DOUBLE POINTS AND THE PROPER TRANSFORM IN SYMPLECTIC GEOMETRY

Transcription

1 DOUBLE POINTS AND THE PROPER TRANSFORM IN SYMPLECTIC GEOMETRY JOHN D. MCCARTHY AND JON G. WOLFSON 0. Introducton In hs book, Partal Dfferental Relatons, Gromov ntroduced the symplectc analogue of the complex analytc operatons of blowng up and blowng down. Of course, n complex geometry one of the prmary uses of blowng up s to resolve sngulartes. Gromov proposed, n 3.4.4(D) of Partal Dfferental Relatons, a program for resolvng the sngulartes of symplectc mmersons wth symplectc crossngs va blowng up, n exact analogue wth the well known complex analytc technque. The purpose of ths note s to show that ths program cannot work. We show that there are symplectcally mmersed surfaces n symplectc 4-manfolds whch do not have a symplectcally embedded proper transform n any blow up of the four manfold. In partcular, there are double ponts of a symplectcally mmersed surface whch cannot be resolved symplectcally usng blowng up. We produce examples of such surfaces wth only double pont sngulartes. Interestngly, double ponts of mmersed surfaces can be resolved usng blowng up n the topologcal category and n the complex category. Symplectc mmersons, however, are flexble enough to allow negatve double ponts yet rgd enough to satsfy a symplectc verson of the adjuncton formula. Ths combnaton leads to our result. Whle all technques used n ths paper are elementary, the man result (Corollary 4.1) that sngulartes of symplectc mmersons cannot be resolved by blowng up s not obvous. Ths result shows that the resoluton of symplectc sngulartes wll requre further technques. 1. Blowng Up n Symplectc 4-Manfolds In ths secton we brefly revew the essental deas of blowng up (and down) n the case of symplectc 4-manfolds. These deas have been worked out n detal by Gullemn and Sternberg [G-S] and McDuff [McD1]. In complex analyss blowng up replaces a pont p, n a complex surface X 2, by the space of complex lnes through p; that s, by CP 1. More precsely, blowng up constructs a complex manfold X and a holomorphc map Date: Aprl 5, Both authors have been partally supported by NSF grant DMS

2 2 JOHN D. MCCARTHY AND JON G. WOLFSON π : X X such that π 1 (p) = CP 1 and π 1 : M \ p M \ CP 1 s a bholomorphsm. The complex curve π 1 (p) X, called the exceptonal curve and usually denoted E, s an mbedded 2 sphere wth self-ntersecton 1. In the symplectc category t s not possble to carry out such an nfntesmal operaton, rather the constructon must be made locally. The dea s to replace, not a pont p, but a symplectc ball around p, by a neghborhood of CP 1. Precsely, consder CP 1 wth area form σ normalzed so that CP 1 σ = πλ 2. Symplectcally mbed (CP 1, σ) nto a symplectc 4-manfold so that ts self-ntersecton s 1. Denote such an embedded surface by E. By the symplectc neghborhood theorem a suffcently small tubular neghborhood of E s determned, up to symplectomorphsm, by σ and the selfntersecton number 1, alone. Consder the ball B 4 (λ + ɛ) R 4 equpped wth the standard symplectc structure on R 4. Proposton 1.1. For ɛ suffcently small there s a symplectomorphsm φ from B 4 (λ+ɛ)\b 4 (λ) onto N ɛ (E)\E, where N ɛ (E) s a tubular neghborhood of E. The map φ extends contnuously over B 4 (λ) to be the Hopf map. Proof. The proposton s proved by buldng a model for the symplectc structure on the tubular neghborhood N ɛ (E) and then gvng φ explctly. For detals see [G-S] or [McD1]. The blow up of the symplectc 4-manfold (M, ω) at p depends on the choce of a symplectc embeddng ρ : B 4 (λ) M, ρ(0) = p. Gven such an embeddng, extend ρ to a symplectc embeddng ρ : B 4 (λ+ɛ) M. Delete B 4 (λ) and glue n N ɛ (E) va the symplectomorphsm φ. It then follows that the resultng symplectc manfold ( M, ω) s (up to sotopy) ndependent of the choce of extenson ρ of ρ and of ɛ. ( M, ω) s called the blow-up of (M, ω) at p of weght λ. Suppose (M, ω) s a symplectc 4-manfold contanng a symplectcally embedded 2 sphere E of self-ntersecton 1 wth area E ω = πλ2. Then E can be blown down by reversng the above procedure. The resultng symplectc 4-manfold has a symplectcally mbedded ball B 4 (λ) replacng the symplectc surface E. 2. The Adjuncton Formula Let (M, ω) be a symplectc 4-manfold. ω determnes, up to homotopy, an almost complex structure on M and hence the Chern classes c (M, ω) are well-defned. Consder a symplectc mmerson f : (Σ, η) (M, ω). By a small perturbaton of f we can assume that the mmerson has only smple double ponts. Moreover snce the condton that an mmerson be symplectc s open t s possble to keep the map symplectc throughout the perturbaton, f we also allow η to be perturbed. Hence the symplectc mmersons wth only double ponts are a famly of maps of partcular nterest. We have:

3 DOUBLE POINTS AND THE PROPER TRANSFORM 3 Theorem 2.1. Let f : (Σ, η) (M, ω) be a symplectc mmerson of a compact symplectc surface (Σ, η) nto a symplectc 4-manfold (M, ω). Suppose that f has only double ponts. Then: c 1 (M, ω)[σ] = 2 2g + Σ Σ 2D (2.1) where g s the genus of Σ, D s the number of double ponts of f counted wth sgn and Σ Σ s the self-ntersecton of the class represented by f(σ). Proof. Consder f T M as a symplectc vector bundle over Σ. Snce f s a symplectc map T Σ s a symplectc subbundle of f T M. A symplectc complement to T Σ can be canoncally defned usng the symplectc orthogonal to T x Σ at each x Σ. Denote ths bundle ν. Then: f T M = T Σ ν as symplectc vector bundles and consequently as complex vector bundles. It follows that: c 1 (M, ω)[σ] = f c 1 (T M) = c 1 (f T M) = c 1 (T Σ) + c 1 (ν). Notng that ν s the normal bundle to f(σ) we have: c 1 (ν) = χ(ν) = Σ Σ 2D and: The result follows. c 1 (T Σ) = χ(σ) = 2 2g. We wll often abbrevate c 1 (M, ω)[σ] = c 1 (M)(Σ). Formula 2.1 for holomorphcally mmersed curves n Kähler surfaces follows from the classcal adjuncton formula of algebrac geometry. Consequently we wll call 2.1 the symplectc adjuncton formula. Analogous formulas for J-holomorphc curves n symplectc 4-manfolds are gven by McDuff [McD2]. These formulas are, t should be noted, deeper than 2.1. However, a symplectcally mmersed compact surface n a symplectc 4-manfold wth only double ponts need not be J-holomorphc for any almost complex structure J. Hence, formula 2.1 does not follow from these deeper results. 3. Proper Transforms Let (M, ω) be a compact symplectc 4-manfold wth frst Chern class c 1 (M) = c 1 (M, ω). Blowng up M at the ponts p 1 p k constructs a new symplectc 4-manfold that we denote ( M k, ω k ). Each pont p determnes a symplectcally embedded 2 sphere E of self ntersecton 1 n M k. Let

