INDEPENDENCE AND NON-TRIVIALITY OF RIGIDSECONDARY COMPLEX CHARACTERISTIC CLASSES
TARO ASUKE
Abstract. Complex secondary characteristic classes for transversely holomorphic foliations are discussed. Examining a complexiﬁcation of an example of Baker [3], we will present a family of rigid, linearly independent
complex secondary classes. Such a family is ﬁrstly given by Hurder [8],from which ours is diﬀerent. The content of this article is based on [2], inwhich the details including proofs can be found. This is also based on atalk given at ‘Integrated Research on Complex Dynamics’ held at RIMS,
Kyoto University in 2012.

Introduction
Let BΓq be the classifying space for real codimension-q foliations. Given areal codimension-q foliation F of a manifold M , we can ﬁnd a classifying mapping from M to BΓq . Similarly, there exists the classifying space for transversely holomorphic foliations which is denoted by BΓqC , where q denotes thecomplex codimension. As in the case of vector bundles, cohomology classes
in H ∗ (BΓqC ) are characteristic classes for transversely holomorphic foliations,where coeﬃcients are chosen in C in this article. There is a dga (diﬀerential
graded algebra) denoted by WUq (see Section 1) and a homomorphism fromH ∗ (WUq ) to H ∗ (BΓqC ). The classes in the image other than (polynomials of)classical Chern classes are called complex secondary classes, and the homomorphism is called the characteristic homomorphism and denoted by χ. A
natural question is if χ is injective, or surjective. They are still open. In
this article we will focus on the injectivity. Suppose that the kernel of thecharacteristic homomorphism is non-trivial. Then, elements in the kernel arelinear relation among secondary classes. On the other hand, it is known thatsome of secondary classes admit continuous deformations. Namely, they can
Date: June 7, 2012.
2010 Mathematics Subject Classiﬁcation.Primary 58H10; Secondary 53C12, 58H15, 57R32.
Key words and phrases. characteristic classes, holomorphic foliations, rigid classes.
1vary continuously according to deformations of foliations. It is rare that ﬁxed
linear relations are kept under deformations so that the linear independenceof some secondary classes can be shown. This kind of arguments is however
not valid for those classes which are rigid under deformations (such classes arecalled rigid classes). In the case where complex normal bundles are trivial,
secondary characteristic classes are constructed by the classifying space BΓqC
and a dga WqC (see Section 1 for a brief explanation). Hurder studied themapping H ∗ (WqC ) → H ∗ (BΓqC ) [8], and obtained a family of rigid, linearly
independent classes. Hurder’s method is based on homotopies. We will discuss the case where complex normal bundles are not necessarily trivial. It will
be done by using an example of transversely holomorphic foliation which is acomplexiﬁcation of an example of Baker [3]. By evaluating secondary classesof the foliation, one can obtain a family of linearly independent classes in theimage of H ∗ (WUq ) in H ∗ (BΓqC ). Some of classes in the family are rigid (andsome others are variable), and are also products of secondary classes. These
classes form a family of rigid classes which is diﬀerent from Hurder’s one. The
evaluation is based on a result of Baker [3].
This article is an announcement of [2], in which the details including proofs
can be found.
Acknowledgements
This article is based on a talk given at ‘Integrated Research on Complex Dynamics’ held at RIMS, Kyoto University in 2012 and also on [2]. The author
expresses his gratitude to the organizers.
1. Preliminaries
First we introduce secondary characteristic classes.
Definition 1.1. Let C[v1 , . . . , vq ] be the polynomial ring generated by v1 , . . . , vq ,where the degree of vj is set to be 2j. We denote by Iq the ideal generated bymonomials of degree greater than 2q, and set Cq [v1 , . . . , vq ] = C[v1 , . . . , vq ]/Iq .

