23
Other Titles in This Series
Volume 7 And y R. Magid
Lectures on differential Galois theory 1994
6 Dus a McDuff and Dietmar Salamon /-holomorphic curves and quantum cohomology 1994
5 V . I. Arnold Topological invariants of plane curves and caustics 1994
4 Davi d M. Goldschmidt Group characters, symmetric functions, and the Hecke algebra 1993
3 A . N. Varchenko and P. I. Etingof Why the boundary of a round drop becomes a curve of order four 1992
2 Frit z John Nonlinear wave equations, formation of singularities 1990
1 Michae l H. Freedman and Feng Luo Selected applications of geometry to low-dimensional topology 1989
http://dx.doi.org/10.1090/ulect/006
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University
LECTURE Series
Volume 6
J-holomorphic Curve s and Quantu m Cohomolog y
Dusa McDuf f Dietmar Salamo n
American Mathematical Societ y Providence, Rhode Island
Editorial Committee Jerry L. Bona Donal d S . Ornstein Theodore W. Gamelin Leonar d L. Scott (Chair )
1991 Mathematics Subject Classification. Primar y 53C15; Secondary 58F05, 57R57.
Library of Congress Cataloging-in-Publication Dat a McDuff, Dusa , 1945 -
J-holomorphic curve s and quantum cohomology/Dus a McDuff , Dietma r Salamon . p. cm . — (University lectur e series, ISSN 1047-3998 ; v. 6)
Includes bibliographical reference s an d indexes . ISBN 0-8218-0332- 8 1. Symplecti c manifolds . 2 . Holomorphic functions . 3 . Homology theory . I . Salamon , D .
(Dietmar) II . Title. III . Series: University lecture series (Providence, R.I.) ; v. 6. QA649.M42 199 4 516.3'62—dc20 94-2541 4
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Contents
1 Introductio n 1 1.1 Sympleeti c manifold s 1 1.2 J-holomorphi c curve s 3 1.3 Modul i space s 4 1.4 Compactnes s 5 1.5 Evaluatio n map s 6 1.6 Th e Gromov-Witte n invariant s 8 1.7 Quantu m eohomolog y 9 1.8 Noviko v rings an d Floe r homolog y 1 1
2 Loca l Behaviou r 1 3 2.1 Th e generalise d Cauchy-Rieman n equatio n 1 3 2.2 Critica l point s 1 5 2.3 Somewher e injectiv e curve s . 1 8
3 Modul i Space s an d Transversalit y 2 3 3.1 Th e mai n theorem s 2 3 3.2 Ellipti c regularit y 2 5 3.3 Implici t functio n theore m 2 7 3.4 Transversalit y 3 3 3.5 A regularity criterio n 3 8
4 Compactnes s 4 1 4.1 Energ y 4 2 4.2 Remova l o f Singularitie s 4 3 4.3 Bubblin g 4 6 4.4 Gromo v compactnes s 5 0 4.5 Proo f o f Gromov compactnes s 5 2
5 Compactificatio n o f Modul i Space s 5 9 5.1 Semi-positivit y 5 9 5.2 Th e imag e o f the evaluatio n ma p 6 2 5.3 Th e imag e o f the p-told evaluatio n ma p 6 5 5.4 Th e evaluatio n ma p fo r marke d curve s 6 6
v
vi CONTENTS
6 Evaluatio n Map s an d Transversalit y 7 1 6.1 Evaluatio n map s ar e submersions 7 1 6.2 Modul i space s o f iV-tuples o f curves 7 4 6.3 Modul i space s o f cusp-curves 7 5 6.4 Evaluatio n map s fo r cusp-curve s 7 9 6.5 Proof s o f the theorem s i n Section s 5. 2 an d 5. 3 . . 8 1 6.6 Proo f o f the theore m i n Sectio n 5. 4 8 3
7 Gromov-Witte n Invariant s 8 9 7.1 Pseudo-cycle s 9 0 7.2 Th e invarian t $ 9 3 7.3 Example s 9 8 7.4 Th e invarian t # . . . . 10 1
8 Quantu m Cohomolog y 10 7 8.1 Witten' s deforme d cohomolog y rin g 10 7 8.2 Associativit y an d compositio n rule s 11 4 8.3 Fla g manifold s 11 9 8.4 Grassmannian s 12 3 8.5 Th e Gromov-Witte n potentia l 13 1
