Selected Title s i n Thi s Subserie s

45 Ya . Eliashberg , D . Puchs , T . Rat iu , an d A . Weinste in , Editors , Norther n

California symplecti c geometr y semina r (TRANS2/196 )

44 Alexande r Astashkevic h an d Serg e Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y

Collection) (TRANS2/194 )

43 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematica l

Physics (TRANS2/191 )

42 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

(TRANS2/190)

41 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection ) (TRANS2/189 )

40 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

(TRANS2/187)

39 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations (TRANS2/186 )

38 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e

Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) (TRANS2/185 )

37 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s

(Dedicated t o Seli m Krein ) (TRANS2/184 )

36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 )

35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )

34 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n

the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

(TRANS2/177)

31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Ne t sve taev , Editors , Mathematic s

in St . Petersbur g (TRANS2/174 )

29 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pes in , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )

26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r (TRANS2/169 )

25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )

24 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations (TRANS2/167 )

23 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

(TRANS2/165)

22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )

Published Earlie r a s Advances i n Sovie t Mathematic s 21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 199 4

20 R . L . Dobrushin , Editor , Probabilit y contribution s t o statistica l mechanics , 199 4

(Continued in the back of this publication)

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Northern Californi a Symplectic Geometr y Seminar

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 19 6

Advances in the Mathematical Sciences —45 (Formerly Advances in Soviet Mathematics)

Northern Californi a Symplectic Geometr y Seminar

Ya. Eliashber g D. Fuch s T. Rati u A. Weinstei n Editors

5BEM47.,

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/trans2/196

A D V A N C E S I N T H E M A T H E M A T I C A L S C I E N C E S

E D I T O R I A L C O M M I T T E E

V . I . A R N O L D

S. G . G I N D I K I N

V . P . M A S L O V

2000 Mathematics Subject Classification. P r i m a r y 53Dxx , 58Dxx , 58Jxx .

ABSTRACT. Th e Norther n Californi a Symplecti c Geometr y Semina r wa s establishe d i n th e Fal l of 198 9 t o encourag e ongoin g interactio n betwee n geometer s a t Stanfor d Universit y an d th e Uni -versity o f Californi a campuse s a t Berkeley , Davis , an d Sant a Cruz . Th e lecture s ove r th e years , by visitors , loca l faculty , an d advance d graduat e students , hav e provide d a panoram a o f develop -ments i n symplecti c an d contac t geometr y an d topology , Poisso n geometry , quantizatio n theory , and applications .

The presen t volum e contain s paper s b y som e o f th e semina r participants . The y reflec t th e diversity o f topic s o f th e seminar , rangin g fro m symplecti c topolog y t o Hamiltonia n dynamics , from quantu m cohomolog y an d mirro r symmetr y t o infinite-dimensiona l symplecti c geometry , th e theory o f Hamiltonia n grou p actions , an d quantization .

The boo k ca n b e use d b y researcher s an d graduat e student s workin g i n symplecti c geometr y and topology , an d applications .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-2075- 3

ISSN 0065-929 0

Copying an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu -cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg -ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permission@ams.org .

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© 199 9 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

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Contents

Introduction i x

Quantization o f Symplecti c Orbifold s an d Grou p Action s ANA CANNA S D A SILV A AN D VICTO R GUILLEMI N 1

Symmetric Spaces , Kahle r Geometr y an d Hamiltonia n Dynamic s S. K . DONALDSO N 1 3

Hamiltonian Dynamica l System s Withou t Periodi c Orbit s VIKTOR L . GINZBUR G 3 5

The Mirro r Formul a fo r Quinti c Threefold s ALEXANDER GIVENTA L 4 9

Stabilisation o f Symplecti c Inequalitie s an d Application s FRANgois LALOND E AN D CHARLE S PESTIEA U 6 3

The Virtua l Modul i Cycl e DUSA M C D U F F 7 3

Engel Deformation s an d Contac t Structure s RICHARD MONTGOMER Y 10 3

Floer Homology , Noviko v Ring s an d Complet e Intersection s MARCIN POZNIA K 11 9

Surgery, Quantu m Cohomolog y an d Birationa l Geometr y YONGBIN RUA N 18 3

Quantum Product s fo r Mappin g Tor i an d th e Atiyah-Floe r Conjectur e DIETMAR A . SALAMO N 19 9

