3.10,KerP =KerP
isnitedimensionaland
ImP=(KerP)
?
. ItthenfollowsthattherestritionofPto(KerP)
?
isabijetive
operator. Onedenes Gto be 0P 1
relativeto the orthogonaldeomposition
C 1
(M;E)=KerP (KerP)
?
. TherelationsPG+H =GP +H =Idarethen
obvious, aswellastheuniqueness ofG. Moreover,Gisontinuousin theFrehet
1
thefat thatthere existsapseudo-dierentialoperatorQof order Æwhihisan
inverseof P modulo R,i.e. PQ=Id+R ; R2R. Itthenfollowsthat
Q=(GP+H)Q=G(Id+R )+HQ=G+GR+HQ;
whereGRandHGareregular. (H isaregularoperatorofniterankdenedbythe
kernel P
'
s (x)'
s
(y),if('
s
)isabasisofeigenfuntionsofKerPC 1
(M;E).)
ConsequentlyG=QmodRandGisapseudodierentialoperatoroforder Æ.
3.16. Corollary. Underthehypothesesof3.15,theeigenvaluesofP forma
realsequene
k
suhthatlim
k !+1 j
k
j=+1,the eigenspaesV
k
of P arenite
dimensional, andonehas aHilbert spaediretsum
L 2
(M;E)= d M
k V
k :
Foranyintegerm2N, anelementu= P
k u
k 2L
2
(M;E)isinW mÆ
(X;E)ifand
onlyif P
j
k j
2m
jju
k jj
2
<+1.
Proof. TheGreen's operatorextendsto aself-adjointoperator
~
G:L 2
(M;E)!L 2
(M;E)
whihfatorsthroughW Æ
(M;E),andisthereforeompat. Thisoperatordenes
an inverse to
~
P : W Æ
(M;E) ! L 2
(M;E) on (KerP)
?
. The spetral theory of
ompat self-adjoint operators shows that the eigenvalues
k of
~
G form a real
sequene tending to 0and that L 2
(M;E) is a diret sum of Hilbert eigenspaes.
Theorrespondingeigenvaluesof
~
P are
k
= 1
k if
k
6=0andaordingto the
elliptiityofP
k
Id,theeigenspaesV
k
=Ker(P
k
Id)arenitedimensional
and ontained in C 1
(M;E). Finally, if u = P
k u
k 2 L
2
(M;E), the Garding
inequalityshowsthatu2W mÆ
(M;E)ifandonlyif
~
P m
u2L 2
(M;E)=W 0
(M;E),
whiheasilygivestheondition P
j
k j
2m
jju
k jj
2
<+1.
4. Hodge theoryof ompat Riemannianmanifolds
The establishment of Hodge theory as a well developed subjet, was arried
outby W.V.D Hodgeduring the deade 1930-1940(see [Hod41℄, [DR55℄). The
prinipal goal of the theory is to desribethe de Rham ohomology algebraof a
Riemannian manifoldin termsofits harmoniforms. Theprinipal resultis that
anyohomologylasshasauniqueharmonirepresentative.
4.A. Eulidean struture of the exterior algebra. Let(M;g)beanori-
ented Riemannian C 1
manifold of dimension m, and let E ! M be a Hermit-
ian vetor bundle of rank r on M. We denote respetively by (
1
;:::;
m ) and
(e
1
;::: ;e
r
)orthonormalframesofT
M
andofEonaoordinatehartM,and
let (
1
;:::;
m ), (e
1
;::: ;e
r
) be the orresponding dual oframes of T
M
; E
re-
spetively. Further,letdV betheRiemannianvolumeelementonM. Theexterior
algebra
T
M
isendowedwithanaturalinnerproduth;i, givenby
(4.1) hu
1
^^u
p
;v
1
^^v
p
i=det(hu
j
;v
k i)
1j;k p
; u
j
;v
k 2T
M
for all p, with
T
M
= L
p
T
M
an orthogonal diret sum. Thus the family of
ovetors
I
=
i1
^^
ip
; i
1
<i
2
<<i
p
, denes anorthonormal basisof
T
. Onedenotesbyh;i theorrespondinginner produton
T
E.