9 Noviko v Ring s an d Calabi-Ya u Manifold s 14 1 9.1 Multiply-covere d curve s 14 2 9.2 Noviko v ring s 14 4 9.3 Calabi-Ya u manifold s 14 8
10 Floe r Homolog y 15 3 10.1 Floer' s cochai n comple x 15 3 10.2 Rin g structur e 15 9 10.3 A compariso n theore m 16 0 10.4 Donaldson' s quantu m categor y 16 2 10.5 Closin g remark . 16 5
A Gluin g 16 7 A.l Cutof f function s 16 8 A.2 Connecte d sum s o f J-holomorphi c curve s 17 0 A.3 Weighte d norm s 17 1 A.4 A n estimat e fo r th e invers e 17 3 A.5 Gluin g 17 6
B Ellipti c Regularit y 18 1 B.l Sobole v space s 18 1 B.2 Th e Calderon-Zygmun d inequalit y 18 5 B.3 Cauchy-Rieman n operator s 19 0 B.4 Ellipti c bootstrappin g 19 2
Bibliography 19 7
Indexes 20 3
Index o f Nota t ion s 20 9
Preface
The theor y o f J-holomorphi c curve s ha s bee n o f grea t importanc e t o symplecti c topologists eve r sinc e it s inceptio n i n Gromov' s pape r [26 ] o f 1985 . It s appli -cations includ e man y ke y result s i n symplecti c topology : see , fo r example , Gro -mov [26] , McDufl f [42 , 45] , Lalonde-McDuf f [36] , and th e collectio n o f article s i n Audin-Lafontaine [5] . I t wa s als o on e o f th e mai n inspiration s fo r th e creatio n of Floe r homolog y [18 , 19 , 73] , an d recentl y ha s caugh t th e attentio n o f mathe -matical physicist s throug h th e theor y o f quantu m cohomology : see Vaf a [82 ] an d Aspinwall-Morrison [2] .
Because of this increased interest on the part o f the wider mathematical commu -nity, i t i s a good time to write an expository accoun t o f the field, which explains th e main technical steps in the theory. Althoug h all the details are available in the liter -ature in some form o r other, the y ar e rather scattered . Also , some improvements i n exposition ar e now possible. Ou r accoun t i s not, o f course, complete, bu t i t i s writ -ten with a fai r amoun t o f analytic detail , an d shoul d serv e as a usefu l introductio n to th e subject . W e develop the theor y o f the Gromov-Witte n invariant s a s formu -lated b y Rua n i n [64 ] and giv e a detailed accoun t o f their application s t o quantu m cohomology. I n particular , w e giv e a ne w proo f o f Ruan-Tian' s theore m [67 , 68] that th e quantu m cup-produc t i s associative .
Many people have made usefu l comment s which have added significantl y t o ou r understanding. I n particular , w e wisj i t o than k Giventa l fo r explainin g quantu m cohomology, Rua n fo r severa l useful discussion s and fo r pointin g ou t t o us the con -nection between associativit y o f quantum multiplicatio n an d th e WDVV-equation , Taubes fo r hi s elegan t contributio n t o Sectio n 3.4 , an d especiall y Gan g Li u fo r pointing ou t a significan t ga p i n a n earlie r versio n o f th e gluin g argument . W e are als o gratefu l t o Lalond e fo r makin g helpfu l comment s o n a first draf t o f thi s manuscript. Th e first autho r wishe s t o acknowledg e th e hospitalit y o f the Univer -sity o f California a t Berkeley , an d th e gran t GER-935007 5 unde r th e NS F Visitin g Professorship fo r Wome n progra m whic h provide d partia l suppor t durin g som e of the wor k o n thi s book .
