aa r X i v : . [ m a t h . DG ] N ov INTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES
STEVEN GINDI
Abstract.
We introduce integrable complex structures on twistor spaces fiberedover complex manifolds. We then show, in particular, that the twistor spacesassociated with generalized Kahler, SKT and strong HKT manifolds all nat-urally admit complex structures. Moreover, in the strong HKT case we con-struct a metric and three compatible complex structures on the twistor spacethat have equal torsions.
Contents
1. Introduction 12. Complex Structures on Twistor Spaces 42.1. Preliminaries 42.2. Horizontal Distributions and Splittings 52.3. The Complex Structures 62.4. (1,1) Curvature 92.5. Other Curvature Conditions 103. Examples 113.1. Chern Connections 113.2. ¯ ∂ -operators 123.3. Three Forms 134. Hermitian Structures on T and their Torsion 154.1. Hermitian Pairs with Equal Torsion 174.2. Strong HKT Manifolds 184.3. Bihermitian Manifolds 195. Twistors and Grassmannians 205.1. Embedding the Fibers 205.2. The Holomorphic Embedding 215.3. Proof of Proposition 5.8 235.4. Corollaries of the Embedding 246. Acknowledgments 27References 271. Introduction
In the 1970’s, Atiyah, Hitchin and Singer introduced a tautological almostcomplex structure on a certain twistor space that along with its generaliza-tions have had, until today, a major impact on differential and complex geometry [3, 18]. The twistor space that they considered was T + ( T M ),the bundle of complex structures fibered over an oriented Riemannian fourmanifold that are compatible with the metric and orientation; the almostcomplex structure was J ∇ taut , where ∇ is the Levi Civita connection, and isdefined in Section 2.3. The importance of J ∇ taut was found to lie in the timeswhen it was integrable, which was when the four manifold was anti-selfdual,and one of its many applications was the construction of instantons on S [2].Given its success in four dimensions, J ∇ taut was generalized in [4, 17] tothe twistor space C ( T M ) = { J ∈ EndT M | J = − } , where M is any evendimensional manifold and ∇ is any connection. Its integrability conditionswere then explored with the hope that J ∇ taut would again lead to majorresults about the base manifold. However, it was found that these conditionsimposed severe restrictions on the curvature of the connection and in almostall cases J ∇ taut was not integrable, thus limiting its applications in differentialgeometry.Of course this did not prevent mathematicians from taking advantage ofthe rare times when it was integrable on either C ( T M ) or on submanifoldswithin, and applying J ∇ taut to advance, for example, the theory of harmonicmappings, integrable systems and hyperkahler geometry. With all of itssuccesses, however, the fact remains that the rarity of the integrability of J ∇ taut has greatly hindered its use in deriving results about the geometryof the base manifold M in higher dimensions. And it is natural to won-der whether there exist other almost complex structures on twistor spaceswhose integrability conditions are more easily satisfied—especially in everydimension—and at the same time can be used to derive results about thebase manifold.The purpose of this paper is to demonstrate that such almost complexstructures do indeed exist if we assume that M is itself equipped with a com-plex structure I . Whereas J taut , which will stand for J ∇ taut for an unspecifiedconnection, can only be defined on twistor spaces that are associated to T M ,the almost complex structures that we introduce in Section 2.3 are definedon more general twistor spaces that are associated to any even rank realvector bundle. Denoting such a bundle by E , in that section, we definethe almost complex structure J ( ∇ ,I ) on C ( E ) = { J ∈ EndE | J = − } ;it depends on a choice of a connection ∇ on E , similar to the defini-tion of J taut . However, unlike J taut , the conditions on the connection ∇ for J ( ∇ ,I ) to be integrable are easily fulfilled. By computing its Nijen-huis tensor, we prove in Theorem 2.17, that J ( ∇ ,I ) on C ( E ) is automati-cally integrable if the curvature of ∇ , R ∇ , is (1,1) with respect to I , i.e., R ∇ ( I · , I · ) = R ∇ ( · , · ). Moreover if g is a fiberwise metric on E and ∇ g = 0then J ( ∇ ,I ) on T ( E, g ) = { J ∈ C ( E ) | g ( J · , J · ) = g ( · , · ) } is integrable if andonly if R ∇ is (1,1) (Theorem 2.23). Under these conditions, the projection NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 3 map π : ( C ( E ) , J ( ∇ ,I ) ) −→ ( M, I ) becomes a holomorphic submersion, aproperty that would never hold if we were to replace J ( ∇ ,I ) by J taut .While we will use ( C ( E ) , J ( ∇ ,I ) ) to derive results about the base manifold( M, I ) in [10], the focus of our present paper is to describe various examplesof vector bundles that admit connections with (1,1) curvature and to studythe resulting holomorphic structures on the twistor spaces. For instance, inSection 3.2 we demonstrate that a general holomorphic Hermitian bundle,(
E, g, J ), admits many such connections. The Chern connection is of coursean example, but as we show, ∂ closed sections D ∈ Γ( T ∗ , ⊗ ∧ E ∗ , ) and α ∈ Γ( T ∗ , ⊗ ∧ E , ) can also be used to define connections on E with(1,1) curvature. In the case when E = T M , D is a ∂ closed (2,1) form andour goal in Section 3.3 is to describe how such forms naturally appear onwell known classes of Hermitian manifolds. For example, SKT manifolds,bihermitian manifolds—also known as generalized Kahler manifolds—as wellas strong HKT manifolds [6, 8, 1, 13, 11] all admit ∂ closed (2,1) forms andthus complex structures on their twistor spaces.In Section 4, we construct a metric on T ( T M, g ) that is compatible with J ( ∇ ,I ) and compute the associated torsion d c w ( ∇ ,I ) in terms of the curvatureand torsion of ∇ . ( w ( ∇ ,I ) is the fundamental two form.) We then focus onthe case when the base manifold is strong HKT and construct a metric andthree compatible complex structures on its twistor space that have equaltorsions.Given E −→ ( M, I ) and a connection ∇ with (1,1) curvature, in Section 5we further study ( C ( E ) , J ( ∇ ,I ) ) by holomorphically embedding it into a morefamiliar complex manifold. The key in finding a suitable manifold is to no-tice that if we C -linearly extend ∇ to a complex connection on E C := E ⊗ R C then R ∇ is (1,1) if and only if ∇ , is a ∂ -operator on E C . Hence given ∇ on E with (1,1) curvature, we have two associated complex analytic manifolds:the first is ( C ( E ) , J ( ∇ ,I ) ) and the other is the holomorphic Grassmannianbundle Gr n ( E C ) ( rankE = 2 n ), and in Section 5 we holomorphically embed C ( E ) into this latter bundle. By then considering C ( E ) as a complex sub-manifold of Gr n ( E C ), we derive a number of corollaries about the holomor-phic structure of twistor spaces. For example, we derive conditions on twoconnections ∇ and ∇ ′ that are defined on E −→ ( M, I ) with (1,1) curvature,so that the twistor spaces ( C ( E ) , J ( ∇ ,I ) ) and ( C ( E ) , J ( ∇ ′ ,I ) ) are equivalentunder a fiberwise biholomorphism. We then use this to prove that certaincomplex structures that we defined in Section 3.2 on the twistor spaces asso-ciated to Hermitian bundles are in fact biholomorphic. As another corollary,given a holomorphic Hermitian bundle ( E, g, J ) −→ ( M, I ) we construct awell defined map from the Dolbeault cohomology group H , (Λ E ∗ , ) tothe isomorphism classes of complex structures on T ( E, g ). Other corollariesof the holomorphic embedding are given in Sections 5.4.1 and 5.4.2.
STEVEN GINDI
Having in this paper introduced, given examples and explored differentproperties of ( C ( E ) , J ( ∇ ,I ) ), in [10] we will use it to study certain Poissonstructures on bihermitian manifolds.2. Complex Structures on Twistor Spaces
Preliminaries.
Let V be a 2 n dimensional real vector space and let C ( V ) = { J ∈ EndV | J = − } be one of its twistor spaces. To describesome of the properties of C ( V ), consider the action of GL ( V ) on EndV viaconjugation: B · A = BAB − . As C ( V ) is a particular orbit of this action,it is isomorphic to GL ( V ) /GL ( V, I ) , where I ∈ C ( V ) and GL ( V, I ) = { B ∈ GL ( V ) | [ B, I ] = 0 } ∼ = GL ( n, C ). Itthen follows that the dimension of C ( V ) is 2 n and that if we consider C ( V )as a submanifold of EndV then T J C = [ EndV, J ] = { A ∈ EndV | {
A, J } = 0 } . With this, we may define a natural almost complex structure on C ( V ) thatis well known to be integrable: I C A = J A, for A ∈ T J C . If we now equip V with a positive definite metric g then another twistorspace that we will consider is T ( V, g ) = { J ∈ C ( V ) | g ( J · , J · ) = g ( · , · ) } . Inthis case, T is an orbit of the action of O ( V, g ) on
EndV by conjugation, andis thus isomorphic to the Hermitian symmetric space O ( V, g ) /U ( I ) , where I ∈ T and U ( I ) ∼ = U ( n ). It then follows that the dimension of T is n ( n − T as a submanifold of EndV then T J T = [ o ( V, g ) , J ] = { A ∈ o ( V, g ) | { A, J } = 0 } . As I C naturally restricts to T J T , T is a complex submanifold of C .2.1.1. Twistors of Bundles.