4 4 JOHN D. MCCARTHY AND JON G. WOLFSON [E ] denote the ntegral homology class determned by E and denote ts Poncare dual by [E ]. Then t follows that: c 1 ( M k k ) = c 1 (M) [E ] (3.1) =1 Further, there are neghborhoods N (E ) M k of the E and neghborhoods B (p ) M of the p such that there s a symplectomorphsm: π : Mk \ N (E ) M \ B (p ). Each B can be taken to be the symplectc ball around p that determnes the blow-up at p, and each N (E ) s then the tubular neghborhood of E determned by the symplectomorphsm of Proposton 1.1. Let f : (Σ, η) (M, ω) be a symplectc mmerson of the surface (Σ, η) nto (M, ω). Let f : ( Σ, η) ( M k, ω k ) be a symplectc mmerson of the surface ( Σ, η) nto ( M k, ω k ). Denote the mmersed surfaces by f(σ) M and f( Σ) M k, respectvely. We say f s a proper transform of f f: π( f( Σ) \ ( f( Σ) N (E ))) = f(σ) \ (f(σ) B (p )). A proper transform s not unque, unlke the analogous defnton n complex geometry. A resoluton of the sngulartes of the mmerson f through blowng up s a symplectc embeddng f : ( Σ, η) ( M, ω) where ( M, ω) s some blow up of (M, ω), such that f s a proper transform of f. To resolve the sngulartes of a gven symplectc mmerson f we can, as remarked above, assume that f has only solated double ponts, we henceforth make ths assumpton. Each double pont p has a sgn whch we wll call ts ndex and denote (p). Thus, n formula 2.1, f {p 1 p r } denote the double ponts of the mmerson then r D = (p j ). j=1 Theorem 3.1. Let f : (Σ, η) (M, ω) be a symplectc mmerson of a compact surface nto a symplectc 4-manfold such that f has only solated double ponts, {p 1 p r }. If (p j ) = 1 for some j then there does not exst a resoluton of f through blowng up. Proof. Snce blow up s a local constructon to prove the theorem t suffces to show that there does not exst a proper transform of f whose only double ponts are {p 1 ˆp j p r }. In other words t suffces to show that blowng up cannot resolve a double pont wth ndex 1. Applyng the symplectc adjuncton formula to f(σ) we have c 1 (M)[Σ] = 2 2g + Σ Σ 2D (3.2) where g = g(σ) s the genus of Σ and D = r l=1 (p l ). Suppose M s obtaned from M by blowng up the pont p j and that f : ( Σ, η) ( M, ω)

5 DOUBLE POINTS AND THE PROPER TRANSFORM 5 s a resoluton of the sngularty of f at p j. In other words, suppose that f s a proper transform of f wth double ponts {p 1 ˆp j p r }. Applyng the adjuncton formula to f we have c 1 ( X)( Σ) = 2 2 g + Σ Σ 2 D (3.3) where g = g( Σ) s the genus of Σ and D = D + 1. (3.4) M contans a new class represented by the exceptonal curve E = E pj. Snce f s a proper transform of f the homology classes represented by f and f satsfy: [ f( Σ)] = [f(σ)] + k[e] (3.5) for some nteger k. The adjuncton formula 3.3 becomes usng 3.1, 3.4 and 3.5 (c 1 (X) [E] )([f(σ)] + k[e]) = 2 2 g + ([f(σ)] + k[e]) 2 2(D + 1). (3.6) Hence, Usng 3.2 ths becomes: c 1 (X)[Σ] + k = 2 2 g + Σ Σ k 2 2D 2. (3.7) Thus, regardless of the value of k, we have: g g = 1 (k(k + 1)) 1. (3.8) 2 g < g. (3.9) Therefore the proper transform f has strctly smaller genus than f. Ths means that by a local operaton we have reduced the number of handles of Σ. Ths s clearly mpossble. Remark 3.1. In the topologcal category, double ponts of mmersed surfaces can be resolved usng blowng up, regardless of whether they are postve or negatve. In ths category, blowng up corresponds to connect sum wth CP Examples In ths secton we produce examples of symplectc mmersons of compact surfaces nto symplectc four manfolds whch have only solated double ponts, some of whch must have negatve ndex. Let (T 2, ω 1 ) be the 2 torus wth area form ω 1 and (S 2, ω 2 ) the 2 sphere wth area form ω 2. Set (M, ω λ ) = (T 2 S 2, ω 1 λω 2 ) where λ > 0. Let Σ be a 2 torus and defne F = (f n, g k ): F : Σ M x (f n (x), g k (x))

