Normal almost contact structures and nonKähler compact complex manifolds
Abstract.
We construct some families of complex structures on compact manifolds by means of normal almost contact structures (nacs) so that each complex manifold in the family has a nonsingular holomorphic flow. These families include as particular cases the Hopf and CalabiEckmann manifolds and the complex structures on the product of two normal almost contact manifolds constructed by Morimoto. We prove that every compact Kähler manifold admitting a nonvanishing holomorphic vector field belongs to one of these families and is a complexificacion of a normal almost contact manifold. Finally we show that if a complex manifold obtained by our constructions is Kählerian the Euler class of the nacs (a cohomological invariant associated to the structure) is zero. Under extra hypothesis we give necessary and sufficient conditions for the complex manifolds so obtained to be Kählerian.
Key words and phrases:
complex structure, normal almost contact structure, Kähler metric1991 Mathematics Subject Classification:
Primary 32C10, 32C16; Secondary 32C171. Introduction
Most of the known examples of complex manifolds, in particular all projective manifolds, are of Kähler type. Nevertheless, the existence of a Kähler metric imposes strong topological restrictions on the manifold, for instance its odd Betti numbers are even. Riemann surfaces are always Kählerian and compact complex surfaces if and only its first Betti number is even (c.f. [6],[17]). For higher dimensions there is not a simple characterization of Kähler manifolds, however one would expect that they are rather the exception than the rule. For instance a corollary of a result by Taubes (c.f [26]) implies that every finite presentation group is the fundamental group of a nonKähler compact complex 3manifold. Historically the first examples of nonKähler manifolds were constructed by H. Hopf as a quocient of for by a contracting holomorphism of which fixes the origin. Later, E. Calabi and B. Eckmann described a class of nonKähler complex structures on the product for such that the corresponding complex manifold is the total space of a holomorphic elliptic principal bundle over . In [20] J.J. Loeb and M. Nicolau generalized CalabiEckmann and Hopf structures by the construction of a class of complex structures on the product that contains the precedents. Similar techniques have been used by S.López de Medrano and A.Verjovsky in [21] to construct another family of nonKählerian compact manifolds and later generalized by L.Meersseman in [22].
The second section of the paper is devoted to describe a general procedure to obtain complex manifolds by means of elementary geometrical constructions. We depart from odddimensional manifolds admitting a normal almost contact structure (nacs for shortness), i.e. a CRstructure of maximal dimension and a transverse CRaction of . More precisely we consider three cases: (A) products of two real manifolds endowed with a nacs, (B) principal bundles over a manifold with a nacs (with an extra restriction on the bundle) and (C) suspensions of a manifold with a nacs by a suitable automorphism. In particular we generalize Morimoto’s construction of a complex structure on a product of two normal almost contact manifolds (c.f. [23]). The constructions of cases A and C produce a compact complexification of the original normal almost contact manifold , i.e. a compact complex manifold such that is a real submanifold of so that its CRstructure is compatible with the complex structure of and a holomorphic vector field on whose real part is the vector field of the CRaction on . Moreover we prove that given a compact Kähler manifold admitting a holomorphic vector field without zeros its complex structure can be recovered by the construction of case C. Therefore, every compact Kähler manifold admitting a nonvanishing holomorphic vector field is a compact complexification of a nacs. We also show that double suspensions of compact complex manifolds by two commuting automorphisms (see p.8), which can be obtained by means of the construction of case C, present a remarkable property. Namely, every compact Kähler manifold admitting a nonvanishing holomorphic vector field can be endowed with a complex structure on the underlying smooth manifold arbitrarily close to the original one which turns it into a double suspension.
In the third section we study criteria to determine when the complex manifolds of the above three families are Kählerian in terms of properties of the departing nacs. The common feature of all the complex manifolds that we construct is the existence of a holomorphic vector field without zeros. Using this fact we will show that for these complex structures to be Kählerian the Euler class of the nacs must be zero. When the flows associated to the nacs are isometric we prove that a compact complex manifold obtained by the constructions of cases A or B is Kähler if and only if the Euler class (or classes) is zero and the flows (or flows) is transversely Kählerian. For suspensions (case C) of a normal almost contact manifold by an automorphism we give a complete characterization when the CRstructure is Leviflat and acting on . Finally, we prove that a double suspension is Kählerian if and only if the departing manifold is Kählerian and the two automorphisms preserve a Kähler class.
Throughout the paper all manifolds are supposed to be smooth and connected and all differentiable objects to be of class .
I would like to thank my advisor, Marcel Nicolau, for having proposed me this problem, for having guided me during these years and for many useful comments regarding this paper. I am also grateful to Marco Brunella for suggesting the characterization for double suspensions and to Aziz El Kacimi for pointing out its validity in a more general context.
2. Compact complex structures defined from nacs
2.1. Normal almost contact structures (nacs)
Recall that a complex subbundle of dimension of the complexified tangent bundle of a manifold is called a CRstructure on M of dimension (cf. [16] or [3]) if:

;

is involutive, i.e. .
The complex bundle induces a real subbundle of . We define an endomorphism imposing that for every . Note that we can determine the CRstructure by giving . Setting we have a decomposition where and are the eigenspaces of (extended by complex linearity to ) of eigenvalue and respectively. We denote by the subset of of maps such that preserves and commutes with . Let be the flow induced by a smooth action on M. We say that defines a CRaction if for each . When we call the action transverse to the CRstructure if the smooth vector field is everywhere transverse to , i.e. has real dimension at every point.
Definition 2.1.1.
A normal almost contact structure (or nacs) on a manifold of odddimension is a pair where is a CRstructure of maximal dimension and a flow induced by a smooth action defining a transverse CRaction. Given a nacs we define its characteristic 1form by the conditions and (where and ) and its associated flow as the flow induced by , which is transversely holomorphic.
Alternatively a nacs can be determined by an endomorphism on the tangent space, a vector field and a 1form . The tercet is called an almost contact structure on if: (1) , (2) rank , (3) , (4) and for every tangent vector field on . There is an almost contact structure on given by . If has an almost contact structure then there is an almost complex structure on defined by is called normal if is integrable (cf. [1]). It is not difficult to see that the two definitions are equivalent. . The almost contact structure on
Recall that a form is called basic with respect to a foliation if for every vector field tangent to the leaves of .
Lemma 2.1.2.
Let be a transversely holomorphic flow on a compact manifold generated by a real vector field without zeros and let be a form such that . Set and let be the almostcomplex structure on induced by . Then is a nacs if and only if and the basic form is of type with respect to the holomorphic structure transverse to .
Proof.
Let the vectors in of type with respect to . The vector field preserves , i.e. if and only if . In this case defines a CRstructure, i.e. , if and only if is of type with respect to the complex structure transverse to . It is clear that then defines a transverse CRaction. ∎
Let be a compact manifold and a transversely holomorphic isometric flow on defined by a Killing vector field . If there exists a characteristic 1form (which verifies and ) such that is of type (1,1) then admits a nacs. Analogously a principal bundle over a compact complex manifold admits a nacs provided that we can choose a connection 1form such that its curvature form is of type . It is well known that a principal bundle admits such a connection form if and only if it is the unit bundle associated to a hermitian metric on a holomorphic line bundle. More generally, compact Seifert fibrations over a complex orbifold also provide examples of transversely holomorphic isometric flows, therefore they admit a nacs provided that there exists a suitable characteristic 1form. Notice that if the last condition of the lemma is always fulfilled since every 2form on a compact Riemann surface is of type .
It is also known that a compact connected Lie group of odd dimension greater than one always admits a non leftinvariant nacs (cf. [19]).
Let now be a CRstructure on and suppose that the distribution is a contact structure, i.e. the characteristic 1form verifies . Then is called a strictly pseudoconvex CRstructure on M. The couple of a strictly pseudoconvex CRstructure of maximal dimension and a transverse CRaction of on an odddimensional manifold is also known as a normal contact structure. For compact connected 3manifolds it is known that if they admit a normal contact structure the vector field defining the CRaction is Killing (c.f. [1]). The opposite situation to a strictly pseudoconvex CRstructure from the point of view of the real integrability of the distribution is Leviflatness, that is, the condition where is a 1form such that or equivalently where is the characteristic 1form of the nacs. In this case we can easily construct examples of nacs such that its associated flow is not isometric. Recall that the suspension of a compact manifold by is the compact manifold given by where . When is a compact complex manifold and the suspension carries a natural Leviflat CRstructure defined by and a transverse CRaction induced by . If we choose such that it is not an isometry for any metric on N, for instance and with such that , the flow generated by is clearly not isometric for any Riemannian metric on .
2.2. The Euler Class
We introduce here the notion of Euler class, which generalizes the classical notion of Euler class of an isometric flow.
Definition 2.2.1.
Let M be a compact manifold endowed with nonvanishing vector field and a 1form such that and . We denote by the flow induced by . We define the Euler class of the pair as the basic cohomology class given by
Recall that the basic cohomology is defined as the cohomology of the differential complex of basic forms for . Note that the class depends on the vector field but not on the 1form, provided that it verifies the above conditions. In particular we can consider the Euler class of a nacs. As the vanishing of the Euler class will be a necessary condition for the complex manifolds that we construct to be Kählerian we will discuss some criteria to determine when it is zero.
Lemma 2.2.2.
Let be a compact manifold endowed with nonvanishing vector field and a 1form such that and . We denote by the flow induced by . Then the following conditions are equivalent:

.

There exists a closed 1form on such that .

There exists a distribution transverse to of maximal dimension and invariant by the flow which is integrable.
Proof.
To prove note that if then there exists a basic 1form tal que . Then is a closed 1form such that . To see that it is enough to remark that the distribution is given by and since the form is invariant by the flow. To prove one defines the 1form imposing that it vanishes on the distribution and . As we have . ∎
Corollary 2.2.3.
Let be a compact manifold endowed with a nacs with Leviflat CRstructure. Then .
Proposition 2.2.4.
Let be a compact manifold endowed with a nacs. If then is a fiber bundle over . In particular and is not simply connected.
Proof.
The first statement is a consequence of a theorem by D. Tischler [27] that states that if is a compact manifold admitting a nonvanishing closed 1form then is a fibre bundle over . The second statement is an immediate consequence of the homotopy exact sequence associated to a fibration. ∎
Proposition 2.2.5.
Let be a compact manifold endowed with a nacs and the vector field inducing the CRaction. If there exists a contact form on such that and then .
The proof is analogous to the one of the corresponding statement for isometry flows due to Saralegui (cf. [25]).
Corollary 2.2.6.
Let be a compact manifold endowed with a normal contact structure. Then .
2.3. Geometrical constructions of complex structures
Proposition 2.3.1.
(Case A) Let and be two manifolds endowed with a nacs. There exists a 1parametric family of complex structures on the product for so that the complex manifold admits a nonvanishing holomorphic vector field .
Proof.
Let us denote by and the vector fields and the characteristic 1forms of the nacs on and respectively. The 2foliation on generated by and is transversely holomorphic and . We set a distribution given by and we define a complexvalued 1form . The 2form is basic with respect to and is of type with respect to the transverse holomorphic structure of . We define an almost complex structure on imposing that is compatible with the transverse holomorphic structure of and that is of type . Using NewlanderNirenberg theorem one can check that is integrable, i.e. a complex structure, if and only , which holds in our case. Moreover the vector field is of type , therefore it is holomorphic if and only if where denotes the subbundle of of vector fields of type with respect to , which holds as a consequence of the equality . ∎
Example 2.3.2.
Let be a compact connected real Lie group of odd dimension. Since admits a nacs the previous proposition describes a complex structure on the product . As is also a Lie group this can be seen as a particular case of a result by Samelson (cf. [24]) which states that every compact connected Lie group of even dimension admits a leftinvariant complex structure. Nevertheless one obtains more complex structures by this construction. Indeed, when we can assume that the nacs is noninvariant. Moreover, by topological reasons cannot be Kählerian except when is a real torus.
Proposition 2.3.3.
(Case B) Let be a manifold endowed with a nacs and its associated flow. Let be a principal bundle over with Chern class , where is a 1form on such that is the pullback of a closed form on basic for the flow . Then there exists a 1parametric family of complex structures on for so that the complex manifold admits a nonvanishing holomorphic vector field .
Proof.
Let be the characteristic 1form of the nacs and let be the vector field inducing the CRaction. We denote by the vector field on contained in such that and we define the 1form . Let denote the fundamental vector field of the action corresponding to the principal bundle such that . We apply now the same arguments as in proposition 2.3.1 using the vector fields and , the distribution and the transverse holomorphic structure for the flow induced by the CRstructure of . We define a complexvalued 1form by imposing , and . The holomorphic vector field is for . The hypothesis and imply that is of type , thus the complex structure is integrable. ∎
Definition 2.3.4.
Let be a compact manifold with a nacs . We define
Proposition 2.3.5.
(Case C) Let be a manifold endowed with a nacs . Given the suspension of by , i.e. where , admits a 1parametric family of complex structures for so that the complex manifold admits a nonvanishing holomorphic vector field induced by .
Proof.
The proof is straightforward using the same arguments as in the previous cases, the distribution is induced by the CRstructure on . ∎
Definition 2.3.6.
Let be a manifold endowed with a nacs, the vector field defining the CRaction and its associated flow. We say that the pair of a compact complex manifold and a nonsingular holomorphic vector field is a compact complexification of the pair if:

is a real submanifold of .

The CRstructure of is compatible with the complex structure of .

There exists such that .
Both the constructions of case A and case C produce complex manifolds that are compact complexifications of the departing normal almost contact manifolds. Indeed, if for then . and
Theorem 2.3.7.
Every compact Kähler manifold admitting a nonvanishing holomorphic vector field can be obtained as a suspension by of proposition 2.3.5 (case C). In particular it is a compact complexification of a compact manifold endowed with a nacs.
Proof.
Let be a compact Kähler manifold admitting a nonvanishing holomorphic vector field . By a result by CarellLieberman (c.f. [7]) there exists a holomorphic 1form over such that . As is compact we can assume and as is Kählerian we have . Assume . Let be closed paths giving a basis of modulus torsion and let be the dual basis of closed 1forms. Fix a basis of . By Hodge’s decomposition theorem we have
where is a differentiable function, for , and . By Stokes theorem the two sets of 1forms and have the same periods. In particular is a basis of dual of . Since is a dense subset in we can choose for arbitrarily small so that is a closed 1form and for . Moreover by construction the 1form is of the form:
with for . It follows that is constant and close to 1 by construction, set . In an analogous way is constant and close to 0, set . Therefore is finitely generated and it is contained in , thus . Fixing a base point the differentiable map
over the elliptic curve is well defined. Furthermore is a proper submersion and thus a fibration. The real vector fields and are transverse to the fibres of and preserve the fibration (because , are constants close to 1 and respectively and is closed). Since we can find a linear map such that and . Let be the induced fibration. The composition is a fibration over the circle. The fibres of , denoted by , admit a CRstructure induced by the complex structure on . There exists such that the real vector field is tangent to . As the flow associated to is holomorphic the vector field preserves the CRstructure of and induces a transverse CRaction. On the other hand there exists such that the vector field projects over the vector field on . The flow of preserves the CRstructure over and clearly . Finally setting we obtain , where . Taking the automorphism over induced by the flow of for time the compact complexification of case C for the preceding gives rise to the original complex structure. ∎
We have shown the suspension of a complex manifold by an automorphism admits a nacs. Applying the construction of case C to such a manifold is equivalent to consider the quotient of by the subgroup generated by and , where so that , which we will call double suspension. In this case there is a holomorphic fibration such that the vector field induced by is transverse to the fibers.
Assume that is a compact Kähler manifold admitting a holomorphic vector field without zeros and the flow induced by . In the previous theorem we have seen that as a consequence of CarrellLieberman theorem (c.f. [7]) there exists a closed holomorphic 1form on such that . Then the complex structure on is the only one compatible with the transverse holomorphic structure of such that is of type .
Theorem 2.3.8.