Let now E −→ M be an even rank real vec-tor bundle fibered over an even dimensional smooth manifold. General-izing the previous discussion to vector bundles, we will define C ( E ) = { J ∈ EndE | J = − } , which is a fiber subbundle of the total space of π : EndE −→ M with general fiber C ( E x ), for x ∈ M . Since the fibersof π C : C ( E ) −→ M are complex manifolds, C ( E ) naturally admits thecomplex vertical distribution V C ⊂ T C ( E ), where V J C = T J C ( E π ( J ) ) ∼ =[ EndE | π ( J ) , J ]. Using the section φ ∈ Γ( π ∗C EndE ) defined by φ | J = J , wewill then identify V C with the subbundle [ π ∗C EndE, φ ] of π ∗C EndE .Letting g be a positive definite fiberwise metric on E , we will also consider T ( E, g ) = { J ∈ C ( E ) | g ( J · , J · ) = g ( · , · ) } . Similar to the case of C ( E ), T ( E, g ) naturally admits the complex vertical distribution V T , defined by V J T = T J T ( E π ( J ) ) ∼ = [ o ( E π ( J ) , g ) , J ]. If we denote the projection mapfrom T ( E, g ) to M by π T then we will identify V T with the subbundle[ π ∗T o ( E, g ) , φ ] of π ∗T EndE , where now φ ∈ Γ( π ∗T EndE ). NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 5
Notation 2.1.
As was done above and will be continued below, we will attimes denote C ( E ) by C and T ( E, g ) by T ( E ) , T ( g ) or just T . Moreover,there are also times when we will denote π C or π T by just π . Horizontal Distributions and Splittings.
With this background,we will now take the first steps in defining integrable complex structureson C ( E ) and T ( E, g ) in the case when M is a complex manifold. Given aconnection ∇ on E we will define the horizontal distribution H ∇ C in T C , sothat this latter bundle splits into V C ⊕ H ∇ C . Similarly, in the case when g is a fiberwise metric on E and ∇ is a metric connection, we will describehow to split T T into V T ⊕ H ∇ T . Once we have described these splittingswe will define the desired complex structures on the above twistor spaces inSection 2.3.To begin, let, as above, E −→ M be a vector bundle, though the basemanifold is not yet assumed to be a complex manifold, and let ∇ be anyconnection. As C is a fiber subbundle of the total space of π : EndE −→ M ,we will find it convenient to split its tangent bundle by first splitting T EndE .Although there are other ways to define this splitting the basic idea hereis to use parallel translation with respect to ∇ . First, if A ∈ EndE and γ : R −→ M satisfies γ (0) = π ( A ) then the parallel translate of A along γ will be denoted by A ( t ). The horizontal distribution H ∇ EndE in T EndE is then defined as follows.
Definition 2.2.
Let H ∇ A EndE = { dA ( t ) dt | t =0 | for all γ, γ (0) = π ( A ) } .It is straightforward to show that H ∇ EndE is a complement to the ver-tical distribution:
Lemma 2.3.
T EndE = V EndE ⊕ H ∇ EndE.
Remark 2.4.
The above procedure can actually be used to split the tangentbundle of any vector bundle with a connection. Another way to define sucha splitting is to consider the bundle as associated to its frame bundle andthen use the standard theory of connections. These two methods yield thesame splittings and are essentially equivalent.
Now if J ∈ C ⊂ EndE and γ : R −→ M is a curve that satisfies γ (0) = π ( J ) then it is clear that the associated parallel translate J ( t ) lies in C forall relevant t ∈ R . It then follows that H ∇ J EndE lies in T J C , so that wehave: Lemma 2.5. T J C = V J C ⊕ H ∇ J C , where H ∇ J C = H ∇ J EndE . Similarly, if g is a fiberwise metric on E and ∇ g = 0 then the paralleltranslate of J ∈ T along γ lies in T . We thus have Lemma 2.6. T J T = V J T ⊕ H ∇ J T , where H ∇ J T = H ∇ J EndE.
STEVEN GINDI
With the above splittings, it will be useful for later calculations to derivea certain formula for the vertical projection operator P ∇ : T EndE −→ V EndE ∼ = π ∗ EndE , which, upon suitable restriction, will also be valid forthe corresponding projection operators for T C and T T . The formula willdepend on the tautological section φ of π ∗ EndE that is defined by φ | A = A : Proposition 2.7.
Let X ∈ T A EndE, then P ∇ ( X ) = ( π ∗ ∇ ) X φ, where we are considering P ∇ to be a section of T ∗ EndE ⊗ π ∗ EndE .Proof of Proposition . . Let { e i } be a local frame for E over some open set U ⊂ M about the point π ( A ), where A ∈ EndE , and let { e i ⊗ e j } be thecorresponding frame for EndE . Then for X ∈ T A EndE ,( π ∗ ∇ ) X φ = ( π ∗ ∇ ) X φ ij π ∗ ( e i ⊗ e j )(2.1) = dφ ij ( X ) e i ⊗ e j | π ( A ) + A ij ∇ π ∗ X e i ⊗ e j . (2.2)Let us now consider the following two cases.A) Let X be an element of V A EndE , which for the moment is not iden-tified with
EndE | π ( A ) , so that π ∗ X = 0. Also let A(t) be a curve in EndE | π ( A ) such that A (0) = A and dA ( t ) dt | t =0 = X. Then by Equation 2.2,( π ∗ ∇ ) X φ = dA ( t ) ij dt | t =0 e i ⊗ e j | π ( A ) = P ∇ ( X ) ∈ EndE | π ( A ) .B) Let X ∈ H ∇ A EndE so that it equals ddt A ( t ) | t =0 , where A ( t ) is the paral-lel translate of A along some curve γ : R −→ M that satisfies γ (0) = π ( A ).As dφ ij ( X ) = ddt A ( t ) ij | t =0 , Equation 2.2 becomes ddt A ( t ) ij | t =0 e i ⊗ e j | π ( A ) + A ij ∇ dγdt | t =0 e i ⊗ e j , which is zero since A ( t ) is parallel. (cid:3) If we consider the corresponding projection operator P ∇ : T C −→ V C then it follows from the above proposition that P ∇ ( X ) = ( π ∗C ∇ ) X φ , where φ is now a section of π ∗C EndE −→ C . Note that since φ = −
1, ( π ∗C ∇ ) X φ ,for X ∈ T J C , is indeed contained in V J C = { A ∈ EndE | π ( J ) | { A, J } = 0 } .In the case when g is a fiberwise metric on E and ∇ g = 0, an analogousformula holds for P ∇ : T T −→ V T . Remark 2.8.
We respectfully report that similar formulas for the projectionoperators for T C and T T were derived in [17] but with a small error. The Complex Structures.
Now let E −→ ( M, I ) be an even rankreal vector bundle that is fibered over an almost complex manifold and let ∇ be a connection on E . We will define the following almost complex structureon the total space of π : C ( E ) −→ M and will explore its integrabilityconditions in the next section. Definition 2.9. J ( ∇ ,I ) : First use ∇ to split T C = V C ⊕ H ∇ C , NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 7 and then let (1) J ( ∇ ,I ) A = J A (2) J ( ∇ ,I ) v ∇ = ( Iv ) ∇ , where A ∈ V J C ⊂
EndE | π ( J ) and v ∇ ∈ H ∇ J C is the horizontal lift of v ∈ T π ( J ) M. In other words, J ( ∇ ,I ) on V C ⊕ H ∇ C equals φ ⊕ π ∗ I , where we haveidentified V C with [ π ∗ EndE, φ ] and H ∇ C with π ∗ T M .It then follows from the definition of J ( ∇ ,I ) that π is pseudoholomorphic: Proposition 2.10. π : ( C , J ( ∇ ,I ) ) −→ ( M, I ) is a pseudoholomorphic sub-mersion. In the case when g is a fiberwise metric on E and ∇ is a metric connection,the claim is that J ( ∇ ,I ) on C restricts to T , so that T ⊂ ( C , J ( ∇ ,I ) ) is analmost complex submanifold. The reason is that T J T splits into V J T ⊕ H ∇ J T ,where H ∇ J T = H ∇ J C = H ∇ J EndE, as explained in the previous section.
Remark 2.11.
It should be noted that J ( ∇ ,I ) has not yet been studied in thisgenerality in the literature. In [19] , Vaisman did study J ( ∇ ,I ) only in thespecial case when E = T M and ∇ I = 0 and only on certain submanifolds of C ( T M ) . However, for our applications we do not want to restrict ourselvesto E = T M and we especially do not want to require ∇ I = 0 . With J ( ∇ ,I ) defined, let us now compare it to the tautological almostcomplex structures on twistor spaces that are usually considered in the lit-erature [3, 17, 4]. If ∇ ′ is a connection on T M −→ M , where here M is anyeven dimensional smooth manifold, then based on the splitting of T C into V C ⊕ H ∇ ′ C , we define J ∇ ′ taut on C ( T M ) as follows.
Definition 2.12.
