Anti-symplectic involutions on rational symplectic 4-manifolds
aa r X i v : . [ m a t h . S G ] M a y Anti-symplectic involutions on rational symplectic4-manifolds
V. Kharlamov and V. Shevchishin
Dedicated to the memory of V.A. Rokhlin
Abstract.
This is an expanded version of the talk given by the first authorat the conference ”Topology, Geometry, and Dynamics: Rokhlin 100”. Thepurpose of this talk was to explain our current results on classification ofrational symplectic 4-manifolds equipped with an anti-symplectic involution.Detailed exposition will appear elsewhere.
1. Introduction
This paper deals with such questions as classifying rational symplectic 4-mani-folds equipped with an anti-symplectic involution and existence of an equivariantK¨ahler structure on these manifolds.Our motivation was twofold. Firstly, it is a natural, necessary, step in general-izing real algebraic geometry achievements on the range of problems in the spirit ofHilbert’s 16th problem into the realm of symplectic geometry; especially, in whatconcerns the study of interactions between topological and deformation equivalenceinvariants. Secondly, it is very closely connected with the contemporary study ofreal analogs of Gromov-Witten invariants like those discovered by J.-Y. Welschinger[ W ] in early 2000s.By a rational symplectic manifold we mean a symplectic manifold which canbe obtained by a sequence of symplectic blow-ups and blow-downs from C P and C P × C P equipped with their standard symplectic structures λω F S and λω F S × µω F S , where ω F S states for symplectic structures given by Fubini-Studymetrics (see [
McD-P ] for the definition and basic properties of symplectic blow-upsand blow-downs). As is known [
McD-1 ], the rational symplectic 4-manifolds haveno other blow-down minimal models than ( C P , λω F S ) and ( C P × C P , λω F S × µω F S ), and, furthermore, each rational symplectic 4-manifold can be obtained froma minimal model by a simultaneous blow-up of a finite collection of disjoint embed-ded balls.By an anti-symplectic involution on a symplectic manifold (
X, ω ) we call adiffeomorphism c : X → X such that c = id X and c ∗ ω = − ω . We name such atriple ( X, ω, c ) a real symplectic manifold and say that it is
K¨ahler if there exists
Mathematics Subject Classification.
Primary 57R17, 53D99, 14J26, 14P99. an integrable complex structure J with respect to which c is an anti-holomorphicinvolution ( a real structure ) and ω a K¨ahler form on ( X, J ). If (
X, ω, c ) does notadmit any c -equivariant blow-down, we call it c -minimal. We say that (
X, ω X , c X ) is c -equivariant deformation equivalent to ( Y, ω Y , c Y ),if there exists a chain of triples ( X i , ω i , c i ), i = 0 , . . . , k , such that ( X, ω X , c X )is isomorphic to ( X , ω , c ), ( Y, ω Y , c Y ) to ( X k , ω k , c k ), and ( X i , ω i , c i ), for each1 i k , is isomorphic to a triple obtained by a smooth c i − -equivariant variation ω i − ( t ) of ω i − on X i − .Our first result is concerned with existence of K¨ahler structure. Theorem . Every real rational symplectic 4-manifold ( X, ω, c ) is c -equivari-ant deformation equivalent to a K¨ahler one. If, in addition, ( X, ω, c ) is c -minimal,then ( X, ω, c ) itself is K¨ahler. According to Kodaira embedding theorem, each compact K¨ahler surface is de-formation equivalent to a projective one. The equivariant version of this theoremand c -equivariant deformation unicity of K¨ahler structures lead immediately to ournext result. Theorem . Deformation classification of real rational symplectic 4-manifoldscoincides with deformation classification of real rational algebraic surfaces. (cid:3)
