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L’INSTITUT FOURIER
Judith BRINKSCHULTE & C. Denson HILL
Non locally trivializable CR line bundles over compact Lorentzian CR
manifolds
Tome 68, no 1 (2018), p. 101-108.
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Ann. Inst. Fourier, Grenoble
68, 1 (2018) 101-108
NON LOCALLY TRIVIALIZABLE CR LINE BUNDLES
OVER COMPACT LORENTZIAN CR MANIFOLDS
by Judith BRINKSCHULTE & C. Denson HILL (*)
Abstract. — We consider compact CR manifolds of arbitrary CR codimen-
sion that satisfy certain geometric conditions in terms of their Levi form. Over
these compact CR manifolds, we construct a deformation of the trivial CR line
bundle over M which is topologically trivial over M but fails to be even locally CR
trivializable over any open subset of M . In particular, our results apply to compact
Lorentzian CR manifolds of hypersurface type.
Résumé. — On considère une variété CR compacte de codimension CR quel-
conque qui vérifie certaines conditions géométriques en terme de sa forme de Levi.
Sur ces variétés CR compactes, on construit une déformation du fibré en droites CR
trivial sur M qui est topologiquement trivial sur M mais qui n’admet même pas de
trivialization CR locale sur un ouvert arbitraire de M . En particulier, nos résultats
s’appliquent au cas de variétés CR compactes Lorentziennes du type hypersurface.
1. Introduction
In geometry, the concept of vector bundles is quite important. In both
categories: smooth real vector bundles over differentiable real manifolds,
and holomorphic complex vector bundles over complex manifolds, there
is a theorem which says that any such vector bundle always has a local
trivialization. In the first situation, the transition functions, or matrices,
are C∞ smooth, and in the second situation, they are holomorphic. For this
reason one usually ignores the intrinsic definition of a vector bundle and
works directly with a local trivialization. It is a somewhat suprising fact,
which we address here, that in the category of smooth CR vector bundles
over a smooth CR manifold, the analogous theorem does not always hold.
Keywords: CR vector bundles, local frames, Lorentzian CR manifolds.
2010 Mathematics Subject Classification: 32V05, 32G07.
(*) The first author was supported by Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation, grant BR 3363/2-1).
102 Judith BRINKSCHULTE & C. Denson HILL
Thus one can have a perfectly good intrinsically defined CR vector bundle
for which there might not be the corresponding CR transition functions,
or matrices.
Our main results are as follows:
Theorem 1.1. — Let M be a compact (abstract) Lorentzian CR hy-
persurface of CR dimension n, n 6= 3. Then there exists a family (La)a>0
of complex CR line bundles La −→ M converging to the trivial CR line
bundle L0 = M × C, as a tends to 0, such that La is differentiably trivial
over M , but La is not locally CR trivializable over any open set U of M
for a > 0.
An easy example satisfying the hypothesis of the above theorem is Pen-
rose’s null twistor space, which is the 5-dimensionalM of type (2,1) in CP3
given in homogeneous coordinates by
|z0|2 + |z1|2 = |z2|2 + |z3|3 .
For CR manifolds of CR dimension n = 2 but arbitrary codimension,
we obtain the following result:
Theorem 1.2. — Let M be a compact (abstract) CR manifold of type
(2, k) which is 1-pseudoconcave. Then there exists a family (La)a>0 of com-
plex CR line bundles La −→ M converging to the trivial CR line bundle
L0 = M ×C, as a tends to 0, such that La is differentiably trivial over M ,
but La is not locally CR trivializable over any open set U of M for a > 0.
An example of a CR manifold of type (2, 2) satisfying the hypothesis
of the above theorem is the twistor space of the Fubini-Study metric on
CP2, which is the 6-dimensional M ⊂ CP2 × CP2 defined by the complex
equation
z0w0 + z1w1 + z2w2 = 0 ,
where (z0 : z1 : z2) are homogeneous coordinates in the first factor, and
(w0 : w1 : w2) are homogeneous coordinates in the second factor.