Let J ∇ ′ taut = φ ⊕ φ , where we have identified V C with[ π ∗ EndT M, φ ] and H ∇ ′ C with π ∗ T M , and where the first φ factor acts byleft multiplication.To compare it to J ( ∇ ,I ) , note that J ∇ ′ taut does not require M to admitan almost complex structure, while the former one does. On the otherhand, J ∇ ′ taut is only defined for the bundle E = T M whereas J ( ∇ ,I ) is de-fined for any even rank real vector bundle. Also, given ( M, I ), the projec-tion map ( C ( T M ) , J ∇ ′ taut ) −→ ( M, I ) is never pseudoholomorphic, whereas( C ( E ) , J ( ∇ ,I ) ) −→ ( M, I ) is always so. Lastly, J ∇ ′ taut is rarely integrable—except in special cases such as when M is an anti-selfdual four manifold([3]), as explained in the Introduction—whereas the integrability conditionsof J ( ∇ ,I ) are very natural to be fulfilled, as we will show below.Having compared the above almost complex structures, let us now returnto the general setup of a vector bundle E −→ ( M, I ) that is fibered over analmost complex manifold and that is equipped with a connection ∇ . The STEVEN GINDI goal is to determine the conditions on I and the curvature of ∇ , R ∇ , that areequivalent to the integrability of J ( ∇ ,I ) not just on C but on other almostcomplex submanifolds C ′ as well. Although these conditions can be workedout for any C ′ , we will focus on the case when the corresponding projectionmap π C ′ : C ′ −→ M is a surjective submersion. If g is a fiberwise metric on E and ∇ g = 0 then as an example we can take C ′ = T ( g ).The method that we will use to explore the integrability conditions of J ( ∇ ,I ) on C ′ is to calculate its Nijenhuis tensor on C .2.3.1. Nijenhuis Tensor.
In this section, let π : C ( E ) −→ M be the projec-tion map and define J := J ( ∇ ,I ) and P := P ∇ : T C −→ V C ⊂ π ∗ EndE , asin Section 2.2. We will presently compute the Nijenhuis tensor, N J , of J that is given by N J ( X, Y ) = [ J X, J Y ] − J [ J X, Y ] − J [ X, J Y ] − [ X, Y ] , in terms of the Nijenhuis tensor of I and the curvature of ∇ , R ∇ . Proposition 2.13.
Let
X, Y ∈ T J C and let v = π ∗ X and w = π ∗ Y . Then π ∗ N J ( X, Y ) = N I ( v, w )2) P N J ( X, Y ) = [ R ∇ ( v, w ) − R ∇ ( Iv, Iw ) , J ] + J [ R ∇ ( Iv, w ) + R ∇ ( v, Iw ) , J ] . Proof of Proposition 2.13, Part 1).
This easily follows from the fact that if X ∈ Γ( T C ) is π -related to v ∈ Γ( T M ) then J X is π -related to Iv . (cid:3) Letting, as above, φ ∈ Γ( π ∗ EndE ) be defined by φ | J = J , the proof ofPart 2 of the proposition, will be based on the following lemma. Lemma 2.14.
Let
X, Y ∈ Γ( T C ) . Then P ∇ ([ X, Y ]) = − [ R π ∗ ∇ ( X, Y ) , φ ] + π ∗ ∇ X P ( Y ) − π ∗ ∇ Y P ( X ) . Proof.
Consider P ∇ ([ X, Y ]) = π ∗ ∇ [ X,Y ] φ = − R ( π ∗ ∇ ,π ∗ EndE ) ( X, Y ) φ + π ∗ ∇ X π ∗ ∇ Y φ − π ∗ ∇ Y π ∗ ∇ X φ, where R ( π ∗ ∇ ,π ∗ EndE ) is the curvature of π ∗ ∇ , which is considered as a con-nection on π ∗ EndE . The lemma then follows from the identity: R ( π ∗ ∇ ,π ∗ EndE ) ( X, Y ) φ = [ R π ∗ ∇ ( X, Y ) , φ ]. (cid:3) Proof of Proposition 2.13, Part 2).
Let
X, Y ∈ Γ( T C ) and consider P N J ( X, Y ) = P ([ J X, J Y ] − J [ J X, Y ] − J [ X, J Y ] − [ X, Y ]) . By using the previous lemma as well as the fact that P J = φP , we canexpress P N J ( X, Y ) as the sum of two sets of terms. The first set involvesthe curvature of π ∗ ∇ :[ R π ∗ ∇ ( X, Y ) − R π ∗ ∇ ( J X, J Y ) , φ ] + φ [ R π ∗ ∇ ( J X, Y ) + R π ∗ ∇ ( X, J Y ) , φ ] . NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 9
When restricted to J ∈ C this gives the expression for P N J ( X, Y ) that iscontained in Part 2 of the proposition.The second set of terms is π ∗ ∇ J X P ( J Y ) − φπ ∗ ∇ J X P ( Y ) − φπ ∗ ∇ X P ( J Y ) − π ∗ ∇ X P ( Y ) − ( X ↔ Y ) . Using P J = φP , it easily follows that the first four terms and the last four,which are represented by ( X ↔ Y ), separately add to zero. (cid:3) Integrability Conditions.
We are now prepared to explore the integra-bility conditions of J ( ∇ ,I ) on C ′ , where, as above, C ′ is any almost complexsubmanifold of ( C ( E ) , J ( ∇ ,I ) ) such that π C ′ : C ′ −→ M is a surjective sub-mersion. As is well known, J ( ∇ ,I ) on C ′ will be integrable if and only if π ∗ N J ( X, Y ) and
P N J ( X, Y ) are both zero ∀ X, Y ∈ T J C ′ and ∀ J ∈ C ′ . ByProposition 2.13, the first condition is equivalent to the vanishing of theNijenhuis tensor of I , while the second is equivalent to[ R ∇ ( v, w ) − R ∇ ( Iv, Iw ) , J ] + J [ R ∇ ( Iv, w ) + R ∇ ( v, Iw ) , J ] = 0 ∀ v, w ∈ T π ( J ) M and ∀ J ∈ C ′ . To analyze this condition, we will express itin terms of R , , the (0,2)-form part of the curvature R ∇ : Lemma 2.15.
The condition [ R ∇ ( v, w ) − R ∇ ( Iv, Iw ) , J ] + J [ R ∇ ( Iv, w ) + R ∇ ( v, Iw ) , J ] = 0 ∀ v, w ∈ T π ( J ) M holds true if and only if [ R , , J ] E , J = 0 . We thus have:
Theorem 2.16. ( C ′ , J ( ∇ ,I ) ) is a complex manifold if and only if I is integrable R , , J ] E , J = 0 , ∀ J ∈ C ′ . Note that the second condition in the above theorem is equivalent to R , : E , J −→ E , J , ∀ J ∈ C ′ . (1,1) Curvature. Assuming henceforth that I is integrable, an impor-tant case of Part 2 of the above theorem that guarantees that ( C ′ , J ( ∇ ,I ) )is a complex manifold is when R (0 , = 0, or equivalently, when R ∇ is (1,1)with respect to I . In particular, we have: Theorem 2.17.
Let E −→ ( M, I ) be fibered over a complex manifold andlet ∇ be a connection on E that has (1,1) curvature. Then J ( ∇ ,I ) is anintegrable complex structure on C ( E ) . In addition, if g is a fiberwise metricon E and ∇ g = 0 then T ( E, g ) is a complex submanifold of ( C ( E ) , J ( ∇ ,I ) ) . If we C -linearly extend ∇ to a complex connection on E C := E ⊗ R C thenthe condition that R ∇ is (1,1) can also be expressed as ( ∇ , ) = 0 . We thushave:
Lemma 2.18.
Let ∇ be a connection on E −→ ( M, I ) . Then R ∇ is (1,1)if and only if ∇ , is a ∂ − operator on E C . In Section 5.2 we will use the fact that ∇ , is a ∂ − operator to holo-morphically embed ( C , J ( ∇ ,I ) ) into a more familiar complex manifold thatis associated with the holomorphic bundle E C —the Grassmannian bundle Gr n ( E C ). Example 2.19 (Pseudoholomorphic Curves) . Let E −→ ( M, I ) be an evenrank real vector bundle fibered over a complex curve. If ∇ is any connectionon E then R , is automatically zero and hence ( C ( E ) , J ( ∇ ,I ) ) is a com-plex manifold. Moreover if g is a fiberwise metric on E and ∇ is a metricconnection then T ( E, g ) is a complex submanifold of ( C ( E ) , J ( ∇ ,I ) ).As an application, let E −→ ( N, J ) be an even rank real vector bundlethat is fibered over an almost complex manifold and let ∇ be any connectionon E . The goal is to show that although ( C ( E ) , J ( ∇ ,J ) ) is only an almostcomplex manifold, it always contains many pseudoholomorphic submanifoldsthat are in fact complex manifolds. The idea is to use the well knownexistence of a plethora of pseudoholomorphic curves in N . Indeed, if we let i : ( S, I ) −→ ( N, J ) be a pseudoholomorphic embedding of a complex curveinto N then the curvature of i ∗ ∇ on i ∗ E is (1,1) and thus ( C ( i ∗ E ) , J ( i ∗ ∇ ,I ) )is a complex manifold. As it is straightforward to show that i induces apseudoholomorphic embedding of C ( i ∗ E ) into C ( E ), C ( i ∗ E ) is one of manyexamples of pseudoholomorphic submanifolds of C ( E ) that are themselvescomplex manifolds.Further connections between twistors and pseudoholomorphic curves willbe explored in the near future. (cid:3) In Section 3, we will use Theorem 2.17 to construct complex structureson the twistor spaces of SKT, bihermitian and strong HKT manifolds.2.5.
Other Curvature Conditions.
