Remarks on symplectic twistor spaces

Remarks on symplectic twistor spaces

Rui Albuquerque
Abstract

We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study self-holomorphic sections of the general twistor space, with which we define a new moduli space of complex structures. We also recall the theory of flag manifolds in order to study the Siegel domain and other domains alike, which are the fibres of various symplectic twistor spaces. We prove they are all Stein. In the context of a Riemann surface, with its canonical symplectic-metric connection and local structure equations, the moduli space is studied again.

R. Albuquerque111The author acknowledges the support of Fundação para a Ciência e a Tecnologia, either through POCI/MAT/60671/2004 and through CIMA-UE., rpa@dmat.uevora.pt,

Departamento de Matemática da Universidade de Évora

Centro de Investigação em Matemática e Aplicações da Universidade de Évora (CIMA-UE), Rua Romão Ramalho, 59, 7000 Évora, Portugal.

Key Words: linear connections, twistor bundle, moduli space.

MSC 2000: Primary: 30F30, 32L25, 53C15; Secondary: 53C28.

These remarks on complex bundles of complex structures have the purpose of recalling the study of the twistor space of a symplectic manifold , as initiated in [1], by showing a collection of recent results. The presence of a preferred symplectic connection is assumed, so we are also studying these differential operators.

Symplectic connections are quite difficult to describe (cf. [5]). Also intriguing is the relevance of the symplectic analogue of the Penrose twistor space, which requires a symplectic connection where the latter requires a metric connection. Applications in physics of symplectic twistors, in the way of the celebrated Riemannian case, have not appeared so far in the literature.

In the construction of bundles with complex structures one may admit to have a connection with torsion. Recall we may only have a covariant derivative with, simultaneously, and vanishing torsion if, and only if, . This is somewhat close to our theorem 1, generalizing another of Koszul-Malgrange since it does not require integrability of the complex structure on the base space.

The pseudo-holomorphic sections of the (general) twistor bundle are considered in section 1.2; satisfying a 1st order equation, we believe they might bring more interesting information in relation with twistors. We found it is possible to define a moduli space of classes of those almost-complex structures , modulo the group of covariantly constant diffeomorphisms.

We prove further details on the manifold topology of symplectic twistor spaces whose sections induce the pseudo-Kählerian metrics on . We recur to the theory of complex symmetric spaces in order to see other characterizations of their standard fibres, which are all Stein domains. With a Grassmannian ambient space it is possible to interchange between signatures. In particular, we prove the fact that, from a symplectic point of view, those bundles with their twistor complex structure all satisfy the respective integrability equations envolving curvature and torsion if, and only if, one of them does.

Finally we initiate some computations for the case of Riemann surfaces and relate the moduli space above to Teichmüller space. We give coordinate structure equations of the sheaf of holomorphic functions.

We acknowledge very fruitful conversations with Prof. John Rawnsley, which led to some results here.

1 Complex bundles of complex structures and a new moduli space

1.1 Twistorial constructions

A twistor space is an almost complex fibre bundle over a real even dimensional manifold . By this definition, following the one introduced in [12], we mean admits a smooth structure of compatible charts and trivializations with a given standard fibre, in the sense of [17], and such that the fibres are almost complex submanifolds. Usually, the complex tangent bundle to the fibres is denoted by due to familiar identification with a vertical distribution.

Let be the frame bundle of , let be the dimension of and let

(1)

The projection from the frames , , is defined by , with some fixed linear complex structure of . is the bundle of complex linear structures on the tangents of , called the general twistor space of when it aquires a Penrose almost complex structure induced by a linear connection on .

Now let be as previously. Then we have a commutative diagram

(2)

where the map is defined as the complex structure on induced by the isomorphism , transporting the complex structure from one space to the other, . This is, is the naturally induced complex structure of the quotient space conjugated by the quotient map .

There are two very distinct cases that the map may assume.