More examples of CR submanifolds satisfying the hypothesis of this the-
orem can be found in Section 4 of [6].
In both theorems the same results apply in the setting of CR vector
bundles of arbitrary rank, because it suffices to deform the vector bundle
in only one fiber direction.
We would like to point out that results in the opposite direction have
been obtained by Webster [8]. Namely he proved that any CR vector bundle
over a strictly pseudoconvex hypersurface of CR dimension n > 7 is locally
trivializable.
ANNALES DE L’INSTITUT FOURIER
CR LINE BUNDLES 103
2. Definitions
Throughout this paper an abstract CR manifold of type (n, k) is a triple
(M,HM,J), where M is a smooth real manifold of dimension 2n+k, HM
is a subbundle of rank 2n of the tangent bundle TM , and J : HM → HM
is a smooth fiber preserving bundle isomorphism with J2 = −Id. We also
require that J be formally integrable; i.e. that we have
[T 0,1M,T 0,1M ] ⊂ T 0,1M where
T 0,1M = {X + iJX | X ∈ Γ(M,HM)} ⊂ Γ(M,CTM),
with Γ denoting smooth sections. The CR dimension of M is n > 1 and
the CR codimension is k > 1.
As usual, we denote by ∂M the tangential Cauchy–Riemann operator on
M . For the precise definition, we refer the reader to [6].
We denote by HoM = {ξ ∈ T ∗M | 〈X, ξ〉 = 0,∀X ∈ Hπ(ξ)M} the
characteristic conormal bundle of M . Here π : TM −→ M is the natural
projection. To each ξ ∈ HopM , we associate the Levi form at ξ :
Lp(ξ,X) = ξ([JX̃, X̃]) = dξ̃(X,JX) for X ∈ HpM
which is Hermitian for the complex structure of HpM defined by J . Here
ξ̃ is a section of HoM extending ξ and X̃ a section of HM extending X.
A Lorentzian hypersurface is a CR manifold M of type (n, 1) such that
its Levi form has one negative and n− 1 positive eigenvalues.
M is called 1-pseudoconcave if for every p ∈M and every characteristic
conormal direction ξ the Levi form Lp(ξ, · ) has at least one positive and
one negative eigenvalue.
For the definition of CR vector bundles, we follow [7]. Namely, a complex
CR vector bundle of rank r over an abstract CR manifoldM of type (n, k)
is a smooth complex vector bundle E π−→M of rank r such that
(i) E has the structure of a smooth abstract CR manifold of type
(n+ r, k),
(ii) π : E −→M is a CR submersion,
(iii) E ⊕ E 3 (ξ1, ξ2) → ξ1 + ξ2 ∈ E and C × E 3 (λ, ξ) → λξ ∈ E are
CR maps.
Let E π−→ M be a complex CR vector bundle of rank r. A CR trivi-
alization of E over an open set U of M is a CR equivalence between E|U
and the trivial bundle U × Cr. We say that E is locally trivializable over
M iff each point of M has an open neighborhood U such that E|U is CR
equivalent to U × Cr.
TOME 68 (2018), FASCICULE 1
104 Judith BRINKSCHULTE & C. Denson HILL
When E is locally CR trivializable, one can arrange an open covering
{Uα} of M , and obtain transition functions, or matrices, {gαβ} which are
CR. Hence in that case the situation is analogous to the case of holomorphic
vector bundles over complex manifolds.
Note that if E is locally CR embeddable, then E is also locally CR trivi-
alizable, by the implicit function theorem. But the local CR embeddability
of the base M does not suffice to make E locally CR trivializable, which is
the main point of this paper.