Although, by Theorem 2.16, the con-dition R (0 , = 0 guarantees the integrability of J ( ∇ ,I ) on C ′ ⊂ C ( E ), it isnot the most general one. The present goal is to demonstrate some of thesemore general conditions for certain C ′ .As a first example, consider a C ′ that satisfies the following condition:given any J ∈ C ′ , − J is also in C ′ . Proposition 2.20. If C ′ satisfies the above condition then ( C ′ , J ( ∇ ,I ) ) is acomplex manifold if and only if [ R , , J ] = 0 for all J ∈ C ′ .Proof. If ( C ′ , J ( ∇ ,I ) ) is a complex manifold then given J ∈ C ′ , it followsfrom Theorem 2.16 that [ R , , J ] E , J and [ R , , J ] E , − J are both zero. Hence[ R , , J ] = 0 for all J ∈ C ′ . As I is already assumed to be integrable, theconverse also follows from Theorem 2.16. (cid:3) In the case when C ′ = C , it is straightforward to show that the condition[ R , , J ] = 0 for all J ∈ C is equivalent to the endomorphism part of R , being pointwise constant. We thus have: NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 11
Theorem 2.21. ( C , J ( ∇ ,I ) ) is a complex manifold if and only if R (0 , = λ ⊗ , where λ is a (0,2) form on M and is the identity endomorphismon E C . Example 2.22.
To take a simple example, let ∇ ′ be a connection on E −→ ( M, I ) that has (1,1) curvature and let ∇ = ∇ ′ + ( w ⊗ ) for some 1-form w . Then ( R ∇ ) , = ( ∇ , ) on E C equals ∂w , ⊗ and hence J ( ∇ ,I ) isa complex structure on C . This complex structure, however, is not newsince J ( ∇ ,I ) is actually equal to J ( ∇ ′ ,I ) . The reason is that although theconnections ∇ and ∇ ′ are not equal on E they are in fact the same on EndE .More interesting examples will be the subject of future work. (cid:3)
For another example of a C ′ of the above type, let g be a fiberwise metricon E −→ ( M, I ) and let ∇ be a metric connection. As in the case for C , itfollows from Proposition 2 .
20 that J ( ∇ ,I ) is integrable on C ′ = T ( g ) if andonly if the endomorphism part of R , is pointwise constant. However, inthis case R , is a (0,2) form that takes values in the skew endomorphismbundle o ( E C , g ), so that its trace is zero. We thus have: Theorem 2.23. ( T , J ( ∇ ,I ) ) is a complex manifold if and only if R (0 , = 0 . Examples
The goal of the next few sections is to describe various connections with(1,1) curvature on holomorphic Hermitian bundles and the resulting complexstructures on the twistor spaces C and T . In particular, we will demonstratethat the twistor spaces of SKT, bihermitian and strong HKT manifoldsnaturally admit complex structures. In Section 4, we will explore propertiesof Hermitian structures on these twistor spaces.We will begin by considering the Chern connections of Hermitian bundles.3.1. Chern Connections.
Let E −→ ( M, I ) be a holomorphic vector bun-dle fibered over a complex manifold. Here, we will view it as a real bundleequipped with a fiberwise complex structure, J . If g is any fiberwise metricon E that is compatible with J then, as is well known, the associated Chernconnection ∇ Ch (considered as a real connection on E ) has (1,1) curvature.We thus have Corollary 3.1. ( C , J ( ∇ Ch ,I ) ) is a complex manifold and T is a complexsubmanifold. Example 3.2.
As a simple example, let (
M, I ) be any complex manifoldthat admits a Kahler metric g . Then the Chern connection, ∇ Ch , on T M is the same as the Levi Civita connection, ∇ Levi . Thus J ( ∇ Levi ,I ) is anintegrable complex structure on C and T . (cid:3) If we now C -linearly extend ∇ Ch to E C then, as a particular case of Lemma2.18, ∇ Ch (0 , is a ∂ -operator for this bundle. To describe this ¯ ∂ -operator in more familiar terms, let us consider the holomorphic bundle E , ⊕ E ∗ , ,where E , is the + i eigenbundle of J . The claim then is that the map1 ⊕ g : E C = E , ⊕ E , −→ E , ⊕ E ∗ , is an isomorphism of holomorphic vector bundles. If we denote the Chernconnection on E , by ˜ ∇ Ch then this follows from the following proposition,whose proof is straightforward. Proposition 3.3. ∇ Ch = ˜ ∇ Ch ⊕ g − ˜ ∇ Ch g , as complex connections on E C = E , ⊕ E , . Thus in particular if { e i } is a local holomorphic trivialization of E , then { e i , g − ( e i ) } is a holomorphic trivialization of E C .Now if g ′ is another fiberwise metric on E that is compatible with J then in Section 5 we will address the question of whether ( T ( g ′ ) , J ( ∇ Ch ′ ,I ) )is biholomorphic to ( T ( g ) , J ( ∇ Ch ,I ) ) by holomorphically embedding twistorspaces into Grassmannian bundles.3.2. ¯ ∂ -operators. In the previous section, we found it useful to describe ∇ Ch (0 , on E C by considering the natural ¯ ∂ -operator ¯ ∂ on E , ⊕ E ∗ , andthe isomorphism1 ⊕ g : E C = E , ⊕ E , −→ E , ⊕ E ∗ , . In this section, we will give more examples of ¯ ∂ − operators on E , ⊕ E ∗ , and use this same isomorphism to transfer them to ones on E C . These inturn will give metric connections on E with (1,1) curvature that can be usedto define complex structures on T .To begin, let ( E, g, J ) −→ ( M, I ) be, as above, a holomorphic Hermitianvector bundle and consider the following natural symmetric bilinear form <, > on E , ⊕ E ∗ , : < X + µ, Y + ν > = ( µ ( Y ) + ν ( X )). A general¯ ∂ -operator that preserves this metric is of the form ¯ ∂ + D ′ , , where D ′ , ∈ Γ( T ∗ , ⊗ so ( E , ⊕ E ∗ , )) . If we now consider the splitting of so ( E , ⊕ E ∗ , ) = EndE , ⊕ ∧ E ∗ , ⊕ ∧ E , then we may decompose D ′ , = (cid:18) A αD − A t (cid:19) , where A, D and α are (0,1) forms with values in EndE , , ∧ E ∗ , and ∧ E , , respectively.Since ¯ ∂ + D ′ , squares to zero, there are differential conditions on thesesections. If we take, for example, the case when D ′ , = D then theseconditions are equivalent to ∂D = 0; a similar statement holds for the casewhen D ′ , = α. To obtain ∂ -operators on E C , consider, as above, the isomorphism,1 ⊕ g : ( E C = E , ⊕ E , , g −→ ( E , ⊕ E ∗ , , <, > ) . NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 13 ∂ + D ′ , on E , ⊕ E ∗ , then corresponds to ∇ Ch (0 , + D , g on E C , where D , g = (cid:18) A αgg − D − g − A t g (cid:19) . As we are interested in real connections on E , note that ∇ Ch (0 , + D , g isthe (0,1) part of the real connection ∇ Ch + D g := ∇ Ch + D , g + D , g , whosecurvature is (1,1). Corollary 3.4. J ( ∇ Ch + D g ,I ) is a complex structure on C and T . For convenience, we summarize the ¯ ∂ -operators and connections that wehave discussed so far in the following table. E E C E , ⊕ E ∗ , ∇ Ch + D g ∇ Ch (0 , + D , g ¯ ∂ + D ′ , If we now take the case when D ′ , = D then in Section 5.4 we will explorehow J ( ∇ Ch + D g ,I ) on T depends on the Dolbeault cohomology class of D in H , ( ∧ E ∗ , ), i.e. if B ∈ Γ( ∧ E ∗ , ) then we will determine whether ∂ + D and ∂ + D + ∂B give isomorphic complex structures on T .Moreover we will also address a question that is a generalization of theone raised in the previous section: if g ′ were another fiberwise metric on E that is compatible with J then given D ′ , ∈ Γ( T ∗ , ⊗ so ( E , ⊕ E ∗ , )), isit true that ( T ( g ′ ) , J ( ∇ Ch ′ + D g ′ ,I ) ) is biholomorphic to ( T ( g ) , J ( ∇ Ch + D g ,I ) )?3.3. Three Forms.
An important case of the above discussion is when E = T M is fibered over a Hermitian manifold (
M, g, I ) that is equippedwith a real three form H = H , + H , of type (1,2) + (2,1), such that ∂H , = 0. In this case, we will let D ′ , = H , , which is defined to bea section of T ∗ , ⊗ so ( T , ⊕ T ∗ , ) by setting H , v w = H , ( v, w, · ), for v ∈ T , and w ∈ T , . It then follows that ∇ Ch (0 , + g − H , , where here g − H , = g − H , (1+ iI ) , is a ∂ -operator on T M C = T , ⊕ T , . As thecorresponding D g in the above table is I [ g − H, I ], we have
Proposition 3.5. ∇ Ch + I [ g − H, I ] is a metric connection on T M with(1,1) curvature.
Hence J ( ∇ Ch + I [ g − H,I ] ,I ) is a complex structure on C and T .As we will now show, natural examples of the above three form H can befound on SKT manifolds, bihermitian manifolds and strong HKT manifolds.3.3.1. SKT Manifolds.
