Compatible Almost Complex Structures on

ESI
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Institute for Mathematical Physics
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A-1090 Wien, Austria
Compatible Almost Complex Structures on
Quaternion–Kähler Manifolds
D.V. Alekseevsky
S. Marchiafava
M. Pontecorvo
Vienna, Preprint ESI 419 (1997)
Supported by Federal Ministry of Science and Research, Austria
Available via anonymous ftp or gopher from FTP.ESI.AC.AT
or via WWW, URL: http://www.esi.ac.at
January 10, 1997
COMPATIBLE ALMOST COMPLEX STRUCTURES
ON QUATERNION-KÄHLER MANIFOLDS
D.V. ALEKSEEVSKY1 - S. MARCHIAFAVA2 - M. PONTECORVO3
Sophus Lie Center (Moscow)1 - Universita’ di Roma I2 - Università di Roma Tre3
Abstract. Let (M 4n , g, Q) be a quaternion-Kähler manifold with reduced scalar curvature ν = K/4n(n+2). Suppose J is an almost complex structure which is compatible
with the quaternionic structure Q and let θ = −δ ◦ J be the Lee form of J. We prove
the following local results: 1) if J is conformally symplectic then it is parallel and
ν = 0; 2) if J is cosymplectic then ν ≤ 0 and ν = 0 if and only if J is parallel; 3) if J
is integrable then dθ is Q-Hermitian and harmonic; 4) if J is conformally balanced but
not symplectic then there exists an associated non-zero Killing vector field. We prove
also that any closed self-dual 2-form ω ∈ Λ2+ = g ◦ Q ⊂ Λ2 is parallel.
When M 4n is compact our main global results are the following: 1) if ν > 0 and
(M 4n , g) is different from the Grassmannian G2 (Cm ) of 2-planes of Cm then there
exists no compatible almost complex structure J; 2) if the first Chern class c1 (J) ≡
c1 (T M ) = 0 for the almost complex structure J on the tangent bundle T M then
ν = 0; 3) if ν 6= 0 then there exists no compatible complex structure J; 4) if ν = 0 a
compatible complex structure J is parallel. The last two results have been proved in
[P2] by twistor methods.
In §5 and §6 we compute the Laplacian and the (real and quaternionic) Weyl tensors
acting on self-dual 2-forms, by relating to the work of Gauduchon in [G1].
1. Introduction and main results
In a previous paper ([AMP]) we studied several problems concerning almost complex structures which are defined on an almost quaternionic manifold (M 4n , Q) and
which are compatible with the almost quaternionic structure Q. We recall that an
almost quaternionic structure Q on M 4n is a rank-3 subbundle Q ⊂ End(T M ) which
is locally spanned by almost hypercomplex structures H = (Jα ); such a locally defined
triple H = (Jα ), where Jα2 = −id and J1 J2 = −J2 J1 = J3 , is called an admissible
basis of Q. An almost
complex structure J on M 4n is compatible with Q if it can be
P
written as J = α cα Jα with respect to any admissible basis H = (Jα ) for suitable
1991 Mathematics Subject Classification. 53C10 - 32C10.
Work done under the program of G.N.S.A.G.A. of C.N.R. and partially supported by M.U.R.S.T
(Italy) and E.S.I. (Vienna).
1
Typeset by AMS-TEX
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
functions cα . By referring to the twistor fibration t : Z −→ M 4n one can say equivalently that J is a section of the twistor space Z, J : M −→ Z, see [S3, pag. 130] and
[AMP].
In this work we will consider the stronger notion of a quaternion-Kähler manifold
(M 4n , g, Q) . This means that g is a Riemannian metric on M which is Q-Hermitian
(i.e. any endomorphism of Q is g-skew-symmetric) and, moreover, that the LeviCivita connection of g preserves Q. This is equivalent to the statement that the
holonomy group of g is contained in Sp(n) · Sp(1). Notice that every oriented Riemannian 4-manifold satisfies the above conditions. For this reason we will assume that
the quaternionic dimension n ≥ 2 and we will systematically treat the 4-dimensional
case in separate statements.
For the basic properties of quaternion-Kähler manifolds we will refer to [S1] and
[S2]. Let us just recall here that any quaternion-Kähler manifold (M 4n , g, Q) with
n ≥ 2 is automatically Einstein. When the scalar curvature K of g is positive
a complete quaternion-Kähler manifold is always compact and simply connected;
the only known examples are the symmetric spaces of Wolf [W]. When K = 0 the
bundle Q is locally parallelizable so that the universal covering of M is a hyperKähler manifold.