Let be a compact Kähler manifold admitting a holomorphic vector field without zeros and the flow induced by and let be a holomorphic 1form such that . Then there exists a closed 1form of type (0,1) arbitrarily small such that the complex structure on compatible with the transverse holomorphic structure of and such that is of type is induced by a double suspension. In particular every compact Kähler manifold admitting a holomorphic vector field without zeros admits a complex structure induced by a double suspension on the underlying smooth manifold arbitrarily close to the original one.
Proof.
We proceed as in theorem 2.3.7 to obtain from a closed 1form with group of periods and a smooth fibration given by . Every fibre of is transverse to the foliation generated by . Therefore admits a complex structure. Note that . Consider the universal covering and the pullback of the fibration by the map . There exists a map such that . The holomorphic vector field is transverse to the leaves of and it preserves the complex structure on . We recall that and that . Fix . We decompose , where and are real vector fields. Then and are transverse to the fibers of , they preserve the fibration and the complex structure on and (for is holomorphic). Finally, they project over and on respectively where . We set as the flows and for time and respectively. Thus is diffeomorphic to the double suspension where , and . Moreover the complex structure on induced by it which is arbitrarily close to the original one. Note that with the new complex structure on the fibration is holomorphic and the fibres are analytic submanifolds. As we can choose arbitrarily close to the starting holomorphic 1form we can conclude. ∎
There is a natural generalization to a suspension of a compact complex manifold by a commutative subgroup of . The resulting manifold has dimension and fibers over the torus . Then one can prove, with the same arguments as in theorem 2.3.8, the result below.
Theorem 2.3.9.
Let be a compact Kähler manifold such that admits an abelian subalgebra of holomorphic vector fields without zeros such that . The underlying smooth manifold admits a complex structure arbitrarily close to the original one, obtained as a suspension over the complex torus .
We denote by the Lie algebra of holomorphic vector fields on and by the Lie algebra of holomorphic vector field with zeros. Recall that . Therefore if and the hypothesis of the above theorem hold. The limit case, i.e. when , is a classical result by Wang’s:
Corollary 2.3.10.
Let be a complex parallisable compact Kähler manifold, then is a complex torus.
2.4. Nacs on compact 3manifolds
When M is a compact manifold of dimension 3 there is a classification due to H.Geiges of the manifolds admitting a nacs based on the classification of compact complex surfaces (see [12]). Using the classification of transversely holomorphic flows on a compact connected 3manifold (see [5] and [13]) together with the condition of the existence of a CRstructure and a transverse CRaction we give an alternative proof of the classification. The main interest of this point of view is that it determines not only the 3manifold but also the flow of the CRaction.
Proposition 2.4.1.
Let be a compact connected 3manifold endowed with a nacs. Then, up to diffeomorphism, the manifold and the vector field inducing the CRaction belong to the following list:

Seifert fibrations over a Riemann surface such that the isometric flow of the action admits a characteristic 1form such that is of type .

Linear vector fields in .

Flows on induced by a singularity of a holomorphic vector field in in the Poincaré domain and their finite quotients, i.e. flows on lens spaces.

