Obtaining the One-Holed Torus from Pants: Duality in an SL(3,C)-Character Variety
aa r X i v : . [ m a t h . AG ] M a r OBTAINING THE ONE-HOLED TORUS FROM PANTS:DUALITY IN AN
SL(3 , C ) -CHARACTER VARIETY SEAN LAWTON
Abstract.
The SL(3 , C )-representation variety R of a free group F r arises nat-urally by considering surface group representations for a surface with boundary.There is a SL(3 , C )-action on the coordinate ring of R . The geometric pointsof the subring of invariants of this action is an affine variety X . The points of X parametrize isomorphism classes of completely reducible representations. Thecoordinate ring C [ X ] is a complex Poisson algebra with respect to a presenta-tion of F r imposed by the surface. In previous work, we have worked out thebracket on all generators when the surface is a three-holed sphere and whenthe surface is a one-holed torus. In this paper, we show how the symplec-tic leaves corresponding to these two different Poisson structures on X relateto each other. In particular, they are symplectically dual at a generic point.Moreover, the topological gluing map which turns the three-holed sphere intothe one-holed torus induces a rank preserving Poisson map on C [ X ]. Introduction
In [8] we describe two competing Poisson structures on the variety of charactersof representations of a rank 2 free group into SL(3 , C ). The purpose of this paper isto show that these two structures generically define symplectically dual symplecticleaves, and that a natural topological mapping relates the two Poisson structuresin a non-trivial fashion.For the remainder of this section we briefly describe character varieties and theirsmooth stratum’s foliation by complex symplectic submanifolds. In these terms weformulate our main theorems. In Section 2, we describe in further detail past resultsnecessary to make sense of the discuss at hand. In particular, for the three-holedsphere and the one-holed torus we explicitly review the algebraic structure and thePoisson structure of the character variety. Lastly, in Section 3 we restate and proveour main theorems.1.1. Algebraic Structure of X (Σ n,g ) . Let Σ n,g be a compact, connected, orientedsurface of genus g with n > g = 0 we assume n ≥
3. Itsfundamental group has the following presentation: π (Σ n,g , ∗ ) = { x , y , ..., x g , y g , b , ..., b n | Π gi =1 [ x i , y i ]Π nj =1 b j = 1 } . The group F r := π (Σ n,g , ∗ ) is always free of rank r = 2 g + n − n,g retractsto a wedge of 2 g + n − G = SL(3 , C ) and let { g , ..., g r } be generators Date : October 30, 2018.2000
Mathematics Subject Classification.
Primary 14L24; Secondary 53D30.
Key words and phrases. poisson, character variety, free group. of F r . The representation variety R = Hom( F r , G ) is bijectively equivalent to G × r given by evaluation: ρ ( ρ ( g ) , ..., ρ ( g r )) , and so inherits the structure of a smooth affine variety from G . The coordinate ringof G is the complex polynomial ring in 9 indeterminates subject to the irreduciblerelation det( X ) − X = ( x ij ) is a generic matrix and x ij are the 9 indeter-minates. There is a polynomial action of G on the coordinate ring of R , denotedby C [ R ], by conjugation in r generic matrices ; that is, for g ∈ G and f ∈ C [ R ]: g · f ( X , ..., X r ) = f ( g − X g, .., g − X r g ) . Work of Procesi from 1976 [10] implies that the ring of invariants C [ R ] G is generatedby { tr ( W ) | w ∈ F r , | w | ≤ } . Here W is the word w in F r with its letters replaced bygeneric matrices. Thus, C [ R ] G is a finitely generated domain, and so its geometricpoints are an irreducible algebraic set, X (Σ n,g ) = Spec( C [ R ] G ) = R // G , called the G - character variety of F r . We note that the quotient notation just used meansthat it is a categorical quotient for the G -action (see [1] or [9]).1.2. The Boundary Map and Foliation of the Top Stratum.