We will assume that (M 4n , g, Q) , n ≥ 2, is a quaternion-Kähler manifold and
discuss various properties of a compatible almost complex structure J, under several
assumptions (see definitions 2.2) both from local and global point of view.
Our main results about existence of local compatible almost complex structures
can be summarized as follows.
Let (M 4n , g, Q) be a quaternion-Kähler manifold with reduced scalar curvature ν
and J an almost complex structure which is compatible with the quaternionic structure
Q on M 4n . Let F = g ◦ J, θ = −δF ◦ J be the Kähler form and , respectively, the
Lee form of J. Then:
1) if J is locally conformally symplectic (that is dF = α ∧ F for some 1-form α)
then it is parallel and ν = 0.
2) if J is cosymplectic (that is θ = 0) then ν ≤ 0 and ν = 0 if and only if J is
parallel.
3) if J is integrable then the differential dθ of the Lee form is Q-Hermitian and
harmonic.
4) if J is conformally balanced (i.e. J is integrable and θ is exact) but not symplectic then there exists a non-zero Killing vector field associated to J.
The proofs of the statements will be given through Theorems 2.4, 3.1, 3.2 and
Propositions 4.6, 4.11.
A further local result concerns self-dual forms on M 4n , that is sections of the 3dimensional subbundle Λ2+ = g ◦ Q of Λ2 which corresponds to Q by means of the
metric g. In Theorem 3.1 we prove that any closed self-dual 2-form ω is parallel .
Our main global results can be stated as follows.
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
3
Let (M 4n , g, Q) be compact, then
1) if ν > 0 and (M 4n , g) is different from the Grassmannian G2 (Cm ) of 2-planes
in Cm , then there exists no compatible almost complex structure J.
(1,0)
2) if the first Chern class c1 (J) ≡ c1 (TJ M ) = 0 for some compatible almost
complex structure J, then ν = 0.
3) if ν 6= 0 there exists no compatible complex structure J.
4) if ν = 0 a compatible complex structure J is parallel.
The last two results have been proved in [P2] by twistor methods.
The proof of these statements will be given throughout Proposition 3.7 and its
Corollary 3.8, Theorem 4.3, Proposition 4.6 and Theorem 3.10 where we establish
the relationship between the first Chern classes c1 (J) of (T M, J) and the first Chern
class c1 (J ⊥ ) of the complex line bundle J ⊥ orthogonal to J in Q.
In paragraphs 5 and 6 we compute the Laplacian and the (real and quaternionic)
Weyl tensors acting on self-dual 2-forms. This part is very much related to work of
Gauduchon [G1].
2. Preliminary results
Notations and definitions. Let (M 4n , g, Q) be a quaternion-Kähler manifold,
with n ≥ 2. We recall that the Riemannian metric g is Hermitian with respect
to Q, that is
g(JX, Y ) + g(X, JY ) = 0
for all X, Y ∈ T M 4n and any J ∈ Q, and also it is Einstein, that is
Ric(g) =
K
g
4n
where K is the scalar curvature.
We define the bundle Λ2+ −→ M of self-dual 2-forms as the 3-dimensional bundle
associated to Q by the metric g, that is Λ2+ |x = (g ◦ Q)x , x ∈ M . If H = (Jα ) is an
admissible basis of Q the Kähler forms Fα of the almost Hermitian structures (Jα , g)
Fα = g ◦ Jα = g(Jα ·, ·)
(α = 1, 2, 3)
represent a (local) frame for Λ2+ . The notation is consistent with the one used for an
oriented 4-dimensional manifold (see also [AMP, section 5]).
The Levi-Civita connection ∇ ≡ ∇g of g is a quaternionic connection, that is for
any admissible basis H = (Jα ):
∇Jα = ωγ ⊗ Jβ − ωβ ⊗ Jγ
(2.1.1)
where the ωα , α = 1, 2, 3 are 1-forms and (α, β, γ) is a cyclic permutation of (1,2,3).
Hence
dFα = ωγ ∧ Fβ − ωβ ∧ Fγ
(2.1.2)
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
since ∇Fα = g(∇Jα ·, ·).
We recall that the following fundamental identities hold:
dωα + ωβ ∧ ωγ = −νFα
(α = 1, 2, 3)
(2.2)
where ν = K/4n(n + 2) is the reduced scalar curvature of g. Moreover
νFα =
1
T r[Jα R(X, Y )]
2n
(α = 1, 2, 3)
(2.3)
where R is the curvature tensor of ∇ (see [AM2]).
We recall also that on M 4n there is a globally defined fundamental 4-form µ,
µ=
X
α
Fα ∧ F α
which is parallel and non degenerate, see [Be, pag.419].