Another equivalent definition of CR vector bundles, which is used in [8],
can be given in terms of a connection ∂E on the complex vector bundle E;
that is, we have a linear differential operator of order one,
∂E : C∞0,q(M,E) −→ C∞0,q+1(M,E)
satisfying the Leibniz rule and ∂E ◦ ∂E = 0. E is locally trivializable over
U ⊂ M iff there exists a nonvanishing section s of E over U satisfying
∂Es = 0. For more details, we refer the reader to [7].
3. Construction of a nontrivial global cohomology class
The key point in the construction of the CR line bundles is the existence
of a global ∂M -closed (0, 1) form on M which is not ∂M -exact on any open
subset ofM . First we consider a single point p ∈M , and show the existence
of a global ∂M -closed form which is not ∂M -exact on any neighborhood of
that point. The existence of such a form follows by similar arguments as
in [2] and [3]. Therefore, in the proof of the following Theorem, we sketch
only the most important ingredients.
Theorem 3.1. — Let M be a compact (abstract) CR manifold of type
(n, k). Assume that H0,2(M) is Hausdorff and that there exists a point p
on M at which there exists a characteristic conormal direction ξ such that
Lp(ξ, · ) has one negative and n− 1 positive eigenvalues. Then there exists
a smooth (0, 1)-form f on M satisfying ∂Mf = 0 on M such that f is not
∂M -exact on any neighborhood of p in M .
Proof. — By contradiction, we assume that any smooth (0, 1)-form f on
M satisfying ∂Mf = 0 is ∂M -exact on some neighborhood Uf of p in M .
Using the functional analytic arguments of [1] this implies that there exists
an open neighborhood U of p inM , independent of f , such that any smooth
(0, 1)-form f on M satisfying ∂Mf = 0 on M is ∂M -exact on U . Moreover,
by the open mapping theorem for Fréchet spaces, we also get an a priori
ANNALES DE L’INSTITUT FOURIER
CR LINE BUNDLES 105
estimate of the following form: For a fixed compact K ⊂ U there exists an
integer m1 and a constant C1 > 0 such that the solution u to ∂Mu = f on
U can be chosen to satisfy
‖u‖K,0 6 C1‖f‖m1 .
Here ‖ · ‖m denotes the usual Cm norm on C∞·,· (M), and ‖ · ‖K,0 denotes the
usual C0 norm on K.
But this implies that we have
(3.1)
∣∣∣∣∫
K
f ∧ g
∣∣∣∣ . ‖f‖m1 · ‖∂Mg‖K,0
for all f ∈ C∞0,1(M) ∩Ker∂M and g ∈ C∞n+k,n−1(M) with supp g ⊂ K.
Using the geometric assumption on M at p, namely that there exists
a characteristic conormal direction ξ such that Lp(ξ, · ) has one negative
and n − 1 positive eigenvalues, one can construct forms fλ ∈ C∞0,1(M),
gλ ∈ C∞n+k,n−1(M) with support inK such that ∂Mfλ and ∂Mgλ are rapidly
decreasing with respect to λ in the topology of C∞·,· (M) as λ→∞. On the
other hand we have
(3.2)
∣∣∣∣∫
K
fλ ∧ gλ
∣∣∣∣ > cλ−n− k2
for some constant c > 0. For the details of the construction of these forms,
we refer the reader to Section 5 of [3].
By assumption H0,2(M) is Hausdorff. But this implies that we can solve
the equation ∂Muλ = ∂Mfλ with an estimate
(3.3) ‖uλ‖m1 6 C2‖∂Mfλ‖m2 ,
where C2 is a positive constant and m2 is an integer. Hence ‖uλ‖m1 is
rapidly decreasing with respect to λ. Defining f̃λ = fλ − uλ, we obtain a
global smooth ∂M -closed (0, 1)-form on M .
To get a contradiction to our assumption, we use the estimate (3.1) with
f̃λ and gλ. Namely, (3.1) implies that
∫
M
f̃λ ∧ gλ is rapidly decreasing with
respect to λ, whereas (3.2) implies that |
∫
M
f̃λ∧gλ| is bounded from below
by a polynomial in 1/λ. For the details of these estimates, we again refer
the reader to [3]. This contradiction proves the Theorem. 