The coordinatering of G // G is C [ G // G ] = C [tr ( X ) , tr (cid:0) X − (cid:1) ] . So G // G = C which we parametrize by coordinates ( τ (1) , τ ( − ). We then definethe boundary map b i : X = R // G = Hom( π (Σ n,g , ∗ ) , G ) // G −→ G // G by sending a representation class, [ ρ ] [ ρ | b i ] = ( τ i (1) , τ i ( − ), to the class corre-sponding to the restriction of ρ to the boundary b i . Subsequently we define b n,g = ( b , ..., b n ) : X = G × r // G −→ ( G // G ) × n . The map b n,g depends on the surface, not only its fundamental group. We refer toit as a peripheral structure , and the pair ( X , b n,g ) as the relative character variety .Let L = b − n,g (( τ , τ − ) , ..., ( τ n (1) , τ n ( − )) , and X be the complement of the sin-gular locus (a proper closed sub-variety) in X . So X is a complex manifold thatis dense in X . At regular values of b n,g (these are generic since b n,g is dominant), L ∩ X is a submanifold of dimension 8 r − − n = 16( g −
1) + 6 n . It is shownin [8] that the union of these leaves , L = L ∩ X , foliate X by complex symplecticsubmanifolds, making X a complex Poisson manifold. This structure continuouslyextends over all of X .1.3. The Quotient Map and Main Results.
There are two orientable surfaceswith Euler characteristic −
1; the three-holed sphere and the one-holed torus. Bothof these surfaces have fundamental groups free of rank 2. Moreover, there is anatural topological quotient mapping (independent of orientation) which maps thethree-holed sphere (hereafter referred to as pants) to the one-holed torus q : Σ , → Σ , . In [8] we work out explicitly (with respect to a coordinate system for X and choicesof orientation for the surfaces) the Poisson structures for the pants and the one-holed torus.We now state our main theorems: NE-HOLED TORUS FROM PANTS 3
Theorem 1.
Depending on the choice of orientation, q ∗ : C [ X (Σ , )] → C [ X (Σ , )] is generically a rank preserving Poisson (anti) morphism. Let L (Σ , ) and L (Σ , ) be generic symplectic leaves of X . Theorem 2. L (Σ , ) and L (Σ , ) are (generically) transverse and so are symplec-tically dual to each other. Acknowledgments.
I would like to thank my adviser Bill Goldman for intro-ducing me to this subject. Also, this work was inspired by a visit to McMaster’sUniversity while I was visiting Hans Boden. I would like to thank him and McMas-ter’s for that opportunity. Additionally, I would like to thank Instituto SuperiorT´ecnico for hosting me while I was working on the draft of this paper.2.
Past Results and Background
In this section we very briefly review some of the results from [6, 8] that we willneed to prove our theorems.2.1.
Symplectic and Poisson Structure on X (Σ n,g ) . In 1997, Guruprasad,Huebschmann, Jeffrey, and Weinstein [4] showed ω (in the following commutativediagram) defines a symplectic form on the leaf L defined in the introduction: H ( S, ∂S ; g Ad ) × H ( S ; g Ad ) ∪ / / H ( S, ∂S ; g Ad ⊗ g Ad ) tr ∗ (cid:15) (cid:15) H ( S, ∂S ; C ) ∩ [ Z ] (cid:15) (cid:15) H ( S ; g Ad ) × H ( S ; g Ad ) O O ω / / H ( S ; C ) = C With respect to this 2-form, we show in [8] Goldman’s proof [2, 3] of the Poissonbracket (a Lie bracket and derivation) generalizes directly to relative cohomology.Let Σ be an oriented surface with boundary, and α, β ∈ π (Σ , ∗ ). Let α ∩ β bethe set of (transverse) double point intersections of α and β . Let ǫ ( p, α, β ) be theoriented intersection number at p ∈ α ∩ β and let α p ∈ π (Σ , p ) be the curve α based at p .In these terms the bracket is defined on C [ X ] by: { tr( ρ ( α )) , tr( ρ ( β )) } = X p ∈ α ∩ β ǫ ( p, α, β ) (cid:0) tr( ρ ( α p β p )) −