On th e Grou p o f Symplecti c Automorphism s o f C P m x CP n

PAUL SEIDE L 23 7

On th e Cohomolog y Ring s o f Hamiltonian T-space s SUSAN TOLMA N AN D JONATHA N WEITSMA N 25 1

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Introduction

The Norther n Californi a Symplecti c Geometr y Semina r wa s established i n th e Fall of 1989 to encourage ongoing interaction between geometers at Stanford Univer -sity an d th e Universit y o f California campuse s a t Berkeley , Davis , an d Sant a Cruz . Since then , th e semina r ha s me t o n th e firs t Monda y o f each mont h fro m Octobe r to Ma y (excep t January) , alternatel y a t Stanfor d an d Berkeley , wit h tw o lecture s at eac h meeting . Beginnin g i n th e Fal l o f 1992 , afte r th e tragi c deat h o f Andrea s Floer, who was among the original participants i n the seminar, on e of the lectures a t the Octobe r meetin g ha s bee n designate d a s the Andrea s Floe r Memoria l Lecture . The Floe r Memoria l Lecture s hav e bee n give n b y Helmu t Hofe r (1992) , Dus a Mc -Duff (1993) , Yasha Eliashberg (1994) , Jon Wolfson (1995) , Vladimir Arnol d (1996) , Simon Donaldso n (1997) , an d Maxi m Kontsevic h (1998) .

The lecture s ove r th e years , b y visitors , loca l faculty , an d advance d graduat e students, hav e provide d a panoram a o f development s i n symplecti c an d contac t geometry an d topology , Poisso n geometry , quantizatio n theory , an d applications .

All the authors of the current volume , except Marcin Pozniak, were participant s and/or speaker s a t th e seminar .

The pape r b y An a Canna s d a Silv a an d Victo r Guillemi n concern s symplecti c orbifolds with a Hamiltonian action by a group G. Th e authors define a quantization space V whic h i s a representatio n o f G , an d describ e a decompositio n o f V int o irreducible representation s o f G.

Simon Donaldso n i n hi s pape r describe s an d studie s th e infinit e dimensiona l symmetric spac e structur e whic h arise s naturall y o n th e spac e o f Kahle r metric s with a fixed cohomolog y class .

The pape r b y Vikto r Ginzbur g i s a surve y o f counter-examples t o th e "Hamil -tonian Seifer t conjecture " an d mor e generall y o f Hamiltonian system s withou t pe -riodic orbi t o n a compac t energ y level .

Alexander Giventa l present s a "shortcut " t o hi s celebrated origina l proof o f th e Mirror Theore m fo r the case of quintic threefolds, thu s giving a proof o f the formul a of Candelas , d e l a Ossa , Gree n an d Parke s fo r th e th e numbe r o f rational curve s of degree d o n a quinti c threefol d i n C 4.

Francois Lalonde and Charle s Pestieau prov e that th e basic symplectic inequal -ities - th e non-squeezin g theore m an d th e energy-capacit y inequalit y - ar e i n a certain sens e stabl e unde r produc t wit h an y close d symplecti c manifold .

Dusa McDuff' s pape r i s devoted t o a constructio n o f the virtua l modul i cycle . Virtual modul i cycle s ar e essentia l fo r th e theor y o f Gromov-Witte n invariants , Floer homolog y theory , etc . McDuff' s pape r provide s furthe r clarificatio n o f thi s important theory .

ix

x INTRODUCTIO N

The pape r b y Richar d Montgomer y ma y serv e a s a goo d introductio n t o th e emerging theor y o f the geometr y an d topolog y o f Enge l structure s o n 4-manifolds . In i t Montgomer y establishe s som e foundationa l fact s i n thi s theory .

Yongbin Rua n contribute d t o th e volum e a n expositor y articl e abou t som e newly discovere d relation s betwee n quantu m cohomolog y an d birationa l geometry , and th e surger y technique s whic h pla y a fundamenta l rol e i n these discoveries .