Theorem 3.2. — Let M be a compact (abstract) CR manifold of type
(n, k). Assume that H0,2(M) is Hausdorff and that for each p ∈ M there
exists a characteristic conormal direction ξ such that Lp(ξ, · ) has one neg-
ative and n−1 positive eigenvalues. Then there exists a smooth (0, 1)-form
ω on M satisfying ∂Mω = 0 on M such that ω is not ∂M -exact on any
neighborhood of any point p ∈M .
TOME 68 (2018), FASCICULE 1
106 Judith BRINKSCHULTE & C. Denson HILL
Proof. — We choose a countable dense set of points on M , say S =
(pj)j∈N. Assume by contradiction that for every ∂M -closed form f on M
there exists j ∈ N such that f is ∂M -exact on a neighborhood Uj of pj .
For fixed j, let (U jm)m∈N denote a fundamental sequence of open neigh-
borhoods of pj in M . To abbreviate notations, we set Z(M) = C∞0,1(M) ∩
Ker∂M . Now for each m, j ∈ N we define
Gjm = {(f, uj) ∈ Z(M)× C∞(U jm) | f = ∂Muj on U jm} .
This, as a closed subspace of Z(M) × C∞(U jm), is also a Fréchet space.
Let πjm : Gjm −→ Z(M) be the natural projection. Our assumption implies
that
Z(M) =
⋃
m,j∈N
πjm(Gjm) .
By Baire’s category theorem one of the spaces, πs` (Gs`), must be of second
category. Then, by the Banach open mapping theorem, the linear continu-
ous map
πs` : Gs` −→ Z(M)
is surjective. But this contradicts Theorem 3.1. 
4. Construction of CR line bundles
Theorem 4.1. — Let M be an (abstract) CR manifold of type (n, k),
and assume that there exists a smooth (0, 1)-form ω onM satisfying ∂Mω =
0 on M such that ω is not ∂M -exact on any neighborhood of any point
p ∈ M . Then there exists a family (La)a>0 of complex CR line bundles
La −→M converging to the trivial CR line bundle L0 = M×C, as a tends
to 0, such that La is differentiably trivial over M , but La is not locally CR
trivializable over any open set U of M for a > 0.
Proof. — The arguments in this section follow [5].
On the differentiably trivial complex line bundle La = M × Cz, we con-
sider the CR structure whose T 0,1La is defined as follows: Let U be an
open set of M such that T 0,1M is spanned over U by L1, . . . , Ln. We de-
fine T 0,1La to be spanned over U × Cz by the basis
(4.1)

X0 =
∂
∂z
Xj = Lj + aω(Lj)
∂
∂z
, j = 1, . . . , n
This gives well defined CR structure on La . The CR line bundle La is
differentiably trivial over M and converges to the trivial CR line bundle
ANNALES DE L’INSTITUT FOURIER
CR LINE BUNDLES 107
over M as a tends to zero. The associated connection of La is defined
as follows: Since La is smoothly trivial, every s ∈ C∞0,q(M,La) is globally
defined by a form σ ∈ C∞0,q(M) and a smooth frame e ∈ C∞(M,C), s = σ⊗e.
Then ∂Las = (∂Mσ + ω ∧ σ)⊗ e. Since ∂Mω = 0, this connection satisfies
the integrability condition ∂La ◦ ∂La = 0.
However, for a 6= 0, a local CR trivialization of La near a point of M
forces the existence of a local smooth solution u of ∂Mu = ω.
Indeed, a local nonvanishing ∂La -closed section means that we have a
local nonvanishing smooth function σ on some nonempty open subset ofM
satisfying ∂Mσ+ωσ = 0. After shrinking U , we can assume that u = − log σ
is well defined on U . But u satisfies ∂Mu = −∂Mσσ = ω, which contradicts
the assumption on ω. Therefore for a 6= 0, La is not locally CR trivializable
over any open subset of M . 