Remark 2.1. For n = 1, an oriented Riemannian 4-manifold (M 4 , g) always satisfies (2.1.1), (2.1.2). The fundamental identities (2.2) hold if g is anti-self-dual and
Einstein. M
Now we define different types of compatible almost complex structures J ∈ Γ(Q)
on a quaternion-Kähler manifold (M 4n , g, Q) .
Definitions 2.2. Let J be an almost complex structure on the Riemannian manifold
(M 4n , g). Then J is said to be:
(1) parallel if ∇J = 0 where ∇ is the Levi-Civita connection of g.
Moreover let assume that g is J-Hermitian. Then J is called to be
(2) symplectic if the Kähler form F = g ◦ J is closed, dF = 0;
(3) locally conformally symplectic if dF = α ∧ F for some 1-form α;
(4) cosymplectic if F is coclosed, δF = 0 .
Remark 2.3. If n = 1 dF = α ∧ F always holds (see also (2.5) below); for n ≥ 2 the
same equality does not hold in general and it implies that α is closed. M
Parallel complex structures on a quaternion-Kähler manifold. The following
theorem describes a parallel complex structure on a quaternion-Kähler manifold.
Theorem 2.4. Let (M 4n , g, Q) be a simply connected complete irreducible quaternion-Kähler manifold with reduced scalar curvature ν. Let J be a parallel complex
structure on M 4n .
P3
P 2
(1) If J is compatible then ν = 0, and J = α=1 aα Jα where aα = const,
aα =
1 and H = (Jα ) is a parallel hypercomplex structure,
(2) otherwise (M 4n , g) is isometric to the quaternionic symmetric space G2 (Cm )
or to its dual non compact symmetric space, and J is the unique parallel
complex structure.
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
5
Proof. Remark that the parallel complex structures on M 4n correspond to the complex structures in the tangent space of a point x which are invariant under the holonomy group Hx . If ν = 0 the manifold (M 4n , g, Q) has the holonomy group Sp(n) and
the first statement holds. If ν 6= 0 then either the holonomy group Hx = Sp(n)·Sp(1)
or the manifold is locally symmetric (see [Be]): the first case cannot occur since there
is no Sp(n) · Sp(1) -invariant complex structure , and investigation of all holonomy
groups of symmetric quaternion-Kähler manifolds gives the second statement. ¤
Some local formulas. Now we derive some formulas which will be useful for studying cosymplectic and integrable almost complex structures.
For a tensor T = (Tij···k ), where Tij···k are the components of T with respect to
an orthonormal basis, we denote
kT k2 =
X
(Tij···k )2
i,j,··· ,k
(In the terminology of Gauduchon k k is the tensorial norm | |T ; [G1, p. 5]).
For example,
kFα k2 = 4n
(α = 1, 2, 3)
From now on we will assume that J is a compatible almost complex structure on
(M 4n , g, Q) with Kähler form F , H = (J = J1 , J2 , J3 ) is a local admissible basis and
ωα are the corresponding 1-forms. We define the Lee form θ of J by
θ = −(δF ) ◦ J
(2.4)
where δ is the codifferential. It is known that
d(F 2n−1 ) = θ ∧ F 2n−1
(2.5)
Proposition 2.5. For a compatible almost complex structure J on a quaternionKähler manifold the following formulas hold:
k∇F k2 = 4n(kω2 k2 + kω3 k2 )
(2.6)
kdF k2 = 12(n − 1)(kω2 k2 + kω3 k2 ) + 6kω2 ◦ J2 + ω3 ◦ J3 k2
(2.7)
1
1
kdF k2 − k∇F k2 = −kω2 ◦ J2 − ω3 ◦ J3 k2
6
2
(2.8)
θ = ω 2 ◦ J2 + ω 3 ◦ J3
(2.9)
Moreover, by denoting h·, ·i = g(·, ·) to simplify,
δθ = 4nν − 2hω2 ◦ J2 , ω3 ◦ J3 i
(2.10)
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
and
1
1
kdF k2 − k∇F k2 + kθk2 + 2δθ = 8nν
6
2
(2.11)
Remarks 2.6. 1) The inequality kdF k2 ≤ 3k∇F k2 which comes from (2.8) was previously established in [G1]. The equality holds if and only if J is integrable (see also
Lemma 4.1 below).
2) For n = 1, the (2.7) still holds: kdF k2 = 6kω2 ◦ J2 + ω3 ◦ J3 k2 .
3) For n = 1 the (2.9) still holds since dF1 = θ ∧ F1 and, as it is easy to see, the
identity −θ∧F1 +ω3 ∧F2 −ω2 ∧F3 = 0 holds if and only if −θ◦J1 +ω3 ◦J2 −ω2 ◦J3 = 0.