4.2. Hodgestar operator. TheHodge-Poinare-deRham?operatoristhe
endomorphismof
T
M
denedbyaolletionoflinearmapssuhthat
?: p
T
M
!
m p
T
M
; u^?v=hu;vidV; 8u;v2 p
T
M :
The existeneand uniqueness of this operator follows easily from theduality
pairing
p
T
M
m p
T
M
!R
(4.3) (u;v)7!u^v=dV =
X
(I;{I)u
I v
{I
;
where u= P
jIj=p u
I
I
; v= P
jJj=m p v
J
J
, andwhere (I;{I)isthe signofthe
permutation (1;2;:::;m) 7!(I;{I) dened by I followed bythe omplementary
(ordered)multi-indies{I. Fromthis, wededue
(4.4) ?v=
X
jIj=p
(I;{I)v
I
{I :
Moregenerally,thesesquilinearpairingf;gdenedby(1.6)induesanoperator
?onthevetor-valuedforms,suhthat
(4.5) ?:
p
T
M
E!
m p
T
M
E; fs;?tg=hs;tidV;
(4.6) ?t=
X
jIj=p;
(I;{I)t
I;
{I e
; 8s;t2 p
T
M E;
fort= P
t
I;
I e
. Sine(I;{I)({I;I)=( 1) p(m p)
=( 1) p(m 1)
,weimme-
diatelyobtain
(4.7) ??t=( 1)
p(m 1)
t on p
T
M E:
Itislearthat?isanisometryof
T
M
E. Wewillalsoneedavariantofthe?
operator,namelytheantilinearoperator
#: p
T
M
E !
m p
T
M
E
dened by s^#t =hs;tidV, where theexteriorprodut ^ is ombinedwith the
anonialpairingEE
!C. Wehave
(4.8) #t=
X
jIj=p;
(I;{I)t
I;
{I e
:
4.9. Contrationbya vetoreld. Assumegivenatangentvetor2T
M
andaformu2 p
T
M
. Theontrationyu2 p 1
T
M
isdened by
yu(
1
;:::;
p 1
)=u(;
1
;:::;
p 1
);
j 2T
M :
Intermsofthebasis(
j
); yisthebilinearoperatorharaterizedby
l y(
i1
^^
ip )=
(
0 ifl62fi
1
;:::;i
p g,
( 1) k 1
i
1
^
^
i
k
i
p
ifl=i
k :
Thissameformulaisalsovalidwhen(
j
)is notorthonormal. An easyalulation
showsthatyisaderivation oftheexterioralgebra,i.e. that
degu
Moreover,if
~
=h;i2T
M
, theoperatoryistheadjointof
~
^,i.e.,
(4.10) hyu;vi=hu;
~
^vi; 8u;v2
T
M :
Indeed,thispropertyisimmediatewhen=
l
; u=
I
; v=
J .
4.B. Laplae-Beltrami operator. Let E bea Hermitianvetorbundleon
M, and letD
E
be a Hermitian onnetionon E. We onsider the Hilbert spae
L 2
(M;
p
T
M
E) of p-forms on M with values in E, with the given L 2
salar
produt
hhs;tii= Z
M hs;tidV
alreadyintroduedin (2.5). Here hs;tiisthespei salarproduton p
T
M
E
assoiated totheRiemannian salarproduton p
T
M
and theHermitianpairing
onE.
4.11. Theorem. The formal adjoint of D
E
ating onC 1
(M;
p
T
M
E)is
given by
D
E
=( 1) mp+1
?D
E
?:
Proof. Ifs2C 1
(M; p
T
M
E)andt2C 1
(M;
p+1
T
M
E)haveompat
support,wehave
hhD
E s;tii=
Z
M hD
E
s;tidV = Z
M fD
E s;?tg
= Z
M
dfs;?tg ( 1) p
fs;D
E
?tg=( 1) p+1
Z
M fs;D
E
?tg
byanappliationofStokestheorem. Asaonsequene,(4.5)and(4.7)imply
hhD
E
s;tii=( 1) p+1
( 1) p(m 1)
Z
M
fs;??D
E
?tg=( 1) mp+1
hhs;?D
E
?tii:
Thedesiredformulafollows.