As is proved in [ DK ], two rational real algebraic surfaces are deformationequivalent if, and only if, their complex conjugation involutions are diffeomorphic.Thus, we get as an easy consequence the following result. Theorem . Real rational symplectic 4-manifolds ( X, ω X , c X ) and ( Y, ω Y , c Y ) are c -equivariant deformation equivalent if, and only if, the involutions c X and c Y are diffeomorphic. (cid:3)
2. Tools
The tools that play the key role in the proofs of our main results are theequivariant versions of: (1) the technique of J -holomorphic curves invented byM. Gromov, (2) the relations between Gromov-Witten and Seiberg-Witten invari-ants established by C. Taubes and enhanced by T. J. Li and A.Liu, and (3) theinflation technique invented by F. Lalonde and D. McDuff. Transition to a versionequivariant with respect to an anti-symplectic involution is often rather easy andstraightforward, but in certain cases (and this concerns especially applications ofthe inflation technique) it rises additional difficulties caused by the need to under-stand codimension-one events.To formulate principal technical statements, we need to fix a few definitionsand notations.We denote by Ω( X ) the space of symplectic forms on X , by C = C ( X ) ⊂ H ( X, R ) the set of cohomology classes representable by symplectic form, and foran element A or a subset U ⊂ C ( X ) we denote by Ω( X, A ) and resp. by Ω(
X, U ) thesubspace of symplectic forms whose cohomology class equals A or resp. lies in U .In the case when c : X → X is an anti-involution we denote by Ω( X, c ), Ω(
X, A, c ),and resp. C ( X, c ), the spaces of c -anti-invariant forms (in the given cohomologyclass A ), and the space of their cohomology classes.For a symplectic form ω we write Ω( X, ω ) instead of Ω( X, [ ω ]), and Ω( X, ω, c )instead of Ω( X, [ ω ] , c ). Denote by Ω ( X, ω, c ) (resp. by Ω ( X, ω )) the connected
NTI-SYMPLECTIC INVOLUTIONS ON RATIONAL SYMPLECTIC 4-MANIFOLDS 3 component of the space Ω(
X, ω, c ) (resp. of Ω(
X, ω )) containing the form ω , andby π Ω( X, ω, c ) (resp. by π Ω( X, ω )) the whole set of connected components ofΩ(
X, ω, c ) (resp. by π Ω( X, ω )).By E = E ( X ) ⊂ H ( X, Z ) we denote the set of homology classes represented bysmoothly embedded 2-spheres with self-intersection − exceptionalspheres ). Theorem
L-L ]) . Let X be a rational symplectic -manifold and Ω ∈ H ( X, R ) a cohomology class. Then Ω is represented by a symplectic form if and only if it satisfies the following condi-tions: (Ω0) R X Ω > ; (Ω1) R E Ω = 0 for every exceptional sphere E ⊂ X .Furthermore, two symplectic forms ω , ω have equal Chern classes, c ( X, ω ) = c ( X, ω ) , if and only if there exists a family Ω t ∈ H ( X, R ) of cohomology classeswhich satisfy the conditions (Ω0 , Ω1) and connect the classes of the forms, i.e. , Ω = [ ω ] and Ω = [ ω ] . Given a symplectic manifold (
X, ω ) with an ω -tamed almost complex structure J , we call the curve cone the cone in H ( X ; R ) generated by the classes Poincar´edual to J -holomorphic curves in X , and denote this cone by C X,J . When, inaddition, (