A natural example of a real three form on any Her-mitian manifold, (
M, g, I ), is H = − d c w = i ( ∂ − ∂ ) w , where w ( · , · ) = g ( I · , · ).If we take its (2,1) part, H , , then it is straightforward to check that it is ∂ closed if and only if dH = 0. Manifolds whose H satisfy this conditionare known in the literature as strong Kahler with torsion (SKT) mani-folds [7, 6]. One of the associated ∂ -operators on T M C = T , ⊕ T , is ∇ Ch (0 , − g − H , . As a corollary of the above discussion, this ∂ -operatorleads to complex structures on the twistor spaces C and T that can be de-scribed as follows. First note that ∇ Ch (0 , − g − H , is the (0,1) part of thereal connection ∇ Ch − I [ g − H, I ] which can be shown to be equal to ∇ − := ∇ Levi − g − H , where ∇ Levi is the Levi Civita connection. The connection ∇ − is closely related to the Bismut connection, ∇ + := ∇ Levi + g − H (seebelow for a general definition as well as [5, 9]). Theorem 3.6. If ( M, g, I ) is SKT then ( C , J ( ∇ − ,I ) ) is a complex manifoldand T is a complex submanifold. The Bismut connection that was mentioned above is actually defined forany almost Hermitian manifold:
Definition 3.7.
Let (
M, g, I ) be an almost Hermitian manifold. The Bismutconnection is the unique connection, ∇ + , on T M that satisfies1) ∇ + = ∇ Levi + 12 g − H, where H is a 3-form2) ∇ + I = 0 . It can be shown that H is (1,2) +(2,1) if and only if I is integrable andin this case it equals − d c w [9, 12].3.3.2. Bihermitian Manifolds.
A source of SKT manifolds is bihermitianmanifolds. They were first introduced by physicists in [8], motivated bystudying certain supersymmetric sigma models, and were later found to beequivalent to (twisted) generalized Kahler manifolds [12, 15] (see also [1]).A bihermitian manifold is by definition a Riemannian manifold (
M, g ) thatis equipped with two metric compatible complex structures J + and J − thatsatisfy the following conditions ∇ + J + = 0 and ∇ − J − = 0 , where ∇ ± = ∇ Levi ± g − H , for a closed three form H .It then follows from Definition 3.7 that ∇ + and ∇ − are the respectiveBismut connections for J + and J − . Thus an equivalent way to express theabove bihermitian conditions is H = − d c + w + = d c − w − and dH = 0 . Since dH is assumed to be zero, ( g, J + ) and ( g, J − ) are two SKT structuresfor M and hence by Theorem 3.6 the associated twistor space T admits thefollowing two complex structures that depend on the three form H : Theorem 3.8. J ( ∇ − ,J + ) and J ( ∇ + ,J − ) are two complex structures on C and T . NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 15
Strong HKT Manifolds.
Another source of SKT manifolds is stronghyperkahler with torsion (strong HKT) manifolds [11]. Let (
M, g, I, J, K ) bea strong HKT manifold so that
I, J and K are metric compatible complexstructures that satisfy • { I, J } = 0 and K = IJ • ∇ + I = 0 , ∇ + J = 0 and ∇ + K = 0 , where ∇ + = ∇ L + g − H , ∇ L isthe Levi Civita connection and H is a closed three form.Setting ∇ − = ∇ L − g − H and using Theorem 3.6, we have Theorem 3.9. J ( ∇ − ,I ) , J ( ∇ − ,J ) and J ( ∇ − ,K ) are three integrable complexstructures on T . Remark 3.10.
The above complex structures on the twistor space of a strongHKT manifold are quite different from the complex structure J taut knownin the literature [11] . First, J taut is integrable generally on S := { aI + bJ + cK | a + b + c = 1 } ⊂ T . Second, it is defined by using the connection ∇ + to split T S = V S ⊕ H ∇ + S and then setting J taut = φ ⊕ φ . Moreover, it isintegrable on S without assuming dH = 0 . This is to be compared with thecomplex structures of Theorem 3.9, which are integrable on all of T and aredefined by using the ∇ − connection to split T T . In Section 4.2, we construct a metric compatible with J ( ∇ − ,I ) , J ( ∇ − ,J ) and J ( ∇ − ,K ) so that the three associated Hermitian structures have equaltorsions. 4. Hermitian Structures on T and their Torsion Let (
M, g, I ) be an almost Hermitian manifold and let ∇ be a metricconnection on T M . We will first build a metric on the total space of π : T ( T M, g ) → M that will be compatible with J ( ∇ ,I ) . To do so, consider thesplitting T T = V T ⊕ H ∇ T into vertical and horizontal distributions. As inSection 2.3, we will identify H ∇ T with π ∗ T M and V T with [ π ∗ o ( T M, g ) , φ ],where φ ∈ Γ( π ∗ End ( T M )) is defined by φ | K = K . Definition 4.1.
Using the splitting T T = V T ⊕ H ∇ T , define g ∇ = − tr ⊕ π ∗ g, where − tr ( A, B ) = − tr ( AB ) for A, B ∈ V K T .Using Theorem 2.17, we have Proposition 4.2. ( g ∇ , J ( ∇ ,I ) ) is an almost Hermitian structure on T .Moreover, it is Hermitian if I is integrable and R ∇ is (1,1) with respectto I . Given ( g ∇ , J ( ∇ ,I ) ), we then define the associated fundamental two form w ( ∇ ,I ) ( · , · ) = g ∇ ( J ( ∇ ,I ) · , · ) . The following gives an expression for the corresponding torsion three form d c w ( ∇ ,I ) ( · , · , · ) := − dw ( ∇ ,I ) ( J ( ∇ ,I ) · , J ( ∇ ,I ) · , J ( ∇ ,I ) · ) in terms of the curva-ture and torsion of ∇ , R ∇ and T ∇ , and the vertical projection opera-tor P ∇ defined by the splitting of T T = V T ⊕ H ∇ T . (As above, φ ∈ Γ( π ∗ End ( T M )) is defined by φ | K = K .) Theorem 4.3.
Let ( M, g, I ) be an almost Hermitian manifold and let ∇ be a metric connection on T M . Given X i ∈ T K T with π ∗ X i = v i and i ∈ { , , } , d c w ( ∇ ,I ) ( X , X , X ) = tr ([ R ∇ ( Iv , Iv ) , φ ] P ∇ ( X ))+ g (( ∇ Iv I ) v , v ) − g ( T ∇ ( Iv , Iv ) , v )+ cyclic (1 , , . The proof will be based on the following proposition whose proof isstraightforward.
Proposition 4.4.
Let ( N, g, J ) be an almost Hermitian manifold and let ∇ be a metric connection on T N . Set w ( · , · ) = g ( J · , · ) and d c w ( · , · , · ) = − dw ( J · , J · , J · ) . Given X i ∈ T x N , for i ∈ { , , } , d c w ( X , X , X ) = g (( ∇ JX J ) X , X ) − g ( T ∇ ( J X , J X ) , X )+ cyclic (1 , , . To apply this proposition to the setting of Theorem 4.3, we will define thefollowing connection on T T that will be compatible with g ∇ . First note thatsince the connection π ∗ ∇ ′ := π ∗ ∇ + ( π ∗ ∇ φ ) φ on π ∗ T M satisfies π ∗ ∇ ′ φ = 0,it extends to a connection on V T . Using the splitting T T = V T ⊕ H ∇ T ,we then define D = π ∗ ∇ + 12 ( π ∗ ∇ φ ) φ ⊕ π ∗ ∇ . Proposition 4.5. D is a connection on T T that is compatible with g ∇ andsatisfies D J ( ∇ ,I ) = 0 ⊕ π ∗ ∇ π ∗ I . We will now express T D in terms of R ∇ and T ∇ and will then proveTheorem 4.3. Theorem 4.6.
For
X, Y ∈ T K T , P ∇ T D ( X, Y ) = [ π ∗ R ∇ ( X, Y ) , φ ]2) π ∗ T D ( X, Y ) = π ∗ T ∇ ( X, Y ) . Proof.
To prove Part 1), consider P ∇ T D ( X, Y ) = P ∇ ( D X Y − D Y X − [ X, Y ])= D X P ∇ ( Y ) − D Y P ∇ ( X ) − P ∇ ([ X, Y ]) . NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 17
Using that P ∇ ( X ) = π ∗ ∇ X φ (see Proposition 2.7) and the definition of D ,this becomes π ∗ ∇ X π ∗ ∇ Y φ − π ∗ ∇ Y π ∗ ∇ X φ − π ∗ ∇ [ X,Y ] φ + 12 [( π ∗ ∇ X φ ) φ, π ∗ ∇ Y φ ] −
12 [( π ∗ ∇ Y φ ) φ, π ∗ ∇ X φ ] . Since the last two terms add to zero, this equals R ( π ∗ ∇ ,Endπ ∗ T M ) ( X, Y ) φ = [ R ( π ∗ ∇ ,π ∗ T M ) ( X, Y ) , φ ] = [ π ∗ R ( ∇ ,T M ) ( X, Y ) , φ ] , where R ( ∇ ′ ,E ) is the curvature associated with a connection ∇ ′ on a vectorbundle E .The proof of Part 2) is straightforward. (cid:3) We will now prove Theorem 4.3.
Proof of Theorem 4.3.
Set J = J ( ∇ ,I ) and let X i ∈ T K T with π ∗ X i = v i and i ∈ { , , } . By Propositions 4.4 and 4.5, d c w ( ∇ ,I ) ( X , X , X ) = g ∇ (( D J ) J X X , X ) − g ∇ ( T D ( J X , J X ) , X ) + cyclic (1 , , . Using Proposition 4.5 and Theorem 4.6, this becomes tr ([ π ∗ R ∇ ( J X , J X ) , φ ] P ∇ ( X )) + g (( π ∗ ∇ J X π ∗ I ) v , v ) − g ( π ∗ T ∇ ( J X , J X ) , v ) + cyclic (1 , , . This equals tr ([ R ∇ ( Iv , Iv ) , φ ] P ∇ ( X )) + g (( ∇ I ) Iv v , v ) − g ( T ∇ ( Iv , Iv ) , v ) + cyclic (1 , , . (cid:3) Hermitian Pairs with Equal Torsion.