Suspensions of a holomorphic automorphism of with a vector field tangent to the flow associated to the suspension.
Moreover, all the previous manifolds admit a nacs such that the CRaction is the one induced by the corresponding vector field.
Proof.
Since the last statement of the proposition is clear it is enough to depart from the classification of transversely holomorphic flows and rule out the two cases that do not admit a nacs: strong stable foliations associated to suspensions of hyperbolic diffeomorphisms of and where for such that with the flow induced by the vertical vector field . The first ones are are examples of nonisometric Riemannian flows (see [8]), if they admitted an invariant CRstructure together with a transverse CRaction they would be isometric (for the flow of a nacs admits a transverse invariant distribution). Let now be where for such that with the flow induced by the vertical vector field . Suppose that admits a CRstructure transverse to and a vector field tangent to inducing a CRaction. Then is a compact complex surface that admits a holomorphic nonvanishing vector field. Therefore it belongs to the following list (c.f. [9], [10]):

Complex tori.

Principal Seifert fibre bundles over a Riemann surface of genus with fiber an elliptic curve.

Ruled surfaces over an elliptic curve.

Almosthomogeneous Hopfsurfaces.
Since is homeomorphic to the complex surface must be a ruled surface over an elliptic curve. However we will see that this is a contradiction. Recall that the universal covering of a ruled surface over an elliptic curve is either or . By construction of the complex structure on we have that is a holomorphic coordinate for some , therefore the analytic universal covering of admits a holomorphic projection defined by with fiber an open subset of . As is compact by the maximum principle it is immersed in the fibers of , which is a contradiction. ∎
3. Kähler criteria
3.1. Conditions on the Euler class
Theorem 3.1.1.
Let be a compact complex manifold with a nonvanishing holomorphic vector field . For every let and be the two real vector fields defined by and , the flows defined by , respectively. If is Kählerian then .
Proof.
Since is a compact Kähler manifold with a holomorphic vector field without zeros by CarrellLiebermann’s theorem there exists a closed holomorphic 1form such that . We decompose where and are real closed 1forms. Using that and , a direct computation shows that for . Thus . ∎
Proposition 3.1.2.
Let be a compact manifold endowed with a nacs, let be the vector field of the CRaction and its associated flow. Assume that the compact complex manifold together with a holomorphic vector field are a compact complexification of . If is Kählerian then .
Proof.
We denote by the holomorphic vector field on and by the complex number such that on . Since is a compact Kähler manifold with a nonsingular vector field there exists a holomorphic closed 1form such that . We can decompose where is a real vector field. Set where are real 1forms on . Then and are closed and on . The closed real 1form verifies (because and ), therefore . ∎
Corollary 3.1.3.
Corollary 3.1.4.
Let be a compact manifold endowed with a nacs, let be the vector field of the CRaction and its associated flow. If admits a normal contact structure compatible with the CRaction induced by then admits no Kähler compact complexification.
It follows from corollary 2.2.6.
Corollary 3.1.5.
Let be a compact manifold endowed with a nacs. If , in particular if is simply connected, then admits no Kähler compact complexification.
It is a consequence of proposition 2.2.4.
Corollary 3.1.6.
Let be a compact connected semisimple real Lie group of odd dimension endowed with a nacs and the vector field of the CRaction. Then admits no Kähler compact complexification.
It is enough to recall that for any such group .
Proposition 3.1.7.
Let be a compact manifold endowed with a nacs and let be a compact complex manifold obtained as a principal bundle over by proposition 2.3.3 (case B). If is Kählerian then and the principal bundle is flat. In particular, if is Kählerian and has no torsion then the principal bundle is topologically trivial. Moreover, if is a connection 1form on such that then in .
Proof.
If is the holomorphic vector field of the complexification there exists a closed holomorphic 1form on such that . The connected group acts holomorphically on (as the group of the action of the principal bundle), therefore the forms and are invariant by the action of . Notice that where is the vector field of the action and, is the vector field contained in such that and . We decompose where are real 1forms. Then and are closed 1forms invariant by the action of (for they are a linear combination of , with constant coefficients) such that and (because and ). Since is a closed real basic invariant 1form it induces a closed 1form on such that , thus . Finally, is a closed connection 1form for the principal bundle , so it is flat. When has no torsion flat bundles are topologically trivial. Moreover, if is a connection 1form on such that then in . ∎