X, ω ) is equipped with an anti-involution c and J is c -anti-invariant,we define the real curve cone R C X,J as the cone generated by Poincar´e duals of c -invariant J -holomorphic curves (including combinations D + cD !). We put R C KD > X,J = { D ∈ R C X,J : KD > } . We say that (
X, J, c ) (respectively, (
X, ω, J, c )) is real rational ruled , if it isequipped with a c -invariant J -holomorphic ruling : that is a c -equivariant smoothmap X → S with J -holomorphic fibers and J -holomoprhic sphere as a genericfiber. Theorem cf. [ Z ]) . Let ( X, ω ) bea closed symplectic manifold equipped with an anti-symplectic involution c and an ω -tamed c -anti-invariant almost complex structure J . Then, there exist countablymany smooth irreducible curves L i with L i = − , , or such that L i + cL i generateextremal rays in the cone R C X,J of real curves on ( X, J, c ) and R C X,J = R C KD > X,J + X R + ( L i + cL i ) . If besides these L i there is one with L i = 1 (respectively, L i = 0 ), then X is C P and c is c st (respectively, ( X, J, ω, c ) is a c -minimal real symplectic manifold witha c -invariant J -holomorphic ruling having L i as a fiber). Lemma c -equivariant embedded curves, cf. [ McD-3 ] ) . Let ( X, ω, c ) be a rational -manifold with anti-involution, J a generic c -anti-invariant ω -tamed almost complex structure, and A ∈ H ( X, Q ) a rational homology classsatisfying c ∗ A = − A , A > and A · [ E ] > for every exceptional sphere E . Thensome multiple mA is represented by a ω -symplectic c -invariant J -holomorphiccurve Σ . Lemma cf. [ McD-2 ]) . Let J be a c -anti-invariant ω -tamed almost complex structure on a real symplectic manfiold ( X, ω, c ) V. KHARLAMOV AND V. SHEVCHISHIN that admits a real non-singular J -holomoprhic curve Z with Z · Z > . Then, thereexists a family ω t , t > , of c -anti-invariant symplectic forms that all tame J andhave cohomology class [ ω t ] = [ ω ] + t PD( Z ) . These two lemmas are crucial for the proof of the next theorem. (For purposesof this paper, it would be sufficient to reduce the statement of this theorem to thecase Y = [0 ,
1] and Y ′ = ∂Y .) Theorem . Let Y bea compact manifold possibly with boundary, { ( X, ψ y , J y ) } y ∈ Y a family of ratio-nal symplectic manifolds, { Ω y } y ∈ Y a continuous family of R -valued -cohomologyclasses on X , and { ω ′ y } y ∈ Y ′ , for a certain closed subset Y ′ ⊂ Y , a continuous familyof symplectic forms on X . Assume that the following conditions are satisfied: (NM0) There exists a continuous map Y × I → C ( X ) ⊂ H ( X, R ) , ( y, t ) A y,t , deforming the classes [ ψ y ] = A y, into the classes Ω y = A y, . (NM1) For each y ∈ Y and each irreducible J y -holomorphic curve C with [ C ] < , one has R C Ω y > ; (NM2) For each y ∈ Y ′ , the form ω ′ y tames J y and has the cohomology class Ω y .Then there exists a family { ω y } y ∈ Y of symplectic forms on X such that ω y = ω ′ y for y ∈ Y ′ and [ ω y ] = Ω y for y ∈ Y . Moreover, there exist families { ˜ J y } y ∈ Y , tamedby { ω y } y ∈ Y which are arbitrarily close to the family { J y } y ∈ Y .Furthermore, if c : X → X is an anti-involution such that the families { J y } y ∈ Y , { Ω y } y ∈ Y , and { ω ′ y } y ∈ Y ′ are c -anti-invariant, then the family { ω y } y ∈ Y also can bechosen c -anti-invariant.
3. Proof Outline
We start from classifying c -minimal models. Applying Theorem 2.2, and usingso-called Gromov’s automatic regularity (see [ HLS ]) for constructing rulings, weget the following result.