As a first application of The-orem 4.3, we will construct a metric and two compatible complex structureson the twistor space that have equal torsions. To do so, let (
M, g, I ) be aHermitian manifold and let ∇ be a metric connection on T M with (1 , J ( ∇ ,I ) , the integrable complex structure −J ( ∇ , − I ) iscompatible with g ∇ . Letting w ( ∇ , and w ( ∇ , be the respective fundamen-tal two forms, we have Theorem 4.7.
Given X i ∈ T K T with π ∗ X i = v i and i ∈ { , , } ,1) d c w ( ∇ , ( X , X , X ) = d c w ( ∇ , ( X , X , X )2) d c w ( ∇ , ( X , X , X ) = tr ([ R ∇ ( v , v ) , φ ] P ∇ ( X ))+ g (( ∇ Iv I ) v , v ) − g ( T ∇ ( Iv , Iv ) , v )+ cyclic (1 , , . Remark 4.8.
In this notation, the “ d c ” in d c w ( ∇ , is with respect to J ( ∇ ,I ) and that in d c w ( ∇ , is with respect to −J ( ∇ , − I ) . Strong HKT Manifolds.
Let (
M, g, I, J, K ) be a strong hyperkahlerwith torsion (strong HKT) manifold so that
I, J and K are metric compat-ible complex structures that satisfy • { I, J } = 0 and K = IJ • ∇ + I = 0 , ∇ + J = 0 and ∇ + K = 0 , where ∇ + = ∇ L + g − H , ∇ L isthe Levi Civita connection and H is a closed three form.Setting ∇ − = ∇ L − g − H and using Theorem 3.9, we have Proposition 4.9. J ( ∇ − ,I ) , J ( ∇ − ,J ) and J ( ∇ − ,K ) are three integrable com-plex structures on T that are compatible with the metric g ∇ − . The associated torsions are equal (see Remark 4.8 on notation):
Theorem 4.10.
Given X i ∈ T K ′ T with π ∗ X i = v i and i ∈ { , , } ,1) d c w ( ∇ − ,I ) ( X , X , X ) = d c w ( ∇ − ,J ) ( X , X , X ) = d c w ( ∇ − ,K ) ( X , X , X )2) d c w ( ∇ − ,I ) ( X , X , X ) = tr ([ R ∇ − ( v , v ) , φ ] P ∇ − ( X )) + cyclic (1 , , − H ( v , v , v ) . The proof is based on Theorem 4.3 and the following lemma.
Lemma 4.11.
Let ( M, g, I ) be a Hermitian manifold and let ∇ − = ∇ L − g − H , where H = − d c w and w ( · , · ) = g ( I · , · ) . For v , v , v ∈ T x M , − H ( v , v , v ) = g (( ∇ − Iv I ) v , v ) − g ( T ∇ − ( Iv , Iv ) , v ) + cyclic (1 , , . Proof.
Using ∇ − I = − [ g − H, I ] and T ∇ − ( v , v ) = − g − H v v , g (( ∇ − Iv I ) v , v ) − g ( T ∇ − ( Iv , Iv ) , v ) + cyclic (1 , , − g ([ g − H Iv , I ] v , v ) + g ( g − H Iv Iv , v ) + cyclic (1 , , . Since H is (1 ,
2) + (2 ,
1) with respect to I , [ g − H Iv , I ] = I [ g − H v , I ], so thatthe above becomes − g ( I [ g − H v , I ] v , v ) + g ( g − H Iv Iv , v ) + cyclic (1 , , g ( g − H v Iv , Iv ) − g ( g − H v v , v ) + g ( g − H Iv Iv , v ) + cyclic (1 , , H ( v , Iv , Iv ) − H ( v , v , v ) + H ( Iv , Iv , v ) + cyclic (1 , , . Using that H is (1 ,
2) + (2 , − H ( Iv , v , Iv ) − H ( Iv , v , Iv ) − H ( Iv , v , Iv )= − H ( Iv , v , Iv ) − H ( Iv , Iv , v ) − H ( v , Iv , Iv )= − H ( v , v , v ) . (cid:3) NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 19
Hyperkahler Manifolds.
If we restrict to the case when (
M, g, I, J, K )is hyperkahler, so that H = 0, then the torsions associated with the metric g ∇ L and the complex structures J ( ∇ L ,I ) , J ( ∇ L ,J ) and J ( ∇ L ,K ) are equal andgenerally nonzero: Theorem 4.12.
Let ( M, g, I, J, K ) be a hyperkahler manifold. Given X i ∈ T K ′ T with π ∗ X i = v i and i ∈ { , , } ,1) d c w ( ∇ L ,I ) ( X , X , X ) = d c w ( ∇ L ,J ) ( X , X , X ) = d c w ( ∇ L ,K ) ( X , X , X )2) d c w ( ∇ L ,I ) ( X , X , X ) = tr ([ R ∇ L ( v , v ) , φ ] P ∇ L ( X )) + cyclic (1 , , . Bihermitian Manifolds.
Let (
M, g, J + , J − ) be a bihermitian mani-fold, so that ∇ + J + = 0 and ∇ − J − = 0 , where ∇ ± = ∇ L ± g − H , ∇ L is the Levi Civita connection and H is aclosed three form. Using Theorem 3.8, we have the following two Hermitianstructures on T : ( g ∇ + , J ( ∇ + ,J − ) ) and ( g ∇ − , J ( ∇ − ,J + ) ).The first step will be to compare the two metrics g ∇ + and g ∇ − in thefollowing proposition, whose proof is straightforward. Proposition 4.13.
1) For
X, Y ∈ H ∇ L K T with π ∗ X = v and π ∗ Y = w , g ∇ + ( X, Y ) = g ∇ − ( X, Y )= g ( v, w ) − tr g − H v , φ ][ g − H w , φ ]) .
2) For
A, B ∈ V K T , g ∇ + ( A, B ) = g ∇ − ( A, B ) .
3) For A ∈ V K T and Y ∈ H ∇ L K T with π ∗ Y = w , g ∇ + ( A, Y ) = − g ∇ − ( A, Y )= − tr A [ g − H w , φ ]) . Consequently, the above constructions do not yield a metric on T thatis compatible with both complex structures J ( ∇ + ,J − ) and J ( ∇ − ,J + ) . Infact, one can show that such metrics do not exist on the twistor space ofa general bihermitian manifold. (Though perhaps they exist on certainsubmanifolds of T .) In the case when ( J + − J − ) is invertible, I have useddifferent constructions to build such metrics on all of T . The key is touse splittings of T T that are not induced from connections on T M . I willpresent these results in a separate paper. Twistors and Grassmannians
In Sections 3.1 and 3.2, we raised several questions about the complexmanifold structure of ( C ( E ) , J ( ∇ ,I ) ), where R ∇ is (1 , C into a morefamiliar complex manifold—a certain Grassmannian bundle. Indeed, as wenoted previously, the condition that R ∇ is (1,1) is equivalent to ∇ , being a ∂ -operator on E C , and if we let rankE = 2 n then the Grassmannian bundlethat we will take will be the holomorphic bundle Gr n ( E C ).To define the embedding, we will first show how to holomorphically embedthe fibers of C into those of Gr n ( E C ).5.1. Embedding the Fibers.
Let V be a 2 n dimensional real vector spaceand let Gr n ( V C ) be the Grassmannians of complex n planes. The map thatwe will consider is ψ : C ( V ) −→ Gr n ( V C ) J −→ V , J ;it has the following properties: Proposition 5.1.
1) The map ψ : C ( V ) −→ Gr n ( V C ) is a holomorphic embedding.2) The image of ψ is { P ∈ Gr n ( V C ) | P ⊕ P = V C } , which is an opensubmanifold of the Grassmannians.Proof. Consider ψ ∗ : T J C ( V ) −→ T V , J Gr n ( V C ) and choose the holomorphicchart End ( V , J ,V , J ) −→ Gr n ( V C ) B −→ Graph ( B ) , where Graph ( B ) = { v , + Bv , | v , ∈ V , J } . If we let A be a generalelement in T J C ( V ) ∼ = { D ∈ EndV |{ D, J } = 0 } then we need to show that ψ ∗ ( J A ) = I ψ ∗ ( A ), where I is the complex structure on the Grassmannians.First consider, ψ ∗ ( J A ) = ddt | t =0 ψ ( exp ( − tA J exp ( tA ddt | t =0 exp ( − tA V , J ) . Using the above chart, ψ ∗ ( J A ) then corresponds to − A , as an element of End ( V , J , V , J ).Similarly we have ψ ∗ ( A ) = ddt | t =0 exp ( − tAJ )( V , J ), so that under theabove chart, I ψ ∗ ( A ) corresponds to − iAJ , which as an element of End ( V , J , V , J )equals − A .The proof of the other parts of the proposition is straightforward. (cid:3) NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 21
If we now choose a positive definite metric, g , on V then by restriction, theabove map, ψ , gives a holomorphic embedding of T ( V ) into Gr n ( V C ). Sincethe metric is positive definite, the image of this map is precisely M I ( V C ) = { P ∈ Gr n ( V C ) | g ( v, w ) = 0 , ∀ v, w ∈ P } , the space of maximal isotropics of V C defined by using the C -bilinearly extended metric. For convenience westate this as a proposition. Proposition 5.2. T ( V ) −→ Gr n ( V C ) J −→ V , J is a holomorphic embedding with image M I ( V C ) . The Holomorphic Embedding.