The (2.10), (2.11) hold under the hypothesis that the metric g is anti-self-dual and
Einstein. M
Proof. Since ∇F = g(∇J1 ·, ·) the (2.1.1) gives ∇F = ω3 ⊗ J2 − ω2 ⊗ J3 . We compute
k∇F k2 , by using an orthonormal frame (E1 , · · · , E4n ). By denoting h·, ·i = g(·, ·),
we have
X
[(∇Ei F )(Ej , Ek )]2
k∇F k2 =
i,j,k
=
X
i,j,k
=
X
i,j,k
[ω3 (Ei )hJ2 Ej , Ek i − ω2 (Ei )hJ3 Ej , Ek i]2
[ω32 (Ei )hJ2 Ej , Ek i2 + ω22 (Ei )hJ3 Ej , Ek i2
− 2ω2 (Ei )ω3 (Ei )hJ2 Ej , Ek ihJ3 Ej , Ek i]
= 4n(kω2 k2 + kω3 k2 )
P
2
2
2
For
P last identity we used k hJ2 Ej , Ek i = kJ2 Ej k = kEj k and
k hJ2 Ej , Ek ihJ3 Ej , Ek i = hJ2 Ej , J3 Ej i = 0. Hence the (2.6) is proved.
Now we prove (2.7). We have
kdF k2 =
=
X
dF (Ei , Ej , Ek )2
i,j,k
X
[(∇Ei F )(Ej , Ek ) + (∇Ek F )(Ei , Ej ) + (∇Ej F )(Ek , Ei )]2
i,j,k
=
X
i,j,k
[ω3 (Ei )hJ2 Ej , Ek i + ω3 (Ek )hJ2 Ei , Ej i + ω3 (Ej )hJ2 Ek , Ei i
− ω2 (Ei )hJ3 Ej , Ek i − ω2 (Ek )hJ3 Ei , Ej i − ω2 (Ej )hJ3 Ek , Ej i]2
By computing the square of the expression in brackets, after a long but straightforward calculation, we get
kdF k2 = (12n − 6)(kω2 k2 + kω3 k2 ) + 12hω2 ◦ J2 , ω3 ◦ J3 i
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
7
This implies (2.7). The (2.8) follows directly from (2.6), (2.7). For a vector field X
and an orthonormal frame Ei , i = 1, . . . , 4n, we have
X
X
hω3 (Ei )J2 Ei − ω2 (Ei )J3 Ei , J1 Xi
h(∇Ei J1 )Ei , J1 Xi = −
(δF )(JX) = −
=−
i
X
ω3 (Ei )hJ2 Ei , J1 Xi +
i
X
i
i
ω2 (Ei )hJ3 Ei , J1 Xi = −ω3 (J3 X) − ω2 (J2 X)
that is
δF = ω3 ◦ J2 − ω2 ◦ J3
(2.12)
Hence (2.9) follows.
Now we compute δθ. We have
∇X θ = (∇X ω2 ) ◦ J2 + (∇X ω3 ) ◦ J3
+ ω2 ◦ (ω1 (X)J3 − ω3 (X)J1 ) + ω3 ◦ (ω2 (X)J1 − ω1 (X)J2 )
Hence
δθ = −
since
P
X
[(∇Ei ω2 )(J2 Ei ) + (∇Ei ω3 )(J3 Ei )
i
− ω1 (Ei )(ω2 ◦ J2 + ω3 ◦ J3 )(J1 Ei ) + 2ω2 (Ei )ω3 (J1 Ei )]
P
i ω3 (J1 Ei )ω2 (Ei ). Now let take into account that
i −ω3 (Ei )ω2 (J1 Ei ) =
(∇X ω2 )Y − (∇Y ω2 )X = dω2 (X, Y )
and hence, putting X = Ei and Y = J2 Ei and summing up over i,
X
X
[(∇Ei ω2 )(J2 Ei ) − (∇J2 Ei ω2 )(Ei )] =
dω2 (Ei , J2 Ei )
i
i
that is
2
X
(∇Ei ω2 )(J2 Ei ) =
X
(∇Ei ω3 )(J3 Ei ) =
i
Analogously
2
X
dω2 (Ei , J2 Ei )
X
dω3 (Ei , J3 Ei )
i
i
i
By the second of integrability conditions (2.2) one has
X
X
dω2 (Ei , J2 Ei ) =
[−(ω3 ∧ ω1 )(Ei , J2 Ei ) + νhEi , J22 Ei i]
i
i
that is
X
i
dω2 (Ei , J2 Ei ) =
X
[−ω3 (Ei )ω1 (J2 Ei ) + ω1 (Ei )ω3 (J2 Ei )] − 4nν
i
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Analogously, one has
X
dω3 (Ei , J3 Ei ) =
i
X
[−ω1 (Ei )ω2 (J3 Ei ) + ω2 (Ei )ω1 (J3 Ei )] − 4nν
i
Hence
2δθ = −
X
[−ω3 (Ei )ω1 (J2 Ei ) + ω1 (Ei )ω3 (J2 Ei )
i
− ω1 (Ei )ω2 (J3 Ei ) + ω2 (Ei )ω1 (J3 Ei )
− 8nν − 2ω1 (Ei )(ω2 ◦ J2 + ω3 ◦ J3 )(J1 Ei ) + 4ω2 (Ei )ω3 (J1 Ei )]
and (2.10) follows immediately. Since by (2.8), (2.9) one has 61 kdF k2 − 12 k∇F k2 =
−kω2 ◦ J2 − ω3 ◦ J3 k2 and kθk2 = kω2 ◦ J2 + ω3 ◦ J3 k2 it follows
1
1
kdF k2 − k∇F k2 + kθk2 = 4hω2 ◦ J2 , ω3 ◦ J3 i
6
2
and, by (2.10), we get (2.11).