4.12. Remark. In the aseof the trivial onnetiond on E =MC, the
formulabeomesd
=( 1) m+1
?d?. Ifmiseven,these formulasredue to
d
= ?d?; D
E
= ?D
E
?:
4.13. Definition. TheLaplae-Beltramioperatoris theseond orderdier-
entialoperatoratingonthebundle, p
T
M
E,suhthat
E
=D
E D
E +D
E D
E :
In partiular, the Laplae-Beltramioperator ating on p
T
M
is = dd
+d
d.
ThislatteroperatordoesnotdependontheRiemannianstruture(M;g).
It is lear that the Laplaian is formally self-adjoint i.e. hh
E
s;tii =
hhs;
E
tii whenever the forms s;t are C 1
and that one of them has ompat
4.14. Calulation of the symbol. ForeveryC 1
funtion f,Leibnitz rule
givese tf
D
E (e
tf
s)=tdf^s+D
E
s. Bydenitionofthesymbol,wethereforend
D
E
(x;)s=^s; 82T
M;x
; 8s2 p
T
M E:
Fromformula(2.8),weobtain
D
E
= (
D
E )
,therefore
D
E
(x;)s=
~
ys
where
~
2T
M
istheadjoint tangentvetorof. Theequality
E
=
D
E
D
E +
D
E
DE
impliesthat
E
(x;)s= ^(
~
ys)
~
y(^s)= (
~
y)s;
E
(x;)s= jj 2
s:
In partiular,
E
is always an ellipti operator. In the speial ase where M is
anopen subsetofR m
withtheonstantmetrig = P
m
i=1 dx
2
i
, allthese operators
d; d
; have onstant oeÆients. They are ompletely determined by their
prinipalsymbol(notermoflowerorderanappear). Oneeasilyomputes:
s= X
jIj=p s
I dx
I
; ds= X
jIj=p;j s
I
x
j dx
j
^dx
I
;
d
s= X
I;j s
I
x
j
x
j ydx
I
;
s=
X
I
X
j
2
s
I
x 2
j
dx
I :
ConsequentlyhasthesameexpressionastheelementaryLaplaianoperator,up
toaminussign.
4.C.Harmoniforms andtheHodgeisomorphism. LetEbeaHermit-
ianvetorbundleonaompatRiemannianmanifold(M;g). WeassumethatEis
givenaHermitianonnetionD
E
suhthat (D
E )=D
2
E
=0. Suh aonnetion
is said to be integrable or at. It is known that this is equivalent to suh an E
givenbyarepresentation
1
(M)!U(r). Suhabundleis alledaat bundleor
aloalsystemofoeÆients. AstandardexampleisthetrivialbundleE=MC
with its obvious onnetion D
E
=d. Our assumption impliesthat D
E
denes a
generalizeddeRhamomplex
C 1
(M;E) DE
!C 1
(M;
1
T
M
E)!!C 1
(M;
p
T
M
E)
DE
!:
Theohomologygroupsofthisomplexaredenoted byH p
DR
(M;E).
Thespaeof harmoniforms ofdegreeprelativetothe Laplae-Beltramiop-
erator
E
=D
E D
E +D
E D
E
isdenedby
H p
(M;E)=
s2C 1
(M;
p
T
M
E);
E s=0 :
Sine hh
E
s;sii = jjD
e sjj
2
+jjD
E sjj
2
, we see that s 2 H p
(M;E) if and only if
D
E s=D
s=0.
4.16. Theorem. For allp, thereexistsanorthogonal deomposition
C 1
(M;
p
T
M
E)=H p
(M;E)Im D
E
ImD
E
; where
ImD
E
=D
E C
1
(M;
p 1
T
M
E)
;
ImD
E
=D
E C
1
(M;
p+1
T
M
E)
:
Proof. It is immediate that H p
(M;E) is orthogonal to the two subspaes
ImD
E
andIm D
E
. The orthogonalityof these twosubspaesisalso obvious, asa
resultofthehypothesisD 2
E
=0,namely:
hhD
E s;D
E
tii=hhD 2
E
s;tii=0:
Wenowapplyth. 3.10totheelliptioperator
E
=
E
atingonthep-forms,i.e.
theoperator
E :C
1
(M;F)!C 1
(M;F)atingonthebundle F = p
T
M E.