Dusa McDuff , Dietma r Salamon , Mathematics Departmen t Mathematic s Institute , SUNY a t Ston y Brook , Universit y o f Warwick , Stony Brook , N Y 11794 , USA Coventr y CV 4 7AL , Grea t Britai n [email protected] [email protected] k
vii
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Index
a prior i estimat e for energ y density , 4 4
action Morse-Novikov theor y for , 15 4 of short loop , 4 5 on loo p space , 15 3
adjunction formula , 39 , 10 1 almost comple x structur e
X-positive, 5 9 iT-semi-positive, 5 9 tu-compatible, 2 5 u;-tame, 2 , 42 condition t o b e regular , 3 8 generic, 5 good, 9 8 integrable, 3 positive, 5 9 regular, 2 4 semi-positive, 59 , 14 3 smooth homotopy , 2 5 tame versu s compatible , 26 , 60
Arnold conjecture , 158 , 16 5 Aronszajn, 15 , 197 Aronszajn's theorem , 14 , 35 Aspinwall, ix , 107 , 151 , 197 Aspinwall-Morrison formula , 15 1 Astashkevich, 119 , 197 Atiyah, 19 7 Atiyah-Floer conjecture , 163 , 165 Audin, ix , 1 , 19 7
Bertram, 123 , 197 bubbling, 41 , 46-49, 15 5
Calabi-Yau manifold , 1 , 12 , 137 , 141, 145, 148-15 1
examples, 148 , 149 Calderon-Zygmund inequality , 186-18 9 Callaghan, 163 , 197
Candelas, 141 , 148, 197 Cauchy-Riemann equation , 2 , 5 , 13 -
15, 19 0 perturbed, 118 , 144 , 19 3
Cauchy-Riemann operator , 2 4 fundamental solution , 19 0
Chern character , 12 8 Chern class , 2
quantum, 12 1 Chern number , 3 8
minimal, 10 , 61 negative, 86 , 16 5 zero, 14 8
Ciocan-Fontanine, 119 , 19 7 cohomology
quantum, 10 8 compactness, 1 , 5- 6 compactness theorem , 51 , 63, 65, 68
proof, 52 , 81-8 7 composition rule , 13 7 condition
fmiteness, see finiteness condtio n (HI), 93 , 101 (F2), 93 , 96 (2*3), 9 6 (£T4), 101 , 103, 114 (iJ5), 101 , 103, 114 (JA3), 6 7 (JA4)y 68 , 86, 14 2 (JAP), 67-69 , 85-87 , 10 1 on dimension, see dimension con-
dition conformal map , 13 , 41 conformal rescaling , 4 1 Conley, 15 5 convergence problem , 137 , 139 , 141 ,
149, 15 1 critical point , 1 5
finite numbe r of , 1 5
203
204 INDEX
stable and unstable manifolds, 16 1 cross-ratio, 10 5 cup produc t
deformed, 110 , 14 7 weakly monoton e case , 14 1
given by tripl e intersection , 10 7 pair-of-pants, 16 0
curve, 3- 4 approximate J-holomorphie , 3 0 as symplecti c submanifold , 3 counting discrete curves, 101,138,
151 critical point , 15-1 8 cusp, see cusp-curv e determined b y oo-jet , 1 4 energy, 6 graph of , 14 2 implicit functio n theore m for , 3 0 injective point , 4 , 1 8 intersections of , 1 7 isolated, 8 9 multiply-covered, 4,18,119,142 -
144, 16 5 of negative Chern number, 86,16 5 parametrized, 3 positivity of intersections, 21 , 39,
100, 10 1 reducible, 4 2 regular, 24 , 38 simple, 4 , 18-2 1 singular poin t o f sequence, 5 3 somewhere injective , 4 , 1 8
is simple, 1 8 weak convergence , 5 0
cusp-curve, 42 , 50 , 62 framing D , 63 , 76 label T , 65 , 81 moduli spac e of , 7 5 type £> , 76 with tw o components , 8 0
Du