¤
3. Compatible almost complex and almost hypercomplex structures
(Conformally) symplectic structures.
Theorem 3.1. Let (M 4n , g, Q) be a quaternion-Kähler manifold with n ≥ 2. Then,
(1) every compatible locally conformally symplectic almost complex structure J
is parallel.
(2) every closed self-dual 2-form ω is parallel
In both cases ν = 0.
Proof. If J is a compatible locally conformally symplectic almost complex structure then dF = α ∧ F . Choose an admissible basis H = (J = J1 , J2 , J3 ) of Q
and denote by Fα , α = 1, 2, 3, the corresponding Kähler forms. Then by (2.1.2)
dF = ω3 ∧F2 −ω2 ∧F3 , hence α∧F −ω3 ∧F2 +ω2 ∧F3 = 0. By [ABM, Lemma 1 p.125]
it follows that α = ω2 = ω3 = 0 so that J is parallel and ν = 0. This proves the first
statement. To prove the second statement we put M̆ = {p ∈ M |ω 6= 0}. On M̆ we
can write ω = f F where f is a never zero smooth function and F is the Kähler form
of a compatible almost complex structure . Without loss of generality, we may
assume that f > 0. Then we have 0 = dω = df ∧ F + f dF and dF = −d log f ∧ F .
Now the first statement implies that f is a constant and F and ω are parallel on M̆
and, hence, on M . ¤
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
9
Cosymplectic almost complex structures.
Theorem 3.2. Let J be a compatible cosymplectic almost complex structure on the
quaternion-Kähler manifold (M 4n , g, Q) , that is θ = 0. Then ν ≤ 0 and kω2 k2 =
kω3 k2 = const = −2nν. Moreover ν = 0 if and only if J is parallel.
Proof. From (2.9) θ = 0 implies ω3 ◦ J3 = −ω2 ◦ J2 and in particular kω2 k = kω3 k.
Then, by (2.10) we have
0 = 4nν + 2kω2 k2
and conclusion is immediate
¤
By combining the expression (2.9) of θ given in Proposition 2.5 with a result of
[AMP, proposition 4.1] it is easy to deduce the following Proposition which for n = 1
was proved by S. Salamon in [S2, Prop. 1].
Proposition 3.3. Let (M 4n , g, Q) be a quaternion-Kähler manifold and H = (Jα )
an admissible basis of Q:
(1) If J1 , J2 are cosymplectic then J3 is integrable;
(2) If J1 is cosymplectic and J2 is integrable then J3 is cosymplectic;
(3) If J1 , J2 are integrable then J3 is integrable.
Proof. We prove (1) as follows. Assume J1 , J2 cosymplectic: then by (2.9) ω2 ◦ J2 +
ω3 ◦ J3 = ω3 ◦ J3 + ω1 ◦ J1 = 0 and hence ω1 ◦ J1 = ω2 ◦ J2 which is equivalent to the
integrability of J3 by proposition 4.1 of [AMP] (see also Lemma 4.1 below). To prove
(2) we remark that by hypothesis one has ω2 ◦ J2 + ω3 ◦ J3 = 0 and ω1 ◦ J1 = ω3 ◦ J3 .