Theorem . Let ( X, ω ) be a non-minimal rational symplectic -manifoldwith an anti-involution c , and let J be an ω -tame c -anti-invariant almost complexstructure. For each exceptional J -holomorphic curve E , set E ′ = c ( E ) and orient E ′ as a J -holomorphic curve. Then: (1) The minimal value of [ E ] · [ E ′ ] lies between − and . (2) If [ E ] · [ E ′ ] = − , the exceptional curve E is c -invariant and there ex-ists a symplectic -manifold with an anti-involution ( X , ω , c ) such that ( X, ω, c ) is the result of the symplectic blow-up of ( X , ω , c ) performedin a c -invariant symplectic ball B . (3) If [ E ] · [ E ′ ] = 0 , the curves E and E ′ are disjoint, and there exists a sym-plectic -manifold with an anti-involution ( X , ω , c ) and disjoint sym-plectic balls B, B ′ ⊂ X with B ′ = c ( B ) , such that ( X, ω, c ) is the resultof the symplectic blow-up of ( X , ω , c ) performed in B, B ′ . (4) If [ E ] · [ E ′ ] = 1 , there exists a unique J -holomorphic ruling pr : X → Y = C P such that E ∪ E ′ is a fiber of this ruling. Moreover, pr ◦ c = c st ◦ pr and y ∗ = pr ( E ∪ E ′ ) is a fixed point of c st , where c st : C P → C P is thestandard complex conjugation involution. (5) If there are no any exceptional curve C with [ C ] · [ C ′ ] < , then either b ( X ) = 7 , K X = 2 , and there exists an exceptional curve E such that NTI-SYMPLECTIC INVOLUTIONS ON RATIONAL SYMPLECTIC 4-MANIFOLDS 5 E + E ′ = − K X , or b ( X ) = 8 , K X = 1 , and there exists an exceptionalcurve E such that E + E ′ = − K X . In the both cases the curve cone R C X,J is generated by K X . (cid:3) Corollary . For c -minimal real rational symplectic 4-manifolds, the listof possible underlying manifolds X and X c = Fix c is as follows: (1) X = C P and X c = R P . (2) X = C P × C P and X c is either S × S , S , or ∅ . (3) X admits a c -equivariant rational ruling with m -singular fibers, m > ,and X c = mS . (4) X has the same homology as a del Pezzo surface with K = 2 , the anti-involution c acts in H ( X ) as a reflection in K , and X c = 4 S . (5) X has the same homology as a del Pezzo surface with K = 1 , the anti-involution c acts in H ( X ) as a reflection in K , and X c = 4 S ⊔ R P . (cid:3) Thus, except four special types of small topological complexity (items (1), (2),(4), (5)), each of other c -minimal real rational symplectic 4-manifolds is rationalruled. Our proof strategy for Theorem 1.1 is to treat first the case of rational ruled4-manifolds, and then to deduce from it the remaining cases.For ( X, ω, c ) with a c -equivariant ruling pr : X → S as in item (3) of Corollary3.2 we observe, first, that: • Up to a c -equivariant diffeomorphism, the mapping pr : X → S isuniquely defined by m . • The c -anti-invariant part of H ( X ) is generated by 2 elements, F and H ,where F is the class of the fiber, F = 0, H = 0, and F · H is equal to 2for m even and 4 for m odd. • The cone of c -anti-invariant symplectic classes [ ω ] is given by aF + bH with a, b > • When (
X, c, pr ) is a generic algebraic real conic bundle, this cone coincideswith the cone of c -anti-invariant K¨ahler classes of X .After that we are left to prove the following theorem. Theorem . Let ( X, c, pr ) be a c -equivariant ruling whose fibers are pseudo-holomorphic with respect to two c -anti-invariant almost complex structures, J and J ′ , tamed by two c -anti-invariant symplectic structures, ω and ω , respectively. If ω and ω represent the same cohomology class, then they are c -equivariant isomor-phic. Proof.