One is precisely when is a Penrose twistor space, a complex subbundle of , with the Penrose almost complex structure . Then is just the inclusion map . The two most familiar subcases are and , respectively, the orientation and metric -compatible, and the symplectic -compatible and tamed complex linear structures on the tangent spaces of, respectively, a Riemannian or an almost-symplectic manifold .

Here is the general procedure to construct . The vertical distribution has a natural symmetric space complex structure: vectors , are endomorphisms of which anti-commute with , hence left multiplication is clearly a map with square . The so called horizontal distribution , suplementary to the vertical one, is given by the kernel of

(3)

where is the canonical section . Up to isomorphism , each becomes the horizontal complex structure itself, thus living with this ubiquitous character. Adding up, we have on preserving such splitting (cf. [3, 4, 12]).

The integrability equations of are well known: the curvature tensor must satisfy , where are the projections inside onto the eigenspaces of . If has torsion, then the condition , must also be fullfield. Note are real tensors, so if we conjugate the equations we get equivalent statements with again.

The second case we would like to describe arises from a space , where is a principal -bundle over with a -connection, is a Lie subgroup of and is a -invariant complex manifold. We suppose furthermore that there exists a fibre preserving smooth map . Then inherits an almost-complex structure exactly by the same procedure as above: we just keep the vertical complex structure invariantly from and put on every horizontal , . Deducing the integrability of this well defined structure seems rather difficult.

The most common situation where the setting occurs is when itself admits an almost complex structure (denoted by the same letter used for the points in twistor space). The map in this case is just a constant on each fibre, ie. . Moreover, if we want to be a pseudo-holomorphic projection, then must be . The integrability equations are given next.

Theorem 1.

Suppose , as above and , the bundle of complex frames. Then is integrable if and only if is integrable and the curvature satisfies , ie. .

Proof.

Notice we just have to prove the result for as a principal -bundle, since the complex connection extends trivially to , and thence the result follows easily with any -holomorphic invariant factor (note may then be a real Lie group). If is odd real-dimensional, then one vertical direction is lost when we pass to a quotient, without affecting the induced complex structure.

Let be the soldering 1-form on , , and let be, respectively, the connection and curvature 1- and 2-forms on . Defining over by the procedure described, makes the components of become generators of the space of -forms. Indeed, is already -valued and its kernel is closed under . For we have

Now recall the complex structure is integrable iff the space of germs of -forms generates a -closed ideal in . On the principal bundle of all frames we have, . With complex frames, we use the projection and find is type iff . On the base this is clearly . However, this is the Nijenhuis tensor . Hence the condition is the same as the integrability of . The well known formula assures us that has no component iff the same happens with . ∎

Exactly by the same proof but with complex charts in the place of , we get the Koszul-Malgrange theorem, which states that any complex vector bundle over a complex manifold with a complex connection such that , is a holomorphic vector bundle. Moreover, the latter is the unique holomorphic bundle structure with a same , namely .

This distinct case is indeed less connected with the Penrose twistor construction case than one would think on a first reading. Notice we required a complex connection and complex Lie group rather than the real setting of twistors. One may consult [12] to notice the differences in detail.

Finally there is a third case to be considered. Again we assume the manifold is furnished with an almost complex structure and a real linear connection reducible to a principal -bundle of frames . For the purpose, the Lie group may be understood as a Lie subgroup of the general linear group. We let be fixed and let .

Then a twistor space may be defined with an almost complex structure denoted : the vertical part is given by the usual fibre structure and the horizontal part assumes the fixed , via . For , this structure is never integrable.

1.2 Pseudo-holomorphic complex structures

We start by a definition. Let be two almost complex structures on the smooth manifold . A linear connection is said to satisfy condition if

(4)

where is the space of -type vector fields for the structure . This definition generalizes the case first studied in [13, 14].

Proposition 1.