Weobtain
C 1
(M;
p
T
M
E)=H p
(M;E)
E (C
1
(M;
p
T
M E));
Im
E
=Im(D
E D
E +D
E D
E
)Im D
E +ImD
E :
Further, sine ImD
E
and Im D
E
are orthogonal to H p
(M;E), these spaes are
ontainedinIm
E
.
4.17. Hodge Isomorphism Theorem. The de Rham ohomology groups
H p
DR
(M;E) arenite dimensional;moreoverH p
DR
(M;E)'H p
(M;E).
Proof. Fromthedeompositionin (4.16),weobtain
B p
DR
(M;E)=D
E (C
1
(M;
p 1
T
M E));
Z p
DR
(M;E)=KerD
E
=(Im D
E )
?
=H p
(M;E)ImD
E :
ThisshowsthatanydeRhamohomologylassontainsauniqueharmonirepre-
sentative.
4.18. Poinare duality. Thepairing
H p
DR
(M;E)H m p
DR (M;E
)!C; (s;t)7!
Z
M s^t
isanon-degenerate bilinearform,and thus denesaduality betweenH p
DR (M;E)
andH m p
DR (M;E
):
Proof. First observe that there is a naturally dened at onnetion D
E
suhthatforalls2C 1
(M;
T
M
E); t2C 1
(M;
T
M
E
),onehas
(4.19) d(s^t)=(D
E
s)^t+( 1) degs
s^D
E t:
ItthenfollowsfromStokestheoremthatthebilinearmap(s;t)7!
R
M
s^tfators
throughtheohomologygroups. Fors2C 1
(M;
p
T
M
E),thereaderaneasily
verifythe followingformulas(use(4.19) in asimilar wayto that whih wasdone
fortheproofofth. 4.11):
(4.20)
D
E
(#s)=( 1) p
#D
s; (D
E )
(#s)=( 1) p+1
#D
E
s;
E
(#s)=#
s
:
21 5. Hermitianand ahler manifolds
Consequently#s2H m p
(M;E
)ifandonlyifs2H p
(M;E). Sine
Z
M
s^#s= Z
M jsj
2
dV =jjsjj 2
;
it follows that the Poinare duality pairing has trivial kernel in the left fator
H p
(M;E)'H p
DR
(M;E). Bysymmetry,italsohastrivialkernelintheright. This
ompletestheproof.
5. Hermitianand Kahlermanifolds
Let X be a omplex manifold of dimension n. A Hermitian metri on X
is a positive denite Hermitian C 1
form on T
X
. In terms of loal oordinates
(z
1
;::: ;z
n
),suhaform anbewritten
h(z)= X
1j;k n h
jk (z)dz
j dz
k
;
where(h
jk
)isapositiveHermitianmatrixwithC 1
oeÆients. Thefundamental
(1;1)-formassoiatedto his
!= Imh= i
2 X
h
jk dz
j
^dz
k
; 1j;kn:
5.1. Definition.
a) AHermitianmanifoldisapair(X;!)where! isapositivedeniteC 1
(1;1)-
formonX.
b) Themetri! issaidtobeKahlerifd!=0.
) X isalled aKahlermanifoldifX hasatleastoneKahlermetri.
Sine! isreal,theonditionsd!=0; d 0
!=0; d 00
!=0areallequivalent. In
loaloordinates,weseethatd 0
!=0ifandonlyif
h
jk
z
l
= h
lk
z
j
; 1j;k;ln:
Asimplealulationgives
! n
n!
=det(h
jk )
^
1jn
i
2 dz
j
^dz
j
=det (h
jk )dx
1
^dy
1
^^dx
n
^dy
n
;
wherez
n
=x
n +iy
n
. Consequentlythe(n;n)form
(5.2) dV =
1
n!
! n
ispositiveandoinideswiththeHermitianvolumeelementofX. IfX isompat,
then R
X
! n
= n!Vol
!
(X) > 0. This simple observation already implies that a
ompatKahlermanifoldmustsatisfyertainrestritivetopologialonditions:
5.3. Consequene.
a) If (X;!) is ompat Kahler and if f!g denotes the ohomology lass of ! in
H 2
(X;R) , thenf!g n
6=0.
b) If X isompat Kahler,thenH 2k
(X;R) 6=0for 0kn. Indeed, f!g k
isa
non-zerolass of H 2k
(X;R).