We start from making the almost complex structures identical andintegrable at a neighborhood of the singular points of the singular fibers. Next, wepick a c -invariant symplectic form on the base, ω B , so that ω ′ = pr ∗ ω B is inducingthe same orientation on the normal bundle of the fibers as ω and ω . Then, weobserve that the forms ω + sω ′ , ω + sω ′ with s ≥ s ∗ >
0, the forms aω ′ + t · ω + (1 − t ) · ω with a ≥ s ∗ , t ∈ [0 , X .This path connecting ω with ω (through ω + aω ′ and ω + aω ′ ) is c -equivariant,but to apply Moser’s argument we need to make it of a constant cohomology class.To handle this, we use Theorem 2.5. Here, a specific for real geometry difficultyarises: we can not avoid appearance of J t -holomorphic ( − t . To overcome such a difficulty we replace the path ( J t , ω t ) by a chain V. KHARLAMOV AND V. SHEVCHISHIN of c -equivairiant pathes ( J kt , ω kt ) , k ∈ { , . . . , r } , t
1, so that all ω k , ω k are ofthe same cohomology class, ( J , ω ) = ( J , ω ), ( J r , ω r ) = ( J , ω ), and ( J k , ω k )is c -equivariant isomorphic to ( J k +10 , ω k +10 ). Our construction goes as follows. Weput the initial path in a generic position with respect to the divisors responsablein the space of almost complex structures for appearance of ( − t where a c -equivariant J t − holomorphic ( − J t with t close to t by structures thatare integrable in a neighborhood of the curve and represent locally one of standardAtiyah-flop families [ A ] of an appropriate signature: x + y + z = − ( t − t ) if c hasno fixed point on the ( − x + y − z = − ( t − t ) otherwise. Recallthat over t = t the latter have a symmetry permuting t = t + ǫ and t = t − ǫ .Next, taking ǫ > J t + ǫ and J t − ǫ outside the insertedAtiyah-flop part to achieve a full symmetry between J t + ǫ and J t − ǫ . After thatthere remain to apply the inflation procedure to ω t + ǫ and ω t − ǫ symmetrically forachieving the cohomology class relation [ ω t + ǫ ] = [ ω t − ǫ ] = [ ω ] . (cid:3) Remark . For all but one, with m = 3, c -minimal real rational ruled sym-plectic 4-manifolds there are no c -equivariant ( − c -equivariant isotopy in Theorem 3.3.The cases when X is C P or C P × C P (items (1) and (2) in Corollary 3.2)can be easily reduced to the case with a ruling: • If X is C P × C P and X c is either R P × R P or ∅ , then ( X, ω, c ) hasa c -equivariant rational ruling without singular fibers. • If X = C P × C P and X c = S , it acquirs a c -equivariant rational rulingwith two singular fibers after a blow up at a pair of c -conjugate points, p and c ( p ) = p . • Similarly, if X = C P then ( X, ω, c ) acquirs a c -equivariant rational rulingwithout singular fibers after a blow up at a c -invariant point, p = c ( p ).The last two cases (items (4) and (5) in Corollary 3.2): K = 2 with X c = 4 S and K = 1 with X c = 4 S ⊔ R P , are more intricate. Here, instead of symplectic( − − − Le ]).To treat the case K = 2 , X c = 4 S we perform a ( − X c :(1) This surgery replaces a neighborhood of such a Lagrangian sphere S ⊂ X c modeled on its cotangent bundle T ∗ ( S ) by a neighborhood of a sym-plectic ( − S new = C P modeled on its T ∗ ( C P ) = O ( −