definition, 2 4 ellipticity of , 2 8 formula for , 2 8 its adjoin t £)* , 29 right inverse , 29 , 173-17 6 surjectivity of , 29 , 38
Darboux's theorem , 2 Daskalopoulos, 123 , 197 deformed cu p product , see cup prod -
uct determinant bundle , 3 3 dimension conditio n
for # , 9 3 for # , 10 2 for a *&, 11 0 for a*b*c, 11 4
Donaldson, 33 , 162 , 167 , 168 , 197 Donaldson's quantu m category , 162 -
165 for Lagrangians , 16 4 for mappin g tori , 16 3
Dostoglou, 30 , 155 , 162, 163 , 197 Dubrovin, 107 , 113 , 131, 139, 198 Dubrovin connection , 113 , 132
and quantu m products , 13 3 explicit formula , 13 4 flatness, 13 6 potential function , 13 5
elliptic bootstrapping, 41 , 43,192-196 elliptic regularity , 25-27 , 29 , 35 , 191,
192 energy, 6 , 42 , 44
bounds derivative , 4 4 conformal invariance , 4 1 identity, 41 , 43
energy level , 14 5 evaluation map , 6-8 , 4 9
and orientations , 9 8 as pseudo-cycle , 94-9 8 domain of , 6 4 for cusp-curves , 79-8 1 for marke d curves , 66-6 9 image of , 62-6 5 is submersion, 71-7 4 p-fold, 8 , 65 , 66
compactification, 8 , 6 5
finiteness condition , 144 , 147 , 156 finiteness result , 15 5 first Cher n class , 2 flag manifold , see quantum cohomol -
ogy
INDEX 205
Floer, ix , 12 , 15 , 41 , 153 , 155-158 , 162, 164 , 19 8
Floer (co)homology , 1 , 11 , 12 , 153 -159
and quantu m cohomology , 16 0 coehain complex , 15 5 Euler characteristic , 16 4 of (if , J ) , 15 6 of symplectomorphism, 16 2 pair-of-pants product , 16 0 ring structure , 159-16 0
Floer-Donaldson theory , 16 3 framing
D fo r cusp-curve , 63 , 76 Fredholm operator , 5
determinant bundle , 3 3 index, 5 , 24 regular value , 5 , 3 3
Fredholm theory , 1 , 4, 5 Frobenius algebra , 113 , 131 Frobenius condition , 13 3
Gilbarg, 186 , 188 , 198 Givental, ix , 11 , 107, 119 , 121 , 198 -Green, 141 , 148, 197 Griffiths, 33 , 38, 100 , 198 Gromov, ix , 7 , 41 , 50 , 89 , 107 , 143 ,
198 Gromov invarian t $ , 66 , 89, 93-9 4
dimension condition , 9 3 for 6-manifolds , 10 0 for conie s i n CP 2 , 9 9 for discret e curves , 10 1 for line s i n C P n , 9 8 on blo w up , 10 0
Gromov's compactnes s theorem , 50 -58
Gromov-Witten invarian t \£ , 66 , 69 , 101-105
dimension condition , 10 2 Gromov-Witten invariants , 1 , 8- 9
compared o n CP 2 , 10 4 composition rule , 11 7
on CP 2 , 11 9 deformation invariance , 9 6 mixed, 90 , 13 8 signs, 9 8 weakly monoton e case , 14 1
well defined for Kahler manifolds , 96
Gromov-Witten potential , 131-13 9 for CP 2 , 13 8 for C P n , 13 8 non-monotone case , 13 9
Grothendieck, 38 , 198 Guillemin, 92 , 19 8
Holder inequality , 16 9 Holder norm , 18 3 Hamiltonian differential equation , 15 3 harmonic function , 18 5
mean valu e property , 18 5 Harris, 33 , 38, 100, 19 8 Hartman, 15 , 198 Heegard splitting , 16 5 Hofer, 12 , 15, 40, 41, 51, 54, 146, 153,
155, 157 , 158, 162 , 198 homology clas s
J-effective, 6 7 framed, 6 3 indecomposable, 4 9 of pure degree , 108 , 131 spherical, 6 , 4 7
Hurewicz homomorphism , 6