It implies ω1 ◦ J1 + ω2 ◦ J2 = 0, that is J3 is cosymplectic. (3) was proved in [AMP,
Remark 2.6]. ¤
Now we associate to an almost complex structure J compatible with Q a globally
defined closed 2-form ΩJ by the formula
ΩJ = dω1
(3.1)
for any admissible basis H = (J1 = J, J2 , J3 ). If H 0 = (J10 = J, J20 , J30 ) is another
such admissible basis and (ω10 , ω20 , ω30 ) are corresponding 1-forms, then
J20 = cosϕJ2 + senϕJ3
,
J30 = −senϕJ2 + cosϕJ3
which implies
ω10 = ω1 + dϕ.
This shows that ΩJ is globally defined and depends only on J.
Proposition 3.4. Let J be a compatible almost complex structure. Assume that
θ = 0, that is J is cosymplectic. If ν < 0, then the 2-form ΩJ is a symplectic
10
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
form which tames J, that is g 0 := ΩJ (·, J·) is a Riemannian metric (which is quasiKählerian with respect to J).
Proof. Since by hypothesis ω3 = ω2 ◦ J1 , for α = 1 the fundamental identity (2.2)
gives
dω1 = −ω2 ∧ (ω2 ◦ J) − νF
hence
dω1 (·, J·) = ω2 ⊗ ω2 + (ω2 ◦ J) ⊗ (ω2 ◦ J) − νg
and the conclusion follows immediately.
¤
Compatible almost complex and almost hypercomplex structures on a
compact quaternion-Kähler manifold. We will now relate the 2-form ΩJ to
the complex line bundle J ⊥ orthogonal to J in Q.
Proposition 3.5. Let (M 4n , g, Q) be a quaternion-Kähler manifold and J a compatible almost complex structure. Then the linear connection ∇0 on the bundle Q which
is defined by
1
1
∇0X = ∇X − J∇X J ≡ ∇X − [ω2 (X)J2 + ω3 (X)J3 ]
2
2
is Riemannian and preserves J (∇0 is the first canonical connection of the almost
Hermitian structure (g, J), [G3, pag. 31]). The 2-form ΩJ is the curvature form of
the Riemannian connection induced by ∇0 on the complex line bundle J ⊥ orthogonal
to J in Q.
1
ΩJ represents the Chern class c1 of J ⊥ and
When M 4n is compact the 2-form 2π
the following conditions are equivalent:
1
ΩJ ] ∈ H 2 (M, Z) vanishes.
(1) The cohomology class c1 (J ⊥ ) = [ 2π
(2) The first Chern class c1 (J) ≡ c1 (T M ) of the tangent bundle T M with respect
to the almost complex structure J vanishes.
(3) There exists a global admissible basis H = (J1 = J, J2 , J3 ) of Q on M 4n .
Proof. Let X, Y be any two vector fields on M . Since (∇J)J + J(∇J) = 0, it is easy
to see that
g((∇X J)Y, JZ) + g(JY, (∇X J)Z) = 0
Hence ∇0 is Riemannian. Moreover,
1
1
(∇0X J)(Y ) = (∇X J)(Y ) − (J∇X J)(JY ) + J 2 (∇X J)(Y )
2
2
1 2
1 2
= (∇X J)(Y ) + (J ∇X J)(Y ) + (J ∇X J)(Y ) = 0
2
2
that is ∇0 leaves J invariant. Hence we can think ∇0 as a connection for the 2-plane
bundle J ⊥ orthogonal complement to the line-bundle RJ in Q. For any admissible
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
11
basis H = (J1 = J, J2 , J3 ) we have a local orthonormal frame (J2 , J3 ) for J ⊥ and the
following identities hold
∇0X J2 = ω1 (X)J3
,
∇0X J3 = −ω1 (X)J2
(3.2)
This shows that the connection ∇0 preserves the complex structure I on J ⊥ , induced
by J:
I(J2 ) = J3 , I(J3 ) = −J2 .
Then the matrix 1-form of the connection ∇0 with respect to the local frame (J2 , J3 )
and matrix 2-form of the curvature are given by
ω=
µ
−ω1
0
0
ω1
¶
,
Ω=
µ
0
dω1
−dω1
0
¶
.
1
Hence, the 2-form ΩJ = 2π
dω1 is the Chern form of ∇0 , as to be proved. To prove
that conditions (1) and (3) are equivalent we observe that, from the classification of
S 1 -bundles over M 4n by elements of H 2 (M, Z), one has c1 (J ⊥ ) = 0 if and only if J ⊥
is trivial or, equivalently, if and only if there exists an admissible basis H = (J1 =
J, J2 , J3 ) of Q = RJ ⊕ J ⊥ globally defined on M 4n . The equivalence of (1) and (2)
will follows immediately from (3.6) of Theorem 3.10 below. ¤
Before giving an application of the last result we recall the following result which
was proved in [AM2].