2) linebundle. More precisely, we cut out a neighborhood symplectomorphic toa tubular ǫ -neighborhood in T ∗ ( S ), replace it by the standard resolution φ : ˜ U → U of U ⊂ Q = { P j =1 z j } ⊂ C , U = Q ∩ { P j =1 | z j | √ ǫ } which we equip with an SO (3)-invariant symplectic structure obtainedfrom φ ∗ ( i P j =1 dz j ∧ dz j ) by a small perturbation in a smaller neighbor-hood.(2) This surgery is compatible with anti-involutions, and the resulting realsymplectic 4-manifold, ( X ′ , ω ′ , c ′ ), turns out to have a c ′ -equivariant ra-tional ruling with m = 3 where S new is a c ′ -invariant bi-section (without NTI-SYMPLECTIC INVOLUTIONS ON RATIONAL SYMPLECTIC 4-MANIFOLDS 7 c ′ -fixed points) and the fibers split off from the representatives of c ( X ′ )intersecting S new .After that the same arguments as in our treatment of manifolds endowed witha rational ruling apply and prove that there exists an integrable structure J ′ , suchthat: • ( X ′ , J ′ , c ′ ) is real algebraic. • ω ′ is J ′ -K¨ahler. • S new is J ′ -holomorphic.Finally, we show that the result of the inverse ( − S new is also K¨ahlerand, due to the symplectic uniqueness of the ( − X, ω, c ) we have started from.The proof in the case K = 1, X c = 4 S ⊔ R P follows the same lines. Weindicate only the difference in surgeries. This time, we use a ( − rational blow-up ) which we apply to the R P -component of X c . Thissurgery transforms a neighborhood of R P ⊂ X c symplectomorphic to a tubular ǫ -neighborhood in T ∗ ( R P ) (which can be seen as the quotient of a neighbor-hood in T ∗ ( S ) by the involution induced by the antipodal map S → S ) into aneighborhood of S new = C P in its O ( −
4) line bundle (which can be obtained bypicking up a neighborhood of C P in O ( −
2) used for ( − − O ( − X ′ , ω ′ , c ′ ), has a c ′ -equivariant rational ruling with m = 4 where the ( − S new produced by the surgery is a c ′ -invariant bi-section (without c ′ -fixed points)and the fibers split off from representatives of c ( X ′ ) intersecting S new . References
A. Atiyah, M. F.
On analytic surfaces with double points.
Proc. Roy. Soc. London. Ser. A247 (1958), 237 - 244.DK. Degtyarev, Alex; Kharlamov, Viatcheslav.
Real rational surfaces are quasi-simple.
J.Reine Angew. Math. 551 (2002), 87 - 99.HLS. Hofer, Helmut; Lizan, Veronique; Sikorav, Jean-Claude.
On genericity for holomorphiccurves in four-dimensional almost-complex manifolds.
J. Geom. Anal. 7 (1997), no. 1,149159.Le. Lerman, E.
Symplectic cuts.
Math. Res. Lett. 2(3) (1995), 247 –258.L-L. Li, T.-J.; Liu, A.-K.:
Uniqueness of symplectic canonical class, surface cone and sym-plectic cone of 4-manifolds with b + = 1 . J. Diff. Geom., 58 (2001), 331–370.McD-1. McDuff, Dusa.
The structure of rational and ruled symplectic 4-manifolds.
J. Amer.Math. Soc. 3 (1990), no. 3, 679712.McD-2. McDuff, Dusa.
Symplectomorphism Groups and Almost Complex Structures.
Essays ongeometry and related topics, Vol. 1, 2, 527 - 556, Monogr. Enseign. Math., 38, Enseigne-ment Math., Geneva, 2001.McD-3. McDuff, Dusa.
From symplectic deformation to isotopy.
Topics in symplectic 4-manifolds(Irvine, CA, 1996), 8599, First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998.McD-P. McDuff, Dusa; Polterovich, Leonid.
Symplectic packings and algebraic geometry.
Invent.Math. 115 (1994), no. 3, 405 - 434.W. Welschinger, Jean-Yves.
Invariants of real symplectic 4-manifolds and lower bounds inreal enumerative geometry.
Invent. Math. 162 (2005), no. 1, 195 - 234.Z. Zhang, Weiyi.
The curve cone of almost complex 4-manifolds.
Proc. London Math. Soc.(3) 115 (2017) 1227 - 1275
V. KHARLAMOV AND V. SHEVCHISHIN
Universit´e de Strasbourg et IRMA (CNRS), 7 rue Ren´e-Descartes, 67084 Stras-bourg Cedex, France
E-mail address : [email protected] University of Warmia and Mazury, ul. Soneczna 54, 10-710 Olsztyn, Poland
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