Given a manifold with a linear connection and three sections , we have:
(a) is -holomorphic if and only if and is satisfied;
(b) in particular, is -holomorphic if and only if is satisfied;
(c) is -holomorphic if and only if and is satisfied.

Proof.

A map is -holomorphic iff . For the horizontal part we push forward this equation using and get . For the vertical part we apply the projection just to find

This is equivalent to . Since and , we get – which is equivalent to condition . The rest of the proof is as easy as the previous. ∎

If is torsion free, condition implies integrable. Conversely, if the setting is that of a pseudo-Riemannian triple and , then the integrability of implies , cf. [2, 14].

Notice is never integrable, so we shall follow on with two results on the general well known twistor space. Recall the action of Diff  on the space of linear connections, by affine transformation, cf. [10]. A diffeomorphism of acts on , the Lie algebra of vector fields, and hence on :

. The subspaces of torsion free or flat connections are preserved. (On a pseudo-Riemannian setting, by uniqueness, the action of isometries in Levi-Civita is trivial; but in a wider context one must treat it with more circumspect.) The diffeomorphism also transforms almost complex structures: .

In virtue of case c of the proposition above, let us call self-holomorphic to those ’s which satisfy condition .

Proposition 2.

Suppose is self-holomorphic and for some diffeomorphism of . Then is self-holomorphic.

Proof.

It is known that a diffeomorphism on the base manifold induces a pseudo-biholomorphism along on the twistor space. Indeed, , , so , and it was proved that (cf. [1, 2] for two distinct proofs). Now notice . Then

as we wished ∎

If we work on the symplectic category, then we may prove the converse: if is a symplectomorphism and is self-holomorphic, then . This follows as easily as above, proving is biholomorphic for and then applying [1, corollary 4.1].

What we have just proved is that there is a good moduli space of self-holomorphic almost complex structures:

(5)

where Diff, the isotropy subgroup of .

For example, let us take a Kähler manifold and let denote the Levi-Civita connection. Then itself shows is non-empty.
Remark: The anti-self-holomorphic almost complex structures in a Riemannian twistor space, endowed with the Levi-Civita connection, correspond to the almost-Kähler complex structures in the classification of Gray-Hervella. Indeed, the equation is

(6)

. These sections give a moduli space just as above and we recall Diff is bigger than the isometry group in general (cf. [11]). Also, notice the twistor space inherits a metric according to its induced tangent bundle decomposition, which, over the image of the embbeding , is given by

(7)

. The induced Kähler form of the second parcel has been studied in the context of Kähler-Einstein manifolds ([15]) and our methods may relate to the Goldberg conjecture — any compact almost-Kähler Einstein manifold is Kähler-Einstein.

2 On the twistor space of a symplectic manifold

2.1 Linear complex structures

Let us take a path towards the description of the complex structures compatible with a non-degenerate 2-form. Let be a fixed real vector space of even dimension and consider the space of all complex structures in . The action of the real shows . As a homogeneous space, the tangent space to at is identified with

(8)

which is closed under left multiplication by itself, hence one may prove the space is a complex symmetric space. According to

(9)

each equals . In sum, has a unique -invariant complex structure whose -tangent space at is .

We now specialise to a subspace of . Suppose is a symplectic form on the real vector space . Let . Then for any vectors ,

so the new imposed condition is the same as being a symplectic linear transformation of , or what is called ‘compatible’ with . Consider the symmetric form . This non-degenerate inner product has even signature, say for some , since any maximal subspace where it is positive definite is also -invariant. We denote by the connected components, as we shall see according to the index , of the disjoint union .

Proposition 3.

(i) None of the are empty.
(ii) The action of on is transitive.

Proof.

(i) It amounts to show that it is possible to find a basis both symplectic and -orthogonal. Recall that a basis is called symplectic if the matrix of is

so that for and is a complex structure.