2

We also deﬁne Cq [v̄1 , . . . , v̄q ] by replacing vj by v̄j . We set∧
WUq = [eu1 , . . . , ueq ] ⊗ Cq [v1 , . . . , vq ] ⊗ Cq [v̄1 , . . . , v̄q ],∧Wq = [u1 , . . . , uq ] ⊗ Cq [v1 , . . . , vq ],∧WqC = [u1 , . . . , uq , ū1 , . . . , ūq ] ⊗ Cq [v1 , . . . , vq ] ⊗ Cq [v̄1 , . . . , v̄q ].
These algebras are equipped with derivatives such that deui = vi − v̄i , dui = vi ,dūi = v̄i and dvi = dv̄i = 0. The degree of uei , ui and ūi are set to be 2i − 1. We
deﬁne Wq by replacing ui and vj by ūi and v̄j in Wq . Then, WqC = Wq ∧ Wq .
It is easy to see that H ∗ (WqC ) is isomorphic to H ∗ (Wq ) ⊗ H ∗ (Wq ), and that
there is a natural inclusion from H ∗ (Wq ) to H ∗ (WqC ). There is also a natural
mapping from H ∗ (WUq ) to H ∗ (WqC ) which maps u
ei to ui − ūi , vi to vi and v̄ito v̄i , respectively. Indeed, this mapping corresponds to the natural mappingfrom BΓqC to BΓqC , which is a part of the homotopy ﬁbration BΓqC → BΓqC →BGL(q; C) and also is the classifying map of the ΓqC -structure of BΓqC , namely,
the map which forgets the triviality of the complex normal bundle.
The following result is classical.
Theorem 1.2. Let BΓqC be the classifying space of transversely holomorphic
foliations of complex codimension q, and let BΓqC the classifying space of transversely holomorphic foliations of complex codimension q with trivialized complex normal bundles. Then, there is a well-defined homomorphism H ∗ (WUq ) →H ∗ (BΓqC ) and H ∗ (WqC ) → H ∗ (BΓqC ).
The above homomorphisms are called the characteristic homomorphisms.
Let M be a manifold and F a transversely holomorphic foliation of M , ofcomplex codimension q. Then, the classifying map induces a homomorphismfrom H ∗ (WUq ) to H ∗ (M ). If the complex normal bundle of F is trivial, thenby ﬁxing the homotopy type of a trivialization, we obtain a homomorphismfrom H ∗ (WqC ) to H ∗ (M ). Elements of H ∗ (WUq ), H ∗ (WqC ) and H ∗ (Wq ) or
their image under the characteristic homomorphisms which involve uei , ui or ūiare called complex secondary characteristic classes.
Let ρ be a mapping from WUq+1 to WUq such that if i ≤ q then ρ(eui ) = uei ,ρ(vi ) = vi , ρ(v̄i ) = v̄i and if i = q + 1 then ρ(eui ) = 0, ρ(vi ) = 0 and ρ(v̄i ) = 0,
respectively. Then, ρ induces a homomorphism ρ∗ : H ∗ (WUq+1 ) → H ∗ (WUq ).
It is known that classes in the image of ρ∗ are rigid under deformations of3

foliations. Such classes are called rigid classes. Similar homomorphisms fromC
Wq+1 to Wq and from Wq+1
to WqC are also deﬁned. The classes in the image
are also rigid under deformations, and called rigid classes. On the other hand,a class ω ∈ H ∗ (WUq ) is said to be variable if ω varies continuously with respect
to a family of foliations.
There are some signiﬁcant classes.
Definition 1.3.
1) The class u1 v1q ∈ H 2q+1 (Wq ) or its image in H 2q+1 (WqC )is called the Bott class.
2) The class ue1 (v1q +v1q−1 v̄1 +· · ·+v̄1q ) ∈ H 2q+1 (WUq ) is called the imaginary
part of the Bott class.
√3) The class −1ue1 v1q v̄1q ∈ H 4q+1 (WUq ) is called the Godbillon-Vey class.
These classes are known to be non-trivial. We have the following.
Theorem 1.4.
1) The image of the imaginary part of the Bott class in∗C
H (Wq ) is equal to u1 v1q − ū1 v̄1q .
2) The image of the Godbillon-Vey class in the sense of Definition 1.3 under
the characteristic homomorphism coincides with the classical GodbillonVey class (for real foliations). Moreover, the Godbillon-Vey class belongs
to the image of ρ∗ : H ∗ (WUq+1 ) → H ∗ (WUq ) so that it is a rigid class.
Remark 1.5. It is known that real secondary classes for transversely holomorphic foliations are complex secondary classes deﬁned in terms of H ∗ (WUq ).
It is in particular the case for the Godbillon-Vey class which is the mostsigniﬁcant real secondary class. The second claim of Theorem 1.4 implies
that the image of the classical Godbillon-Vey class in H 4q+1 (WUq ) is equal to√−1ue1 v1q v̄1q . It is well-known that the Godbillon-Vey class admits continuous
deformations in the category of real foliations [11], [12], nevertheless, it is rigid
in the category of transversely holomorphic foliations (see Theorem 2.3).
2. Independent classes of H ∗ (WUq )We begin with a classical example.
Example 2.1 ([4]). Let X be a holomorphic vector ﬁeld on Cn+1 deﬁned byn∑∂X(z) =
λi zi
,
∂z
i
i=0
4