Theorem 3.6 ([AM2]). Let (M 4n , g, Q) , n > 1, be a quaternion-Kähler manifold
with the reduced scalar curvature ν. Assume that there exists a (globally defined)
almost hypercomplex structure H = (Jα ) which generates Q.
Then either
(1) ν = 0, that is (M 4n , g) is a locally hyper-Kähler manifold.
or
P
(2) the fundamental 4-form µ = α Fα ∧ Fα is exact.
For a compact M 4n only (1) is possible.
Proof. By identities (2.2) one has
ν2µ =
X
α
= d(
(dωα + ωβ ∧ ωγ ) ∧ (dωα + ωβ ∧ ωγ )
X
α
ωα ∧ dωα + 2ω1 ∧ ω2 ∧ ω3 )
where
Pthe 1-forms ωα are globally defined. Hence, if ν 6= 0 the fundamental 4-form
µ = α Fα ∧ Fα is exact. The last case cannot occur if M 4n is compact, since the
form µ is parallel and, hence, harmonic. ¤
12
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Proposition 3.7. Let (M 4n , g, Q) be a compact quaternion-Kähler manifold with
1
a compatible almost complex structure J. If the cohomology class [ 2π
ΩJ ] = 0 ∈
2
H (M, Z) then ν = 0.
1
ΩJ ] vanishes then, by proposition 3.5, J ⊥
Proof. If the Chern class c1 (J ⊥ ) = [ 2π
admits a nowhere vanishing section and Q admits a global admissible basis - i.e.
there exists a global compatible almost hypercomplex structure. Then we can apply
theorem 3.6. ¤
Theorem 3.8. Let (M 4n , g, Q) be a compact quaternion-Kähler manifold with ν > 0.
Assume that it is not homothetic to the standard Grassmannian G2 (Cm ). Then there
is no compatible almost complex structure J on (M 4n , g, Q) .
Proof. By a result of C. LeBrun and S. Salamon [LS] we know that a compact quaternion-Kähler manifold (M 4n , g, Q) with ν > 0, different from the complex Grassmannian G2 (Cm ), has second Betti number b2 (M ) = 0 . Since furthermore M must
be simply connected , we conclude that also H 2 (M, Z) = 0 [LS, proof of 0.2 p. 123].
The result follows now from Proposition 3.7. ¤
Remark 3.9. At present we do not know if there exists any compatible almost complex
structure J on G2 (Cm ). M
Now we prove a fundamental identity between the Chern class c1 (J ⊥ ) and the first
Chern class c1 (J) of the tangent bundle T M 4n endowed with the compatible almost
complex structure J.
We first remark that the curvature tensors R, R0 of the connections ∇, ∇0 are
related by
0
RXY
= RXY
1
1
− (ω2 ∧ ω3 )(X, Y )J − [(dω2 + ω3 ∧ ω1 )(X, Y )J2 + (dω3 + ω1 ∧ ω2 )(X, Y )J3 ]
2
2
that is, by fundamental identities (2.2),
0
RXY
= RXY +
1
νX
Fα (X, Y )Jα + dω1 (X, Y )J
2 α
2
(3.3)
Let now recall that the first Chern form γ1 of the almost complex structure J (on
T M ) with respect to the connection ∇0 is given by (see for ex. [GBNV])
1
0
◦ J)
2πγ1 (X, Y ) = − T race(RXY
2
(3.4)
Theorem 3.10. The Chern 2-forms for J ⊥ , (T M, J) respectively are related by
γ1 = n
1
Ω1
2π
(3.5)
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
Hence
c1 (J) = nc1 (J ⊥ )
13
(3.6)
Proof. By (3.4) we have
2πγ1 (X, Y ) =
1 X
hR0 Es , JEs i
2 s=1,..,4n XY
(3.7)
where X, Y ∈ Tp M, p ∈ M 4n and {E1 , E2 , . . . , E4n } is an orthonormal basis of Tp M .