Dietmar Salamon' s paper complement s his earlier joint wor k with S . Dostoglou , which contained a proof o f the Atiyah-Floer conjectur e base d on the construction of a natura l isomorphis m betwee n instan t on an d symplecti c Floe r homolog y theorie s for th e mappin g tor i o f symplectomophism s o f surfaces . I n the-curren t paper , Salamon prove s that thi s isomorphism interwine s the produc t structure s i n the tw o theories.

The wor k o f Pau l Seide l concern s th e topolog y o f th e grou p o f symplectomor -phisms o f the produc t o f complex projective spaces ; i t als o partially extend s earlie r results of Gromov, Abreu and McDuff abou t the group of symplectic automorphism s of 5 2 x S 2. I n particular , Seide l shows that th e inclusio n o f the group of symplecti c diffeomorphisms int o the grou p o f al l diffeomorphisms induce s a non-injective ma p on som e homotop y groups .

Susan Tolma n an d Jonatha n Weitsma n conside r a symplecti c manifol d wit h a Hamiltonia n toru s action , an d giv e a simpl e proo f o f th e Goresky-Kottwitz -MacPherson formul a fo r th e imag e o f th e homomorphis m induce d betwee n equi -variant cohomolog y o f the manifol d an d th e fixe d poin t se t o f the toru s action .

Marcin Poznia k wa s a Ph D studen t o f Dietma r Salamon . H e die d o n 2 9 No -vember 199 6 a s a resul t o f a tragi c accident . A s Salamo n remembers , "Poznia k began hi s graduat e studie s a t Warwic k i n Octobe r 1989 . Fo r hi s MS c degre e h e worked on triangulations an d Mors e theory, an d hi s PhD thesis , completed i n 1994 , was devote d t o th e theor y o f so-calle d clea n Lagrangia n intersections . Afte r hi s doctoral studies Marci n Poznia k returne d t o hi s position a t th e Jagellonia n Univer -sity of Krakow. Marci n Poznia k wa s a very thoughtful perso n an d a talented youn g mathematician. H e ha d man y friend s amon g th e student s an d staf f a t Warwick . We shal l mis s hi m ver y much. " Becaus e o f hi s tragi c death , Pozniak' s Ph D thesi s remained unpublished , althoug h it s result s hav e bee n use d an d quote d b y man y authors workin g i n th e area . Thu s w e consider i t appropriat e t o publis h Pozniak' s thesis i n thi s volume . Th e manuscrip t wa s prepare d fo r publicatio n b y David e Castelvecchi.

The Editor s

Selected Title s i n Thi s Subserie s (Continued from the front of this publication)

19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4

18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4

17 D m i t r y Puchs , Editor , Unconventiona l Li e algebras , 199 3

16 Serge i Gelfan d an d Simo n Gindikin , Editors , I . M . Gelfan d seminar , Part s 1 and 2 , 1993

15 A . T . Fomenko , Editor , Minima l surfaces , 199 3

14 Yu . S . Il'yashenko , Editor , Nonlinea r Stoke s phenomena , 199 2

13 V . P . Maslo v an d S . N . Samborskii , Editors , Idempoten t analysis , 199 2

12 R . Z . Khasminskii , Editor , Topic s i n nonparametri c estimation , 199 2

11 B . Ya . Levin , Editor , Entir e an d subharmoni c functions , 199 2

10 A . V . Babi n an d M . I . Vishik , Editors , Propertie s o f globa l attractor s o f partia l differential equations , 199 2

9 A . M . Vershik , Editor , Representatio n theor y an d dynamica l systems , 199 2

8 E . B . Vinberg , Editor , Li e groups , thei r discret e subgroups , an d invarian t theory , 199 2

7 M . Sh . Birman , Editor , Estimate s an d asymptotic s fo r discret e spectr a o f integra l an d

differential equations , 199 1

6 A . T . Fomenko , Editor , Topologica l classificatio n o f integrabl e systems , 199 1

5 R . A . Minlos , Editor , Many-particl e Hamiltonians : spectr a an d scattering , 199 1

4 A . A . Suslin , Editor , Algebrai c if-theory , 199 1

3 Ya . G . Sinai , Editor , Dynamica l system s an d statistica l mechanics , 199 1

2 A . A . Kirillov , Editor , Topic s i n representatio n theory , 199 1

1 V . I . Arnold , Editor , Theor y o f singularitie s an d it s applications , 199 0