5. Proof of the main theorems
Proof of Theorem 1.1. — Since M is a Lorentzian hypersurface, it sat-
isfies the classical condition Y (q) for q 6= 1, n−1. If n 6= 3, then M satisfies
in particular the condition Y (2). Hence the ∂M -complex is 12 -subelliptic in
degree (0, 2) (see [4]). It follows that H0,2(M) is finite dimensional, thus
Hausdorff. But then the statement of the theorem follows by combining
Theorems 3.2 and 4.1. 
Proof of Theorem 1.2. — SinceM is 1-pseudoconcave, it follows from the
ε-subelliptic estimates proved in [6] that the top-degree cohomology group
H0,n(M) is finite dimensional. But since n = 2 we thus have that H0,2(M)
is Hausdorff. Therefore we may again conclude by applying Theorems 3.2
and 4.1. 
Remark. — We conjecture that Theorem 1.1 also holds for n = 3. In-
deed, if we start with a CR embedded hypersurface, then we can find many
∂M -closed (0, 1)-forms defined on a neighborhood of a given point which
are not ∂M -exact on any neighborhood of that point (see [1]). However
it seems to be an open problem to show that H0,2(M) is Hausdorff for a
Lorentzian hypersurface of CR dimension 3. Therefore we cannot construct
a global ∂M -closed (0, 1) form on M which is not ∂M -exact on M .
BIBLIOGRAPHY
[1] A. Andreotti, G. Fredricks & M. Nacinovich, “On the absence of Poincaré
lemma in tangential Cauchy-Riemann complexes”, Ann. Sc. Norm. Super. Pisa, Cl.
Sci. 8 (1981), p. 365-404.
TOME 68 (2018), FASCICULE 1
108 Judith BRINKSCHULTE & C. Denson HILL
[2] J. Brinkschulte & C. D. Hill, “Obstructions to finite dimensional cohomology
of abstract Cauchy-Riemann complexes”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 15
(2016), p. 343-354.
[3] J. Brinkschulte, C. D. Hill & M. Nacinovich, “On the nonvanishing of abstract
Cauchy–Riemann cohomology groups”, Math. Ann. 363 (2016), no. 3-4, p. 1701-
1715.
[4] G. B. Folland & J. J. Kohn, The Neumann problem for the Cauchy-Riemann
complex, Annals of Mathematics Studies, vol. 57, Princeton University Press, 1972,
viii+146 pages.
[5] C. D. Hill, “Counterexamples to Newlander-Nirenberg up to the boundary”, in
Several complex variables and complex geometry, Proceedings of Symposia in Pure
Mathematics, vol. 52, American Mathematical Society, 1991, p. 191-197.
[6] C. D. Hill & M. Nacinovich, “Pseudoconcave CR manifolds”, in Complex analysis
and geometry, Lecture Notes in Pure and Applied Mathematics, vol. 173, Marcel
Dekker, 1996, p. 275-297.
[7] ———, “A weak pseudoconcavity condition for abstract almost CR manifolds”,
Invent. Math. 142 (2000), no. 2, p. 251-283.
[8] S. M. Webster, “The integrability problem for CR vector bundles”, in Several com-
plex variables and complex geometry, Proceedings of Symposia in Pure Mathematics,
vol. 52, American Mathematical Society, 1991, p. 355-368.
Manuscrit reçu le 30 juin 2016,
accepté le 15 juin 2017.
Judith BRINKSCHULTE
Universität Leipzig
Mathematisches Institut
Augustusplatz 10
D-04109 Leipzig (Germany)
brinkschulte@math.uni-leipzig.de
C. Denson HILL
Stony Brook University
Department of Mathematics
Stony Brook NY 11794 (USA)
dhill@math.stonybrook.edu
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