13 tr( ρ ( α ))tr( ρ ( β )) (cid:1) . Algebraic Structure of X (Σ , ) and X (Σ , ) . We now review the algebraicstructure of C [ X ] for the pants and one-holed torus and the corresponding Poissonstructures in those cases. Details are available in [6, 8].Let C [ t (1) , t ( − , t (2) , t ( − , t (3) , t ( − , t (4) , t ( − , t (5) ]be a freely generated complex polynomial ring, and let R = C [ t (1) , t ( − , t (2) , t ( − , t (3) , t ( − , t (4) , t ( − ] . S. LAWTON
Define the following ring homomorphism, R [ t (5) ] Π −→ C [ X ]by t (1) tr ( X ) t ( − tr (cid:0) X − (cid:1) t (2) tr ( X ) t ( − tr (cid:0) X − (cid:1) t (3) tr ( X X ) t ( − tr (cid:0) X − X − (cid:1) t (4) tr (cid:0) X X − (cid:1) t ( − tr (cid:0) X − X (cid:1) t (5) tr (cid:0) X X X − X − (cid:1) It can be shown using trace equations that Π is surjective, and hence R [ t (5) ] / ker(Π) ∼ = C [ X ] . The Krull dimension of X is 8 since generic orbits are 8 dimensional. Hence, ker(Π)is non-zero and principal.Let S be the formal sum of the elements in the group generated by the permu-tations (in cycle notation)(1 , − , − , −
4) and (1 , − , − − , R [ t (5) ] / ker(Π). The action is induced bythe following elements of the Out( F ): t = (cid:26) x x x x i = (cid:26) x x − x x The group generated has order 8 and is isomorphic to the dihedral group D . In[6] we show Theorem 3. (1) X = G × // G is a degree hyper-surface in C (2) ker(Π) = ( t − P t (5) + Q ) where P, Q ∈ R (3) There is a D -equivariant surjection (submersion) m : X → C , generically -to- . (4) P and Q are given by: P = S (cid:16) (cid:0) t (1) t ( − t (2) t ( − − t (1) t ( − t ( − + 2 t (1) t ( − + 2 t (3) t ( − (cid:1)(cid:17) − Q = S (cid:16) (cid:0) t ( − t − t t (2) + 4 t t t (3) − t t ( − t (2) − t ( − t ( − t ( − t − t (4) t (3) t (2) t (1) t ( − + 8 t (1) t (3) t − +8 t ( − t (1) t − t t (2) t (1) + 4 t (4) t ( − t + t ( − t ( − t (2) t (1) + t ( − t ( − t (3) t (4) + 4 t ( − t ( − t (3) t (1) + 4 t + 4 t +12 t ( − t ( − t (1) − t ( − t (2) t (3) − t (1) t ( − − t (3) t ( − (cid:1)(cid:17) + 92.3. Poisson Structures of X (Σ , ) and X (Σ , ) . For our purposes a Poissonvariety is an affine variety X over C endowed with a Lie bracket { , } on its coordinatering C [ X ] that acts as a formal derivation (satisfies the Leibniz rule). On thesmooth strata of X (denoted by X ), it makes X a complex Poisson manifold inthe usual sense (by Stone–Weierstrass). For any such Poisson bracket there existsan exterior bi-vector field a ∈ Λ ( T X ) whose restriction to symplectic leaves isgiven by the symplectic form as { f, g } = ω ( H g , H f ) (where H f = { f, ·} is called NE-HOLED TORUS FROM PANTS 5 the Hamiltonian vector field). Let f, g ∈ C [ X ]. Then with respect to interiormultiplication { f, g } = a · df ⊗ dg. In local coordinates ( z , ..., z k ) it takes the form a = X i,j a i,j ∂∂z i ∧ ∂∂z j and so { f, g } = X i,j (cid:18) a i,j ∂∂z i ∧ ∂∂z j (cid:19) · (cid:18) ∂f∂z i dz i ⊗ ∂g∂z j dz j (cid:19) = X i,j a i,j (cid:18) ∂f∂z i ∂g∂z j − ∂f∂z j ∂g∂z i (cid:19) . Denote the bi-vector associated to X (Σ n,g ) by a (Σ n,g ). In the case of the pantsor the one-holed torus, there are 9 generating functions of C [ X ]: t ( ± i ) for 1 ≤ i ≤ t (5) . Since the bi-vector is a Lie bracket and a derivation, its formulation isin these terms. Let a i,j = { t ( i ) , t ( j ) } . In [8] we show the following two structuretheorems. Theorem 4.