implicit functio n theorem , 27-3 3 integrable systems , 113 , 121 isoperimetric inequality , 41 , 45
(J, J')-holomorphic, 2 J-holomorphic curve , see curv e J-holomorphic map , 1 3 John, 185 , 186 , 199
Kahler manifold , 6 1 and symplectic deformations, 10 0 Fano variety , 6 0 Gromov-Witten invariants , 9 6
Kim, 11 , 107, 119 , 121 , 198 Kobayashi, 19 9 Kobayaski, 2 8 Kodaira vanishin g theorem , 3 8 Kontsevich, 90,113,131,137-139,19 9 Kronheimer, 167 , 168 , 197
labelling T fo r cusp-curve , 65 , 81
206 INDEX
Lafontaine, ix , 1 , 19 7 Lalonde, ix , 7 , 19 9 Landau-Ginzburg potential , 126 , 12 7 Laplace's equatio n
fundamental solution , 18 5 Laurent polynomia l ring , 1 0 Lawson, 44 , 19 9 Liu, 118 , 199 Lizan, 4 0 loop
action of , 4 5 loop spac e
of manifold , 146 , 15 4 universal cover , 15 4
Lorek, 40 , 98 , 199
Manin, 90 , 113 , 131, 137-139, 19 9 mapping tor i
and Atiyah-Floe r conjecture , 16 3 marked sphere , 10 5 Maslov index , 15 5 mass
of singular point , 5 3 McDuff, ix , 1 , 2, 7 , 17 , 21, 28, 33, 39,
40, 59 , 62 , 74 , 77 , 89 , 98 -101, 156 , 199 , 200
Micallef, 21 , 200 Milnor, 92 , 129 , 200 minimal Cher n number , 10 , 61 mirror symmetry conjecture, 122,148 ,
149 moduli space , 4- 5
cobordism of , 2 5 compactifications, 59-6 9 complex structur e on , 3 3 integration over , 97 , 10 3 main theorems , 23-2 5 of iV-tuples o f curves, 74-7 5 of cusp-eurves , 75-7 8 of flat connections , 16 3 of two-component cusp-curves , 80 of unparametrized curves , 52 orientation, 32 , 33, 98 regular poin t of , 2 4 top strat a i n the boundary , 8 2 universal, 33 , 77 Yang-Mills, 3 3
monotone, see symplecti c manifol d
Morrison, ix , 107 , 151 , 197 Morse-Novikov theory , 146 , 154 Morse-Witten coboundary , 15 8 Morse-Witten comple x , 161 Moser, 121 , 122, 200
Nijenhuis, 74 , 200 Nijenhuis tensor , 2 8
anti-commutes wit h J , 3 2 Nomizu, 28 , 199 non-squeezing theorem , 7 Novikov, 146 , 154 , 200 Novikov ring , 11 , 12, 141, 144-148
as Lauren t serie s ring, 14 6 number o f curve s
in CP 2 , 105 , 138 in Calabi-Yau manifold , 101 , 151
Oh, 20 0 Ono, 153 , 158, 200 Ossa, 141 , 148, 197
Pansu, 41 , 44, 51 , 200 Parker, 41 , 51, 200 Parkes, 141 , 148, 197 Piunikhin, 157 , 161 , 200 Poisson's identity , 18 5 Pollack, 92 , 19 8 potential function , 13 5 product, see cup produc t pseudo-cycle, 8 , 63, 66, 68, 90-9 3
bordant, 9 0 of dimension & , 90 transverse, 9 1 weak representative , 9 3
quantum category, 162 , 164 Chern classes , 12 1
quantum cohomology , 1 , 9-11 , 108 -110
and Floe r cohomology , 157 , 160 as Lagrangian variety , 12 2 cup product , 110-114 , 14 7
associativity, 114-119 , 16 1 dimension condition , 11 0
Probenius structur e of , 113 , 131 o f C F n , 11 , 113
INDEX 207
of flag manifold , 11 , 110 , 119 -122, 14 6
of Grassmannian , 123-13 0 periodic form , 12 , 14 7 weakly monoton e case , 14 8 with Noviko v rings , 14 6
rational map s Rat m o f CP 1 , 83 , 150 regular
J-holomorphic curve , 24 , 39 value o f Predholm operator , 3 3