Finally for the suspension (case C) of a compact connected Lie group endowed with a nacs an elementary computation of the second cohomology group shows that the resulting complex manifold cannot be Kählerian:
Proposition 3.1.8.
Let be a nonabelian compact connected real Lie group of odd dimension endowed with a nacs, and endowed with the complex structure obtained as a suspension by proposition 2.3.5 (case C). Then is not Kählerian.
Proof.
The suspension admits a finite covering such that where is a compact connected semisimple real Lie group, and the lift of to . Since we conclude that and . Then using MayerVietoris sequence for the De Rham cohomology groups (c.f. [4]) one proves that is isomorphic to
It is not difficult to see that can not be Kählerian and it follows that is not Kählerian. ∎
3.2. Criteria for isometric flows
Theorem 3.2.1.
Let be a compact complex manifold with a nonvanishing holomorphic vector field . For every let and be the two real vector fields defined by and , the flows defined by , respectively. Assume that the flows and are isometric. Then the manifold is Kählerian if and only if and the real foliation is transversely Kählerian.
Proof.
The same argument as in theorem 3.1.1 shows that there are two closed real 1forms and on such that for . Then and are characteristic forms for and respectively and . We denote by and the 1parametric groups associated to and respectively and by the closure in of the abelian group generated by and . If is the Kähler form on then the transverse part with respect to of
where we integrate with respect to the Haar mesure on , is a transverse Kähler form.
As and are isometric we can assume that there exists an invariant transverse distribution of maximal dimension for the real foliation . We denote by and the 1forms on defined by and for . Since there exist such that for . We denote by the basic form . It follows that . We begin by showing that it is enough to find of type such that . Indeed, if exists the form is closed and of type . Adding to a positive multiple of the transverse Kähler form of we obtain a Kähler form on and the proof is complete. We will now show that such a form exists. Since we have , i.e. , and
so is a form which is exact as a basic form and closed. Applying the basic lemma (c.f. [11]) to we obtain a basic function such that
Then is a basic form of type such that . The form is basic, of type and so the conclusion follows. ∎
Every Riemannian holomorphic flow in a compact complex surface is transversely Kählerian. Therefore with the above hypothesis when the complex manifold is Kählerian if and only if
Corollary 3.2.2.
Let be a complex manifold obtained as a product by proposition 2.3.1 (case A) from two manifolds and endowed with a nacs such that the flows and in and respectively associated to the nacs are isometric. Then is Kählerian if and only if and the flows and are transversely Kählerian.
Corollary 3.2.3.
Let be a compact manifold endowed with a nacs such that its associated flow is isometric. Let be a complex manifold obtained as a principal bundle over by proposition 2.3.3 (case B). Then is Kähler if and only if the principal bundle is flat, the flow is transversely Kählerian on and .
Example 3.2.4.
Let be a compact complex surface obtained as a principal bundle over a 3manifold with a nacs by means of the construction of case B. From the last result together with the classification of compact complex surfaces one concludes the following. With the notation of section §2.4 for each of the possibilities for the corresponding surface is: i) an elliptic Seifert principal fibre bundle, ii) a complex torus or nonKählerian elliptic Seifert principal fibre bundle, iii) a Hopf surface and iv) either a Hopf surface or a ruled surface over an elliptic curve.
Proposition 3.2.5.
Let be a complex manifold obtained as a suspension by proposition 2.3.5 (case C) from a manifold endowed with a nacs such that its associated flow is isometric. If is Kählerian then the flow is transverselly Kählerian and there exists a basic Kähler form such that .
Proof.
We choose a Kähler form on and define as the pullback to . We define so that and is a closed form of type with respect to the holomorphic transverse structure. We denote by the 1parameter group on associated to and by the closure in