6 6 JOHN D. MCCARTHY AND JON G. WOLFSON where f n : T 2 T 2 s an n fold coverng map and g k : T 2 S 2 s a smooth map of degree k. Snce f n s an mmerson, F s an mmerson. Moreover: F (ω 1 λω 2 ) = f n(ω 1 ) + λg k(ω 2 ). (4.1) Clearly, choosng λ suffcently small F s a symplectc map. Perturb F so that t remans a symplectc mmerson and has only solated double ponts. The symplectc adjuncton formula 2.1 gves: c 1 (T 2 S 2 )[F (Σ)] = χ(t 2 ) + Σ Σ 2D (4.2) where D = the total number of double ponts counted wth sgn. It follows that : Hence: 2k = 0 + 2n( k) 2D. (4.3) D = k(1 n). (4.4) For n > 1, D s negatve, so there must be double ponts wth negatve ndex. From Theorem 3.1, we see that these surfaces provde counterexamples to Gromov s program for resolvng the sngulartes of symplectc mmersons wth symplectc crossngs va blowng up as outlned n 3.4.4(D) of Partal Dfferental Relatons. Hence, these examples establsh the followng corollary of Theorem 3.1. Corollary 4.1. There exst symplectc mmersons wth symplectc crossngs whose sngulartes cannot be resolved by blowng up. 5. Resolvng Double Ponts wth Postve Index Double ponts of a symplectc mmerson wth ndex = +1 can, unlke double ponts of ndex = 1, be resolved usng blowng up. In fact there are dfferent ways of dong ths reflectng the nonunqueness of the symplectc proper transform. In ths secton we outlne one method whch has a close analogy to resoluton n the complex category. Let p be a double pont of the symplectc mmerson f : (Σ, η) (M, ω) wth ndex = +1. Consder a neghborhhood U of p whch we symplectcally dentfy wth a neghborhood of the orgn n R 4 equpped wth the standard lnear symplectc form ω 0. Such a neghborhood U s called a Darboux neghborhood of p. By perturbng f, f necessary, we can suppose that the mage of f n a neghborhood V U concdes wth the two tangent planes of f passng through the orgn. We are thus led to consder the geometry of a par of symplectc planes n R 4 whch ntersect postvely and transversely n the orgn. Ths problem has been analyzed by McDuff [McD3]. She shows that there s an almost complex structure J tamed by ω 0 so that both planes are J holomorphc

7 DOUBLE POINTS AND THE PROPER TRANSFORM 7 lnes. J n general wll not be ω 0 compatble (.e., ω 0 (J, ) wll not be a hermtan metrc). However, f J s ω 0 compatble, then we have: Proposton 5.1. Let p be a double pont gven locally by the ntersecton of two transverse symplectc planes whch are J holomorphc lnes for a ω 0 compatble almost complex structure J. Then p can be symplectcally resolved by blowng up. Proof. There s a 3 sphere, Sλ 3 = B4 (λ), centered on the orgn so that the planes ntersect Sλ 3 n dstnct fbers of the Hopf fbraton of S3 λ. Any proper transform for the blow up of p of weght λ then must ntersect N (E) n two dstnct fbers of the normal crcle bundle of E. Thus a symplectc proper transform whch resolves p can be defned by extendng across the two fbers of the normal bundle. Remark 5.1. Ths constructon determnes a unque proper transform; closely analogous to the proper transform of complex geometry. In the general case when J s not ω 0 compatble, t can be shown that: Proposton 5.2. Let f : (Σ, η) (M, ω) be a symplectc mmerson wth a postve double pont at p M. Suppose n a Darboux neghborhood V of p that the mage of f s two symplectc planes ntersectng transversely n the orgn. Then there s a smooth famly f t : Σ (M, ω) of symplectc mmersons t [0, 1], such that () f 0 = f () f t M\V = f 0 () f t V \{p} are mbeddngs (v) There s a neghborhood W V such that the mage of f 1 W conssts of two J holomorphc lnes ntersectng transversely n the orgn, where J s a ω 0 compatble almost complex structure. Combnng Propostons 5.1 and 5.2 shows that postve double ponts of symplectc mmersons can be resolved. References [G] Gromov, M., Partal Dfferental Relatons, Sprnger Verlag, Berln, [G-S] Gullemn, V and Sternberg, S., Bratonal equvalence n the symplectc category, Invent. Math. 97 (1989), [McD1] McDuff, D., Blowng up and symplectc embeddngs n dmenson 4, Topology 30 (1991), [McD2] McDuff, D., The Local Behavor of Holomorphc Curves n Almost Complex 4-manfolds, J. Dff. Geom. 34 (1991), [McD3] McDuff, D., Immersed spheres n symplectc 4-manfolds, Ann. Inst. Fourer, Grenoble 42, 1-2 (1992), Department of Mathematcs, Mchgan State Unversty, East Lansng, MI 48824