The Poisson bi-vector on X (Σ , ) is a (Σ , ) = ( P − t (5) ) ∂∂t (4) ∧ ∂∂t ( − + (1 − i ) (cid:18) a , ∂∂t (4) ∧ ∂∂t (5) (cid:19) , where a , = ∂∂t ( − ( Q − t (5) P ) and where i = i ti t is the mapping x i x − i . In D define i = ti t ; the mapping which sends x x − .Additionally, define the following elements of the group ring of D : • Σ = 1 + i − i − i • Σ = 1 + i − t − it .Note 12 Σ Σ = 1 + i − i − i − t − it + i t + i t . Then after doing 28 calculations and observing symmetry along the way, weconclude:
Theorem 5.
The Poisson bi-vector field on X (Σ , ) is a (Σ , ) =Σ (cid:18) a , ∂∂t (1) ∧ ∂∂t (2) (cid:19) + Σ (cid:18) a , ∂∂t (3) ∧ ∂∂t (4) (cid:19) + 12 Σ Σ (cid:18) a , ∂∂t (1) ∧ ∂∂t (3) + a , − ∂∂t (1) ∧ ∂∂t ( − (cid:19) , where: • a , = t (3) − t (1) t (2) • a , = t (1) t (3) − t ( − t (2) + t ( − • a , − = − t ( − + t (1) t ( − • a , = − t + t ( − − t ( − t ( − − t (2) t ( − + t ( − t (2) t ( − − t (3) t (4) . Comment 6.
We note here that the orientations chosen on these surfaces areopposite. Our presentation of the pants has the boundary on the outside whereasthe one-holed torus has the same boundary (after projection) on the inside. Since
S. LAWTON the orientations of the boundaries are the same the surfaces are “inside-out” withrespect to each other, and so the orientations are reversed. Consequently, if thequotient mapping taking the pants to the one-holed torus is to preserve orientationsone of the above two bi-vectors must be multiplied by − . Obtaining the Torus from Pants
Let q : Σ , → Σ , be the quotient map given by identifying two of the bound-aries (call them b and b ). Let x be a fixed base point. And let x ∈ b and x ∈ b be also fixed.Then the third boundary b in Σ , is homotopic to ( b b ) − in π (Σ , , x ).Let γ and γ be paths from x to x and x to x respectively. Then q ( γ γ − ) := β is a non-trivial based loop in Σ , .Moreover, ( γ γ − ) b ( γ γ − ) − is homotopic to b − since γ − b γ is homotopicto γ − b − γ in Σ , .Therefore, q ♯ : π (Σ , , x ) = h b , b , b | b b b = 1 i → π (Σ , , x ) = h α, β, γ | [ α, β ] γ = 1 i is injective and given by b α, b βα − β − , and b [ α, β ] − . Consequently, q ∗ : X (Σ , ) → X (Σ , ) is given by[( A, B )] [( A, BA − B − )] , and q ∗ : C [ X (Σ , )] → C [ X (Σ , )] is given by f f ◦ q ∗ .To be concrete we write the assignments which determine q ∗ : t (1) t (1) , t ( − t ( − , t (2) t ( − , t ( − t (1) ,t (3) t (5) , t ( − tr (cid:0) A − BAB − (cid:1) = P − t (5) ,t (4) tr (cid:0) ABAB − (cid:1) = t ( − t ( − + t ( − + t ( − t (2) − t ( − t (2) t ( − + t (3) t (4) ,t ( − tr (cid:0) A − BA − B − (cid:1) = t ( − t ( − + t (1) + t (3) t ( − − t (1) t (2) t ( − + t (2) t (4) , NE-HOLED TORUS FROM PANTS 7 and t (5) tr (cid:0) ABA − B − A − BAB − (cid:1) = t − − t ( − t ( − t − + t ( − t (2) t − + t (1) t (4) t − + t − t − t ( − − t ( − t (1) t t ( − + t ( − t t ( − + t − t ( − t ( − − t ( − t ( − t ( − + t − t (1) t ( − − t ( − t ( − t ( − t (2) t ( − + 2 t (1) t (2) t ( − + t ( − t (2) t (3) t ( − − t (3) t ( − + t t (4) t ( − − t ( − t (4) t ( − − t ( − t ( − t (1) t (4) t ( − + t ( − t (3) t (4) t ( − + t − + t + t − t ( − t (1) t − t ( − t (1) t + t (2) t (3) t + 2 t ( − t ( − t (1) − t − t ( − t (1) − t ( − t (2) − t ( − t − t (1) t (2) − t ( − t ( − t (1) t (2) + t ( − t − t (3) + 2 t ( − t ( − t (3) − t − t ( − t (2) t (3) + t − t t (4) − t ( − t ( − t t (4) + t ( − t t (4) − t ( − t (4) + t − t ( − t (4) + t ( − t ( − t (2) t (4) + 2 t ( − t (2) t (4) − t (1) t (3) t (4) − t ( − t (1) t (2) t (3) t (4) + 3 . This last three identities are a consequence of recursive trace reduction formulas(see [6], [7]).Let { , } be the bracket corresponding to Σ , and let { , } be the bracket cor-responding to Σ , . Let Q be the image of q ∗ . We now prove Theorem 1 from theIntroduction. Theorem 7. (1) Q is a Poisson subalgebra of C [ X (Σ , )](2) q ∗ is an anti-Poisson morphism; that is, { q ∗ ( f ) , q ∗ ( g ) } = − q ∗ (cid:0) { f, g } (cid:1) (3) at a generic point of X , rank (cid:18) a (Σ , ) (cid:12)(cid:12)(cid:12)(cid:12) Q (cid:19) = rank ( a (Σ , )) Proof.
First we note that (2) implies (1).To prove (2), since q ∗ is an algebra morphism and the bracket is a derivation,it is enough to verify it on all generators of the algebra. One can use the explicitform of the mapping q ∗ and the explicit form of the bi-vectors to verify the result.However, since q preserves transversality of cycles, double points, and does not affectorientation it follows that for any two cycles α and β in Σ , used in computing thebi-vector a (Σ , ) we have: q ∗ (cid:0) { tr( ρ ( α )) , tr( ρ ( β )) } (cid:1) =(1) X q ( p ) ∈ q ( α ) ∩ q ( β ) ǫ ( q ( p ) , q ( α ) , q ( β )) (cid:0) tr( ρ ( q ( α ) q ( p ) q ( β ) q ( p ) )) −
13 tr( ρ ( q ( α )))tr( ρ ( q ( β ))) (cid:1) . However, as already noted earlier in Section 2.3, the intersection numbers ǫ ( q ( p ) , q ( α ) , q ( β ))and ǫ ( p, α, β ) must be reversed since the bracket computations carried out in [8]are with respect to opposite orientations on the surfaces Σ , and Σ , .Thus Equation (1) is exactly −{ tr ( q ( α )) , tr ( q ( β )) } , as was to be shown. S. LAWTON
To prove (3) we first note that the rank of a bi-vector is the rank of the anti-symmetric matrix of functions ( a ij ). Then, from (2), there are only three non-zerocoefficients to a (Σ , ) after restricting to the image of q ∗ . Namely, t (5) and P − t (5) are Casimirs for { , } , and since the mapping is Poisson and t ( ± are fixed, it followsthat they are Casimirs in the images since they are Casimirs in the pre-image. Thuswe are left with the image generators q ∗ ( t ( j ) ) for j = 4 , − ,
5. Since the bi-vectoron the Poisson subalgebra is exactly the induced one, we explicitly formulate thebi-vector matrix and compute its rank; finding it generically 2. In particular, thematrix has the form: a b − a c − b − c where a = { q ∗ ( t (4) ) , q ∗ ( t ( − ) } , b = { q ∗ ( t (4) ) , q ∗ ( t (5) ) } , and c = { q ∗ ( t ( − ) , q ∗ ( t (5) ) } .However any matrix of this form has rank 2 as long as all three of a, b, and c are not 0, in which case the rank is 0. By direct calculation one sees that allpolynomials a, b, c are in terms of only algebraically independent generators, andso none of a, b, c are generically 0 on X . So generically the rank is 2, and the rankof { , } is two since the rank is equal to the dimension of a symplectic leaf. Hence,the mapping is rank preserving as long as it is not completely degenerate (which isgenerically the case). (cid:3) Comment 8.