Let be such that (there exists such a pair). Now, with the non-degenerate restricted, is a symplectic vector space. Assuming the result true by induction for , we find the new basis on and a complex structure on . Rearranging terms together with and we find the full basis we required, the index remaining a simple combinatorial problem. Also becomes the orthogonal complement of . We proved the existence of symplectic bases and the existence of compatible complex structures for any .
(ii) Now let be given and let

(10)

is the matrix of the inner product . So far we have proved there exists which transforms, say, the first symplectic basis into a new one both orthogonal and symplectic. Henceforth satisfies the equations

(The necessity of the sign here becomes obvious when we do .) Thus and hence . It is trivial to see commutes with , so we have a complex structure in the same as and conjugate to . ∎

Therefore

(11)

a quotient by the pseudo-unitary group. Fixing any compatible with index , we have an inner automorphism of which preserves and the respective . Appealing to the theory we observe that the subspaces we have just been describing are symmetric-subspaces of (cf. [11]). Clearly too, so there is a direct sum

(12)

where . One easily checks that is preserved under left multiplication by , thus the are also complex submanifolds of .

Now we recall the Siegel upper half space or Siegel domain

(13)

where the elements are matrices with complex entries. acts transitively on by

(14)

where are square matrices. To see that this is well defined and the action is transitive we appeal to [16]. Easy enough is that the stabiliser of is the subgroup of those for which and , that is, the subgroup . Hence we recover .

Now we look at as an open complex manifold.

Proposition 4.

The map given by

(15)

is a -equivariant anti-biholomorphism.

Proof.

Of course . Suppose is such that

Then must be equal to . Using the well known relations between the four squares inside , which also give , we can use the equation above to write in terms of and . Or rather one can check directly that the matrix presented is a true element of . By construction, is -equivariant.

Since acts by rational maps in variable , hence by holomorphic transformations of , we may conclude that multiplication by in agrees with a -invariant complex structure. It remains to show that this is the same as right multiplication by in , up to the isomorphism

arising from the projection , . If we denote

then the following derivative taking place at point 1 makes sense.

Now the result

is immediate to check. ∎

One may give simple counter-examples to prove that the obvious generalisation of the above to any signature does not hold.

Let us digress in search for a compact symmetric space in which to embed all the . We appeal to the theory of flag manifolds and parabolic subgroups, as explained in [6, 7].

Let be a connected semisimple complex Lie group and its Lie algebra. Recall a parabolic subgroup is the normaliser in of a parabolic subalgebra of , this is, a complex Lie subalgebra which contains a maximal solvable subalgebra of , or a Borel subalgebra. Thus,

The theory shows that Lie and that, if is a compact real form of , then

because acts transitively on ( has closed image on ). These compact and complex spaces are the so called flag manifolds. If is a non-compact real form (thus semisimple) of then it may not act transitively on . On the other hand is everything but solvable. The open orbits of this action are called flag domains.

Applying this theory to with its two canonical real forms we are able to deduce the components of are disjoint flag domains in the flag manifold . We shall prove this in the following lines.

To simplify notation let so that . Notice acts holomorphically on the Grassmannians of complex -planes (but not transitively). Notice that the stabilizer of a point of the Grassmaniann is also a complex subgroup.

We are particularly interested in the case . A quick computation shows that this

(16)

is neither the smallest nor the largest subgroup we can achieve with those actions (perhaps the smallest subgroup is the case when ). However, our is still big enough.

Lemma 1.

The maximal solvable subalgebra in the Lie algebra of is maximal solvable in , ie. is a Borel subalgebra of .

Proof.

Just from the theory of solvable Lie algebras over algebraically closed fields, one concludes that all solvable in preserve some -plane (its elements are all representable in triangular form for a same basis). So any maximal solvable will preserve an -plane. By conjugation with some we find and to be maximal in . To see that inclusion, notice the subgroup coincides with the stabilizer of the -stable -plane. ∎

Thus, is a parabolic subgroup and has the ‘right’ dimension. For the computation of the dimension we apply some ideas which J. Rawnsley explained to us.