Let us now consider a rank 2 n realvector bundle E −→ ( M, I ) that is fibered over a complex manifold. Asdiscussed above, a connection ∇ on E with (1,1) curvature gives rise to twocomplex analytic manifolds: the twistor space ( C , J ( ∇ ,I ) ) and the holomor-phic fiber bundle π Gr : Gr n ( E C ) −→ M . To holomorphically embed C into Gr n ( E C ), we will generalize the map ψ that was defined in the previoussection: Theorem 5.3.
The map ψ : ( C , J ( ∇ ,I ) ) −→ Gr n ( E C ) J −→ E , J is a holomorphic embedding. In the case when E is equipped with a fiberwise metric g and ∇ is a metricconnection, we will define M I ( E C ) to be the space of maximal isotropics in Gr n ( E C ); we then have: Proposition 5.4. ( T , J ( ∇ ,I ) ) −→ Gr n ( E C ) J −→ E , J is a holomorphic embedding with image M I ( E C ) . To prove Theorem 5.3, we will need to describe the complex structureon the Grassmannians similarly to how we defined J ( ∇ ,I ) on C . The firststep will be to define the horizontal distribution H ∇ Gr n on Gr n ( E C ). Butbefore giving the definition, let us first recall that if P ∈ Gr n ( E C ) and γ : R −→ M satisfies γ (0) = π Gr ( P ) then we can use ∇ , considered asa complex connection on E C , to parallel transport P along γ as follows.If we set P = < e , ..., e n > C , so that { e i } is a basis for P , then define P ( t ) = < e ( t ) , ..., e n ( t ) > C , where γ ∗ ∇ e i ( t ) = 0 and e i (0) = e i . Since ∇ isa complex connection on E C , P ( t ) does not depend on the basis { e i } for P that was chosen.With this, let us define the desired horizontal distribution on Gr n ( E C ). Definition 5.5.
Let H ∇ P Gr n = { dP ( t ) dt | t =0 | P ( t ) is the parallel translate of Palong γ, γ (0) = π Gr ( P ) } . Along with H ∇ Gr n , there is also the natural vertical distribution V Gr n ;as it is defined by the fibers of Gr n ( E C ), it is a complex vector bundle andsatisfies π Gr ∗ ( V P Gr n ) = 0, for all P ∈ Gr n ( E C ). It is straightforward toprove that these two distributions are complements to each other: Lemma 5.6. T P Gr n = V P Gr n ⊕ H ∇ P Gr n . We may now use the above lemma to define an almost complex struc-ture on Gr n ( E C ), which we will show in Proposition 5.8 to be the complexstructure that is induced by ∇ , and which we will use to prove Theorem5.3. As the definition of this almost complex structure is similar to that of J ( ∇ ,I ) on C , we will denote it by the same symbol: Definition 5.7.
Let J ( ∇ ,I ) on Gr n ( E C ) be defined as follows. First split T Gr n = V Gr n ⊕ H ∇ Gr n and then let J ( ∇ ,I ) = J V ⊕ π ∗ Gr I, where J V is the standard fiberwise complex structure on V Gr n and wherewe have used the natural identification of H ∇ Gr n with π ∗ Gr T M .If we consider the complex manifold structure of Gr n ( E C ) that is inducedby the ∂ -operator ∇ , on E C , we then have: Proposition 5.8.
The complex structure on Gr n ( E C ) is J ( ∇ ,I ) . We will prove the above proposition for a more general setup in the nextsection; here we will use it to prove Theorem 5.3 by showing that the map ψ : ( C , J ( ∇ ,I ) ) −→ ( Gr n ( E C ) , J ( ∇ ,I ) ), which is given by ψ ( J ) = E , J , isholomorphic. Recalling the splitting of T C = V C ⊕ H ∇ C , as given in Lemma2.5, let us first consider the following: Lemma 5.9.
The map ψ ∗ preserves horizontals: ψ ∗ : H ∇ J C −→ H ∇ E , J Gr n .In fact, ψ ∗ ( v ∇ ) = v ( ∇ ,Gr ) , where v ∇ and v ( ∇ ,Gr ) are the appropriate hori-zontal lifts of v ∈ T x M .Proof. Let γ ( t ) be a curve in M such that γ (0) = x and γ ′ (0) = v . Also let J ( t ) be the parallel translate of J ∈ C ( E x ) along γ (by using ∇ ), so that ψ ∗ ( v ∇ ) = ddt | t =0 ψ ( J ( t )) . The claim then is that ψ ( J ( t )), which is by definition E , J ( t ) , equals E , J ( t ),the parallel translate of E , J along γ . To show this just note that if e ( t ) isthe parallel translate of e ∈ E , J then J ( t ) e ( t ) is also parallel and since J e = − ie , it follows that J ( t ) e ( t ) = − ie ( t ) for all relevant t ∈ R . Hence ddt | t =0 ψ ( J ( t )) = ddt | t =0 E , J ( t ) = v ( ∇ ,Gr ) . (cid:3) NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 23
Assuming Proposition 5.8, we can now prove that ψ is holomorphic: Proof of Theorem 5.3.
Consider ψ ∗ : T J C −→ T E , J Gr n . By Proposition5.8, we need to show that ψ ∗ J ( ∇ ,I ) = J ( ∇ ,I ) ψ ∗ .A) If A ∈ V J C , the vertical tangent space to J , then it follows fromProposition 5.1 that ψ ∗ ( J A ) = J ( ∇ ,I ) ψ ∗ ( A ) , so that ψ ∗ is holomorphic in the vertical directions.B) As for the horizontal directions, let v ∇ ∈ H ∇ J C be the horizontal liftof v ∈ T x M . Then ψ ∗ ( J ( ∇ ,I ) v ∇ ) = ψ ∗ (( Iv ) ∇ ), which by Lemma 5.9 equals( Iv ) ( ∇ ,Gr ) . This in turn equals J ( ∇ ,I ) v ( ∇ ,Gr ) = J ( ∇ ,I ) ψ ∗ ( v ∇ ). (cid:3) Proof of Proposition 5.8.
In this section, we will prove a slightlymore general version of Proposition 5.8; this will then complete the proofof Theorem 5.3. To begin, we will find it useful to describe the complexstructures on holomorphic vector bundles:Let π F : F −→ ( M, I ) be a complex vector bundle that is equippedwith a ∂ -operator, ∂ , and let ∇ be a complex connection on F such that ∇ , = ∂ . Below we will let J ( ∇ ,I ) be the almost complex structure on either F or Gr k ( F ) that is defined in a by now familiar way: use ∇ to split theappropriate tangent bundle into vertical and horizontal distributions, anddefine J ( ∇ ,I ) to be the direct sum of the given fiberwise complex structureon the verticals and the lift of I on the horizontals. Proposition 5.10.
Let ∇ be a complex connection on F such that ∇ , = ∂ .Then the associated complex structure on F is J ( ∇ ,I ) .Proof. Let { f i } (1 ≤ i ≤ rankF ) be a holomorphic frame for F over U ⊂ M and let W be a complex vector space with basis { w i } . To prove theproposition, we need to show that the map σ : ( F | U , J ( ∇ ,I ) ) −→ U × Wa i f i | x −→ ( x, a i w i )is pseudoholomorphic. For this, consider σ ∗ : T f F −→ T σ ( f ) ( U × W ), where π F ( f ) = x .1) Since σ | x is a complex linear isomorphism from F | x to W , σ is holomor-phic in the vertical directions, i.e., σ ∗ ( if ′ ) = iσ ∗ ( f ′ ), where f ′ ∈ V f F = F | x .
2) As for the horizontal directions, we need to show that σ ∗ ( J ( ∇ ,I ) v ∇ ) = I σ ∗ ( v ∇ ), where v ∇ is the horizontal lift of v ∈ T x M to H ∇ f F ⊂ T f F and I is the complex structure on U × W . Let us first consider, σ ∗ ( J ( ∇ ,I ) v ∇ ) = σ ∗ (( Iv ) ∇ )= ddt | t =0 σ ( f ( t )) , where f ( t ) is the parallel translate of f along a curve γ : R −→ M thatsatisfies γ (0) = x and γ ′ (0) = Iv . If we let f ( t ) = a j ( t ) f j | γ ( t ) then the aboveequals ( Iv, da j ( t ) dt | t =0 w j ) . Similarly, σ ∗ ( v ∇ ) = ( v, d ˜ a j ( t ) dt | t =0 w j ) , where ˜ f ( t ) = ˜ a j ( t ) f j | ˜ γ ( t ) is the paralleltranslate of f along a curve ˜ γ : R −→ M that satisfies ˜ γ (0) = x , ˜ γ ′ (0) = v .Now since I σ ∗ ( v ∇ ) = ( Iv, i d ˜ a j ( t ) dt | t =0 w j ) , σ is pseudoholomorphic if and onlyif da j ( t ) dt | t =0 = i d ˜ a j ( t ) dt | t =0 . To show this equality, note that the condition ˜ γ ∗ ∇ ˜ f ( t ) = 0 togetherwith a j := ˜ a j (0) = a j (0) imply that i d ˜ a j ( t ) dt | t =0 f j = − ia j ∇ v f j . This thenequals − a j ∇ Iv f j because ∇ , f j = 0, which in turn equals da j ( t ) dt | t =0 f j since γ ∗ ∇ f ( t ) = 0. Hence σ is pseudoholomorphic. (cid:3) As for the Grassmannians, we have:
Proposition 5.11.