Equation (1) used in the above argument shows much more. For anytwo surfaces Σ n ,g and Σ n ,g with n > n > and χ (Σ n ,g ) = χ (Σ n ,g ) , there isa quotient mapping (identifying pairs of boundary components) q : Σ n ,g → Σ n ,g which gives an injection on fundamental groups and therefore gives a mapping of co-ordinate rings q ∗ : C [ X (Σ n ,g )] → C [ X (Σ n ,g )] . This is true not only for SL(3 , C ) but for any complex algebraic reductive Lie group G . The above argument (Equation (1) ) shows that if the orientations of the surfaces correspond to each other then q ∗ is a Poisson mapping and if the orientations are opposite then it is a anti-Poissonmorphism. Consequently, the image of q ∗ is a Poisson subalgebra of the codomainof q ∗ in general.It does not seem clear whether rank preserving is a general property or not. Let L (Σ n,g ) be a generic symplectic leaf of X (Σ n,g ). We now prove Theorem 2from the Introduction. Theorem 9. L (Σ , ) and L (Σ , ) are transverse (generically).Proof. The mapping from Theorem 3 m : X → C is given by( t (1) , t ( − , ..., t (4) , t ( − , t (5) ) ( t (1) , t ( − , ..., t (4) , t ( − ) . It is surjective, and since the first eight generators are algebraically independent,it is submersive as well.This immediately implies that the mapping b , : X (Σ , ) → C given by( t (1) , t ( − , ..., t (4) , t ( − , t (5) ) ( t (1) , t ( − , ..., t (3) , t ( − )is likewise surjective and submersive. Consequently, for any x ∈ X , L (Σ , ) = b − , ( b , ( x )) has dimension 2. Moreover, we can locally parametrize this leaf bythe coordinates ( t (4) , t ( − ) since the other six coordinates t ( ± i ) for i = 1 , , t (5) is then determined by the defining relation t − P t (5) + Q . NE-HOLED TORUS FROM PANTS 9
In particular, flows through these coordinates determine a dimension 2 subspace T ∗ L (Σ , ) ⊂ T ∗ X of the tangent space.Now consider the mapping b , : X (Σ , ) → C given by( t (1) , t ( − , ..., t (4) , t ( − , t (5) ) ( t (5) , P − t (5) ) . Note that this is in fact the correct mapping since P = tr ([ A, B ]) + tr ([
B, A ]) (see[6]). It is shown in [8] that the boundary mapping is always surjective if g > b , may not be everywhere submersive; in particular, when P − Q =0. However, there is an open dense set of X where d b , is onto (call it U ), since b , is surjective and regular (see [8]). We may assume U ⊂ X .Take any u ∈ U . Then (as is shown in [8]) the leaves L := b − , ( b , ( u )) ∩ X and L := b − , ( b , ( u )) ∩ X are complex symplectic manifolds of dimensions 6and 2 respectively. Consequently, these leaves are properly transverse; that is,dim L + dim L = dim X .We now show dim L ∩ L = 0 and L ∩ L = ∅ . At an intersection point, P = t (5) + t ( − := C and Q = t (5) t ( − := D and t ( ± , t ( ± , t ( ± are all fixed.Moreover, solving P = C for t (4) generically gives t (4) t ( − − t ( − t (2) (cid:18) C + t ( − t ( − t (1) − t ( − t (1) − t ( − t (2) + t ( − t (1) t (2) − t ( − t ( − t (1) t (2) − t ( − t (3) + t ( − t ( − t (3) + 3 (cid:19) . (2)Now, substituting this into Q − D = 0 gives a monic degree six polynomial inthe variable t ( − since the degree in t (4) of Q is 3. So the intersection is non-emptyand of dimension 0 (at most 6 discrete points) if t ( − − t ( − t (2) = 0. Otherwise,setting t ( − = t ( − t (2) and substituting this into Q − D = 0 gives a monic degree3 polynomial in t (4) . Either way, the intersection is non-empty and of dimension 0.We claim that the tangent space to L locally can be determined by flowsthrough { t (1) , t ( − , t (2) , t ( − , t (3) , t ( − } by solving for t (4) and t ( − in terms of t ( ± , ..., t ( ± on an open subset since both P and Q are constant on L .Explicitly, substituting Equation (2) into Q − D = 0, where t ( ± i ) for 1 ≤ i ≤ t ( − , and subsequently t (4) , as functionsof t ( ± i ) for 1 ≤ i ≤
3. Thus the flows through t ( ± i ) , 1 ≤ i ≤ L whenever t ( − − t ( − t (2) = 0.Switching the roles of t (4) and t ( − give a like result for at any point where t (4) − t (1) t ( − = 0.However, the tangent space to any point in L is given by the kernel to themapping M := (cid:0) ∂f i /∂t ( ± j ) (cid:1) where f = t − P t (5) + Q, f = t (5) − a, and f = P − t (5) − b , and C = a + b and D = ab (see [5]). This follows since these threefunctions define the leaf as an algebraic set cut out of C . At any smooth pointin the leaf the dimension of the kernel is 6. So whenever t (4) − t (1) t ( − = 0 and t ( − − t ( − t (2) = 0 using P = C , Q = D , and t (5) = a from above, this matrixhas its entries rational functions of t ( ± , .., t ( ± alone. On the other hand, if anysmooth point also satisfies t (4) − t (1) t ( − = 0 and t ( − − t ( − t (2) = 0 solving for t ( ± again gives M as a matrix in these six variables. Thus the flows through thesesix coordinate functions always determine the tangent space at a smooth point of L . Consequently, the span of the flows through { t (4) , t ( − } and { t ( ± , ..., t ( ± } generically and locally give full dimensional tangent spaces to L and L , respec-tively, at an intersection point u . However, collectively they span a full dimensionaltangent space to X since they are globally independent. Hence, for any point in u ∈ U , T u L + T u L = T u X .Compounded with the fact that the leaves are properly transverse and triviallyintersect, we conclude that for any u ∈ U , T u X = T u L ⊕ T u L ; that is, the leavesare generically transverse. (cid:3) We thus conclude that the tangent spaces T u X are symplectic given by theproduct form. This does not imply that X is complex symplectic since the formmay not be closed. We call two symplectic submanifolds symplectically dual if theirtangent spaces are symplectic duals to each other with respect to this form. Thuswe have Corollary 10.
The symplectic leaves of X are (generically) symplectically dual. Comment 11.
This sort of phenomena is not general. For
SL(2 , C ) the leaves L (Σ , ) and L (Σ , ) are respectively dimension 0 and 2 and the variety X is di-mension 3, so there is not transversality. References
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