The theory above gives a framework for studying these problems.

Lemma 2.

Consider the plane . Then

(17)

Moreover, we may write .

Proof.

Let . By definition,

implies . Also , so

and we find ,  . Equivalently, and with . ∎

Now we take a digression on the real forms of , some of which we do not really need. First we see . The proof is that

if and only if . The matrix is from (10). Since is connected, arguing with dimensions we find

(18)

where , are the non-compact real forms of and is the well known compact real form of . The last equality in (18) relies on the fact that the orbit of in must be open and closed.

On the other hand, orbits of are only locally closed. The open ones, the flag domains, certainly appear when has the lowest possible dimension as varies. From 18 above we know this has to be . Depending on the signature over of the metrics given by , leads to the solutions . The -planes are spanned by

(19)

where is a symplectic basis.

Thus we may claim to have constructed a holomorphic embedding

(20)

and a commutative diagram follows:

(21)

where the map on the top is . We shall call real-Lagrangians the n-dimensional -isotropic -subspaces of such that .

Notice the first in lemma 2 was the stabilizer of a Lagrangian -plane , since , but not a real-Lagrangian.

Proposition 5.

The map is a holomorphic embedding and has image the locally closed manifold of real-Lagrangian subspaces.

Proof.

Since , the map is injective, and by definition is isotropic. Now let . To any we associate a sequence

with only depending on . Therefore where and hence

where stands for . Now for real

Hence satisfies and so

Notice we proved the whole embedding of in is holomorphic.

Now assume . Clearly is non-degenerate, so the maximal dimension an isotropic subspace can attain is precisely . Indeed, we have a general formula, , where is the -anihilator of .

With the above one proves that the hemi-symmetric form on defined by

is non-degenerate for real lagrangian . According to the signature of this pseudo-metric we may then define by , and , hence such that . It is trivial to see is real. For example

We have proved is a biholomorphism onto the aforesaid manifold. Notice also

Here, the first set is open in the Grassmannian and the second is closed. ∎

The above is only part of either the cell or the algebraic structure of the Grassmannian. We will not pursue these in this work. As an example, in every line (1-plane) is Lagrangian and there is a circle in of non-real lines. The open hemispheres are the two Siegel domains and .

Notice also the following result which was not so clear before, due to phenomena like pseudo-convexity.

Corollary 1.

All are Stein spaces, .

Proof.

Conjugating by some non-real element yields a biholomorphism to . The Siegel domain is Stein (it is convex) and being Stein is preserved by biholomorphism, hence the result. ∎

2.2 Integrability equations of symplectic twistor space

Any given -dimensional real manifold endowed with a non-degenerate 2-form has associated to it a bundle of linear complex structures which we call the symplectic twistor space. The standard fibre is , where is the standard symplectic vector space. We have already mentioned .

With a linear connection such that , we may define the Penrose almost complex structure on any symplectic twistor space , cf. section 1. The integrability equations were recalled in the same section. It is not obvious that they are equivalent for different , ie. independent of the connected components of (assuming is connected).

Proposition 6.

The almost complex structure is integrable on some if, and only if, it is integrable on all.

Proof.

Let and . Consider first the torsion equation:

(22)

Fix in this set. Then (22) is saying that takes values in the largest -invariant subspace of torsion-like tensors such that

(23)

Indeed, since , we have that

and hence is in the subspace iff is in the subspace, iff satisfies (22) for . These are ideas from [12].

Notice such subspace of the space of torsion tensors is immersed in a -space of complex linear tensors defined by the same condition (23); such mapping is induced by complexification, ie.

We can now pass to another by acting on with an element of , as we saw in (20). Since is -invariant, this space is the same for all and hence the result.

Finally, analogous arguments follow for the curvature condition, this time with the -subspace sitting in