The complex structure on Gr k ( F ) that is induced by ( F, ∂ ) is J ( ∇ ,I ) . The proof of the above proposition and hence of Proposition 5.8 is justa straightforward generalization of the previous proof. This then completesthe proof of Theorem 5.3 as well.5.4.
Corollaries of the Embedding.
We will now demonstrate some corol-laries of the holomorphic embedding ψ : ( C , J ( ∇ ,I ) ) −→ Gr n ( E C ) , as givenin Theorem 5.3. In particular, we will address certain issues regarding theholomorphic structure of twistor spaces that were raised in Section 3.2.Let E and E ′ be two real vector bundles of even rank that are fiberedover ( M, I ) and that are respectively equipped with connections ∇ and ∇ ′ of (1,1) curvature. Proposition 5.12.
Let A : E −→ E ′ be a bundle map such that its C -extension, A : ( E C , ∇ , ) −→ ( E ′ C , ∇ ′ , ) is an isomorphism of holomorphicvector bundles. Then this map induces a fiber preserving biholomorphismbetween ( C ( E ) , J ( ∇ ,I ) ) and ( C ( E ′ ) , J ( ∇ ′ ,I ) ) .Proof. The isomorphism A : ( E C , ∇ , ) −→ ( E ′ C , ∇ ′ , ) induces the biholo-morphism ˜ A : Gr n ( E C ) −→ Gr n ( E ′ C ) that is defined by ˜ A ( < e , ..., e n > C ) = < Ae , ..., Ae n > C . Since A is a real map, ˜ A restricts to a biholomor-phism between the set { P ∈ Gr n ( E C ) | P ⊕ P = E C | π Gr ( P ) } in Gr n ( E C ) andthe corresponding one in Gr n ( E ′ C ). The proposition then follows from The-orem 5.3 and Proposition 5.1, which show that these sets are respectivelybiholomorphic to ( C ( E ) , J ( ∇ ,I ) ) and ( C ( E ′ ) , J ( ∇ ′ ,I ) ). (cid:3) Now suppose that E and E ′ are also equipped with respective fiberwisemetrics g and g ′ and that the above connections preserve the appropriatemetrics. If we C -bilinearly extend g and g ′ to E C and E ′ C , we then have NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 25
Proposition 5.13.
Let A : ( E C , ∇ , ) −→ ( E ′ C , ∇ ′ , ) be an isomorphism ofholomorphic vector bundles that is orthogonal with respect to g and g ′ . Then A induces a fiber preserving biholomorphism between ( T ( E, g ) , J ( ∇ ,I ) ) and ( T ( E ′ , g ′ ) , J ( ∇ ′ ,I ) ) .Proof. Similar to the proof of Proposition 5.12, the isomorphism A : ( E C , ∇ , ) −→ ( E ′ C , ∇ ′ , ) induces a biholomorphism ˜ A : Gr n ( E C ) −→ Gr n ( E ′ C ). Since A is an orthogonal map, ˜ A maps the space of maximal isotropics, M I ( E C ),in Gr n ( E C ) to the one in Gr n ( E ′ C ). The proposition then follows fromProposition 5.4, which shows that ( T ( E, g ) , J ( ∇ ,I ) ) and ( T ( E ′ , g ′ ) , J ( ∇ ′ ,I ) )are respectively biholomorphic to M I ( E C ) and M I ( E ′ C ). (cid:3) In the following two sections we consider some applications of the abovepropositions.5.4.1.
Cohomology Independence.
Let (
E, g, J ) −→ ( M, I ) be a holomorphicHermitian bundle fibered over a complex manifold and let ∂ be the standard ∂ -operator on E , ⊕ E ∗ , , where E , is the + i eigenbundle of J . If wechoose D ∈ Γ( T ∗ , ⊗∧ E ∗ , ) to satisfy ∂D = 0 then, as described in Section3.2, ∇ Ch (0 , + g − D is a ∂ -operator on E C = E , ⊕ E , and, for ∇ = ∇ Ch + g − D + g − D , the twistor space ( T ( E ) , J ( ∇ ,I ) ) is a complex manifold. If wenow let B ∈ Γ( ∧ E ∗ , ) then ∇ Ch (0 , + g − ( D + ∂B ) is another ∂ -operatoron E C and it is natural to wonder, as in Section 3.2, whether the associatedtwistor space is biholomorphic to the previous one. In other words, does theabove give a well defined mapping from the Dolbeault cohomology group H , ( ∧ E ∗ , ) to the isomorphism classes of complex structures on T ?By using Proposition 5.13, we will show here that such a mapping does in-deed exist. As a first step, let us consider the section of O ( E C , g ) exp ( g − B ),which equals (1 + g − B ) since ( g − B ) = 0. We then have Proposition 5.14.
The map exp ( − g − B ) : ( E C , ∇ Ch (0 , + g − D ) −→ ( E C , ∇ Ch (0 , + g − ( D + ∂B )) is an isomorphism of holomorphic vector bun-dles.Proof. Let ( ∇ Ch (0 , + g − D ) v = 0 and consider( ∇ Ch (0 , + g − ( D + ∂B ))(1 − g − B ) v = −∇ Ch (0 , ( g − Bv ) + ( g − ∂B ) v = − ( ∇ Ch (0 , g − B ) v − g − B ∇ Ch (0 , v + ( g − ∂B ) v. Since the first and last terms cancel, we are left with − g − B ∇ Ch (0 , v = − g − B ( − g − Dv ) = 0. This then proves the proposition. (cid:3) By Proposition 5.13, we can now conclude that the twistor spaces men-tioned above are biholomorphic:
Proposition 5.15. exp ( − g − B ) induces a fiber preserving biholomorphismbetween ( T , J ( ∇ ,I ) ) and ( T , J ( ∇ ′ ,I ) ) , where ∇ , = ∇ Ch (0 , + g − D and ∇ ′ , = ∇ Ch (0 , + g − ( D + ∂B ) . As a corollary, we have
Proposition 5.16.
The map [ D ] −→ [ J ( ∇ ,I ) ] , where ∇ , = ∇ Ch (0 , + g − D , from the Dolbeault cohomology group H , ( ∧ E ∗ , ) to the isomor-phism classes of complex structures on T ( E, g ) is well defined. Changing the Metric.
In the previous example we worked with a fixedmetric g ; but what if we were to choose another metric g ′ on E that is com-patible with J —then is it true that ( T ( g ) , J ( ∇ Ch ,I ) ) and ( T ( g ′ ) , J ( ∇ Ch ′ ,I ) )are biholomorphic? This is part of a more general question that was posed inSection 3.2: in that section we used a fixed metric, g , to define ∂ -operatorson E C and thus complex structures on T ( g )—but if we were to choose an-other metric g ′ then do we obtain new complex manifolds by considering T ( g ′ )?To address these questions, let us first recall some of the details of thatsection. Let ( E, J ) −→ ( M, I ) be a holomorphic vector bundle, consideredas a real bundle with fiberwise complex structure J , that is fibered over acomplex manifold. Defining <, > and ∂ to be the standard inner productand ∂ -operator on E , ⊕ E ∗ , , let us consider the ∂ -operator ¯ ∂ + D ′ , ,where D ′ , ∈ Γ( T ∗ , ⊗ so ( E , ⊕ E ∗ , )) . If g is a fiberwise metric on E that is compatible with J then, as in Section 3.2, we can use the orthogonalisomorphism1 ⊕ g : ( E C = E , ⊕ E , , g −→ ( E , ⊕ E ∗ , , <, > )to obtain the ∂ -operator ∇ Ch (0 , + D , g on E C as well as the complex struc-ture J ( ∇ Ch + D g ,I ) on T ( g ). (Here, D g = D , g + D , g .)Similarly, if g ′ is another fiberwise metric that is compatible with J thenwe have the complex structure J ( ∇ Ch ′ + D g ′ ,I ) on T ( g ′ ). The goal then is touse Proposition 5.13 to show that the complex manifolds T ( g ) and T ( g ′ )are equivalent under a fiberwise biholomorphism.First note, that if we compose the map (1 ⊕ g ) with (1 ⊕ g ′ ) − then weobtain the following isomorphism of holomorphic vector bundles:( E C , ∇ Ch (0 , + D , g ) −→ ( E C , ∇ Ch ′ (0 , + D , g ′ ) v , + v , −→ ( v , + g ′− gv , ) , where we have used the decomposition, E C = E , ⊕ E , . As this is anorthogonal map from ( E C , g ) to ( E C , g ′ ), by Proposition 5.13 we have Proposition 5.17.
There exists a fiber preserving biholomorphism between ( T ( g ) , J ( ∇ Ch + D g ,I ) ) and ( T ( g ′ ) , J ( ∇ Ch ′ + D g ′ ,I ) ) . In particular, if we set D ′ (0 , to zero, we have: NTEGRABLE COMPLEX STRUCTURES ON TWISTOR SPACES 27
Proposition 5.18.
Let ( E, J ) −→ ( M, I ) be a holomorphic vector bundlethat is equipped with two Hermitian metrics g and g ′ . Then ( T ( g ) , J ( ∇ Ch ,I ) ) and ( T ( g ′ ) , J ( ∇ Ch ′ ,I ) ) are biholomorphic. Acknowledgments
I would like to thank Blaine Lawson and Nigel Hitchin for helpful discus-sions. I would also like to thank Alexander Kirillov and Martin Roˇcek.
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