Rellich's theorem , 27 , 18 4 removal o f singularities, 41 , 43-46 reparametrization grou p G , 42 , 62 Riemann-Roch theorem , 24 , 33 Ruan, ix , 9 , 33, 50, 59, 62, 64, 66, 76,
89, 90 , 100 , 101 , 104 , 105 , 107, 117 , 118, 131, 137, 138, 144, 200 , 20 1
Sacks, 41 , 201 Sadov, 119 , 161 , 197, 201 Salamon, ix , 1 , 2 , 7 , 12 , 15 , 30 , 41 ,
44, 51 , 100 , 146 , 153 , 155 -158, 161-163 , 197-20 1
Sard's theorem , 5 , 24 Sard-Smale theorem , 33 , 36, 78 Schwarz, 157 , 158 , 161 , 201 semi-positivity, 5 9 Siebert, 121 , 123, 124, 20 1 sigma model , 8 9 sign
convention fo r Li e bracket, 13 2 of intersection , 9 8
Sikorav, 4 0 singular poin t o f curve, 5 3 Smale, 33 , 201 smoothness
of class W k>p, 181 , 184 smoothness convention , 3 4 Sobolev embeddin g theorem , 27 , 18 4 Sobolev estimat e
borderline case , 6 , 41 , 184 Sobolev space , 181-18 4 spherical, 4 7 Stasheff, 129 , 200 supermanifold, 131 , 139 symplectic action , see actio n
symplectic for m deformation, 9 6
symplectic manifold , 1- 3 monotone, 1 , 8, 60 non-equivalent, 10 0 weakly monotone, 59-61,141,15 3
Arnold conjectur e for , 15 8
Taubes, ix , 33 , 36, 20 1 Tian, ix , 90 , 104 , 107 , 117 , 118 , 121,
123, 124 , 131, 137, 138 , 144, 201
Toda lattice , 119 , 12 1 triple intersectio n product , 10 7 Trudinger, 186 , 188 , 198
Uhlenbeck, 41 , 201 unique continuation , 1 5 universal modul i space , 33 , 77
is a manifold , 34 , 7 7
Vafa, ix , 107 , 201 Verlinde algebra , 123 , 128-13 0
gluing rules , 13 0 Viterbo, 160 , 201
WDVV-equation, 11 , 135, 137 weak convergenc e
definition, 5 0 weak derivative , 18 1 weak solution , 185 , 191 weakly monoton e manifold , see sym -
plectic manifol d Wentworth, 123 , 197 Weyl's lemma , 18 6 White, 21 , 200 Wintner, 15 , 198 Witten, 89 , 107 , 123 , 124 , 130 , 158 ,
202 Wolfson, 41 , 51, 200 Woolf, 74 , 200
Yau, 14 8 Ye, 41, 51, 202
Zehnder, 155-157 , 20 1
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Index o f Notation s
JT{M,w) 2 End(TM,J,w) 3 4
ci (A) 2 ir:M l{A,J)^Jl 3 6
u : (£, j) - > (M, J) 3 &,(« , w) = <t> , w) j 4 2
C = Imu 3 E(u;B r) 4 4
M(A,J) 4 , 23 C = C lUC2U...UCN 5 0
Jreg(A) 4 , 24 u=(u\...,u N) 5 0
{v, w) 5 A{r, R) 5 2
G = PSL(2,C ) 6 J+(M,u), J+(M,u,K) 5 9
W(A,J) = M(A,J)x GCP1....7, 6 2 D = {A 1,... ,A N,j2,... ,j N} 6 3
ej : W(A, J)-*M 7 W(D, J) 64 , 7 9
* x ( a i , . . . , a„ ) 9 , 94 e p : W{A, J,p) - » M* 6 5
* A ( O I , • • • ,otp) 9 , 99 z = («! , . . . , z p ) 6 6
a = PD(a ) 10 , 106 e z = e^j.z : M(A, J) -* AP 6 6
a * 6 10 , 108 e DtT,z : V(£>,T,J,z) -^M " 6 8
QH*(M) 10 , 106 MCA 1 , . . . ,A N , J ) 7 4
QH*(M) = H*{M)®l[q] 10 , 107 &N 7 6
T = lmn 2{M) - f H 2{M) 11 , 142 *D : M{D,J) - » J 7 8
A„ 11 , 142 * * * : i?d(MP, Z) - Z 9 3
Sj 1 3 * 4 , P : -ffd(MP, Z) -* 1 : 10 2
X = Map(E,M ; A) 2 3 M > 108 > 131
Du 24 , 28 {a*b) A I l l
Jies(A) 2 4 $ ^ . s ( a , /3; j, S) 11 6
J(M,u) 2 4 J, A 14 3
(u, J) € M'(A, J) C Xk'P x J* . . .. 34 W k'P(n), W 0fc,p(O) 18 1
209