where each λi is a non-zero complex number. Then, the integral curves of X
form a holomorphic foliation of Cn+1 \ {0}. This foliation is invariant underhomothecies so that a holomorphic foliation Fλ is induced on (Cn+1 \ {0})/ ×2∼= S 1 × S 2n+1 . It is well-known that the complex normal bundle of Fλ is trivv1 vJ (λ0 , . . . , λn )ial, and u1 vJ (Fλ ) =volS 2n+1 , where vJ is a Chern polynomial
λ0 · · · λnof degree n [4] (see also [7]). It follows that the classes u1 vJ are variable andlinearly independent.
We study the following example, which is a complexiﬁcation of Baker’s example [3, Example 1(a)].
Example 2.2. Let G = SL(k + n; C), where n > k > 0 or n = k = 1, H ={(aij ) ∈ G | aij = 0 if i > k and j ≤ k}, K = S(U(k) × U(n)) and T = T k+n−1
the maximal torus in G realized as diagonal matrices. Then, the left cosets of
H induce transversely holomorphic foliations of complex codimension kn on
Γ \G/T and Γ \G/K, where Γ is a discrete subgroup of G such that Γ \G/K
is a closed manifold.
Secondary characteristic classes of this kind of foliations, so-called locallyhomogeneous foliations, are well-studied, e.g. [9], [3], [1]. In particular, if
n = k = 1, then it is a complexiﬁcation of an example of Roussarie [6]. If
k = 1, then the Godbillon-Vey class is studied in [1]. The case where n = 3
and k = 2 is studied by Enatsu [5].
We have a complexiﬁcation of [3, Theorem 5.3] as follows. If I = {i1 , . . . , il }then we set u
eI = uei1 · · · ueil . We denote ueI also by ui1 ,...,il . If I = ∅ then we setu
eI = 1 and regard i1 = +∞.
Theorem 2.3 ([2, Theorem 2.2]). Let I = {i1 , . . . , il } and suppose that k <i1 < · · · < il ≤ n. The classes of the form u
e1,...,k ueI v1kn v̄1kn are non-trivial andlinearly independent in H ∗ (Γ \SL(k +n; C)/T ). These classes are rigid classes,
namely, rigid under deformations of foliations if k = 1 and i1 > 2.
The last claim can be seen as follows. The cochain u
e1,...,k ueI (v1kn+1 v̄1kn−1 +e1,...,k ueI v1kn v̄1kn under ρ : WUq+1 →v1kn v̄1kn +v1kn−1 v̄1kn+1 ) is mapped to the cocycle u
WUq , and is closed if k = 1 and i1 > 2.
Let n = 3 and k = 2 in Example 2.2. Then, we obtain a transversely holou1 (v15 +v14 v̄1 +morphic foliation of complex codimension 6. We set η2,6 = ue2 v16 +e
5· · · + v̄15 )v̄2 . Then ξ6 and η2,6 are non-trivial, because we have ξ6 η2,6 = ue1 ue2 v16 v̄16
and it is non-trivial by Theorem 2.3. Hence we have a non-trivial class which
is a product of secondary classes. On the other hand, if we set µ = ue1 ue2 (v16 v̄15 +v15 v̄16 ), then µ is closed and µv1 = ue1 ue2 v16 v̄16 . Hence µ and v1 are also non-trivial.
Actually v16 is a generator of H 12 (SU(5)/S(U(2)×U(3))). On the other hand, by
examining the degree of generators of H ∗ (SU(5)) ⊗ H ∗ (SU(5)/S(U(2) × U(3))),
we see that the Godbillon-Vey class ue1 v16 v̄16 is trivial. We have u
e1 v16 v̄16 = ξ6 v16 .
Hence, unlike the above cases, a product of non-trivial classes yields a trivial class.
We remark ﬁnally that using a result of Kamber-Tondeur [10], one can show
an analogue of Theorem 2.3 [2, Theorem 2.3].
References[1] T. Asuke, Godbillon-Vey class of transversely holomorphic foliations, MSJ Memoirs, 24,
Mathematical Society of Japan, Tokyo, 2010.
[2], On independent rigid classes in H ∗ (WUq ), to appear in Illinois J. Math.
[3] D. Baker, On a class of foliations and the evaluation of their characteristic classes,
Comment. Math. Helv. 53 (1978), 334–363.
[4] P. Baum and R. Bott, Singularities of holomorphic foliations, J. Diﬀerential Geom. 7(1972), 279–342.
[5] K. Enatsu, A transversely holomorphic foliation whose complex secondary characteristicclasses are non-trivial, master thesis, the University of Tokyo, Tokyo, Japan, 2011.
[6] C. Godbillon and J. Vey, Un invariant des feuilletages de codimension 1, C. R. Acad.
Sci. Paris Sér. A–B 273 (1971), A92–A95.
[7] J. Heitch, Independent variation of secondary classes, Ann. Math. 108 (1978), 421–460.
[8] S. Hurder, Independent rigid secondary classes for holomorphic foliations, Invent. Math.
66 (1982), 313–323.
[9] F. W. Kamber and P. Tondeur, Foliated Bundles and Characteristic Classes, Lecture
Notes in Mathematics, No. 493, Springer-Verlag, 1975.
[10], On the linear independence of certain cohomology classes of BΓq , Studies inalgebraic topology, Adv. in Math. Suppl. Stud., vol. 5, Academic Press, New YorkLondon, 1979, pp. 213–263.
[11] W. P. Thurston, Noncobordant foliations of S 3 , Bull. Amer. Math. Soc. 78 (1972),
511–514.
[12], Foliations and groups of diﬀeomorphisms, Bull. Amer. Math. Soc. 80 (1974),
304–307.
Graduate School of Mathematical Sciences, University of Tokyo, 3–8–1,
Komaba, Meguro-ku, Tokyo 153–8914, Japan
E-mail address: asuke@ms.u-tokyo.ac.jp6