By using (3.3) we have
X
s=1,..,4n
0
hRX,Y
Es , JEs i =
X
s=1,..,4n
hRX,Y Es , JEs i
X
ν X
1 X
Fα (X, Y )hJα Es , JEs i +
dω1 (X, Y )(JEs , JEs )
2 α=1,2,3 s=1,..,4n
2 s=1,..,4n
+
that is
X
s=1,..,4n
0
Es , JEs i =
hRX,Y
X
s=1,..,4n
hRXY Es , JEs i + 2nνF (X, Y ) + 2ndω1 (X, Y ) (3.8)
On the other hand we get the following. By curvature identity,
X
s=1,..,4n
hRX,Y Es JEs i = −
X
s=1,..,4n
hRJEs ,X Es , Y i −
X
s=1,..,4n
hRY,JEs Es , Xi
Moreover, by using the known identity for the curvature tensor of a quaternionKähler manifold (see for ex [Be, pag. 403]; also [M, pag. 423]),
hRJZ
T U, V
i = −hRZJT U, V i + ν(−hJ3 U, V ihJ2 Z, T i + hJ2 U, V ihJ3 Z, T i)
where U, V, Z, T ∈ Tp M 4n , we get
X
s=1,..,4n
=
hRX,Y Es JEs i
X
s=1,..,4n
+
X
s=1,..,4n
hREs JX Es , Y i + ν(hJ3 Es , Y ihJ2 Es , Xi − hJ2 Es , Y ihJ3 Es , Xi)
hRJY Es Es , Xi) − ν(h−J3 Es , XihJ2 Y, Es i + hJ2 Es , XihJ3 Y, Es i)
and, by taking into account the identity
hRU V Z, T i = hRJU JV JZ, JT i
14
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
(see [M, pag. 423]), it results
X
hRXY Es , JEs i
s=1,..,4n
=
X
s=1,..,4n
=
X
s=1,..,4n
hREs ,JX Es , Y i + 2νhY, JXi +
hREs JX Es , Y i −
Hence
X
s=1,..,4n
that is
X
s=1,..,4n
X
s=1,..,4n
hRJY Es Es , Xi) − 2νhX, JY i
hRY,JEs JEs , JXi + 4νhJX, Y i
hRX,Y Es , JEs i = −2Ric(Y, JX) + 4νhJX, Y i
= −2(n + 2)νhY, JXi + 4νhJX, Y i
X
s=1,..,4n
hRX,Y Es , JEs i = −2nνhJX, Y i
(3.9)
Hence, by substitution in (3.8), we get
X
0
Es , JEs i = −2nνF (X, Y ) + 2nνF (X, Y ) + 2ndω1 (X, Y )
hRX,Y
s=1,..,4n
= 2ndω1 (X, Y )
and, finally,
2πγ1 = ndω1
that is (3.5) and (3.6) hold.
¤
4. Compatible complex structures
A quaternion-Kähler manifold M 4n locally admits many compatible almost complex structures which are integrable, see [S3, pag. 130] and [AMP]. Any such structure
J defines a local section J : U ⊇ M 4n −→ Z, U 3 x 7→ Jx ∈ Z of the twistor fibration t : Z −→ M 4n such that the image X = J(U) is a complex submanifold of Z.
Conversely, any complex submanifold X ⊂ Z such that the projection t : X −→ M
is a diffeomorphism defines a compatible complex structure on U = t(X) ⊆ M 4n .
Now we refine some of the formulas of §2 in the special case when J = J1 is an
integrable almost complex structure on the quaternion-Kähler manifold (M 4n , g, Q) .
Let us first recall the following result.
Lemma 4.1 ([AMP]). A compatible almost complex structure J on (M 4n , g, Q) is
integrable if and only if
ω 2 ◦ J2 = ω 3 ◦ J3
for an admissible basis (J = J1 , J2 , J3 ) with connection 1-forms (ω1 , ω2 , ω3 ).
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
15
Proposition 4.2. Assume that the compatible almost complex structure J on M
with Kähler form F and Lee form θ is integrable. Then one has
k∇F k2 = 2nkθk2
and
, kdF k2 = 6nkθk2
,
kδF k2 = kθk2
(4.1)
kθk2 + 2δθ = 8nν
(4.2)
1
4θ = − d(kθk2 ) + δdθ
2
(4.3)
Proof. The three identities (4.1) and (4.2) follow from (2.5), (2.7), (2.9) and from
(2.10) respectively, by considering that for J = J1 one has θ = 2ω2 ◦ J2 = 2ω3 ◦ J3 .
The (4.3) is obtained by differentiating (4.2). ¤
As a consequence of our previous results we deduce the following theorem obtained
in [P2] by twistor methods.
Theorem 4.3. On a compact quaternion-Kähler manifold (M 4n , g, Q) every compatible complex structure is necessarily parallel.
Proof. If the scalar curvature is non-positive, by integrating (4.2) on M 4n we get
θ = ν = 0 and ∇J = 0 by first of (4.1).
Otherwise ν > 0, by Corollary 3.3 the existence of a compatible almost complex structure J implies that (M 4n , g) is homothetic to the Grassmannian G2 (Cm )
equipped with the symmetric metric. By a strong result which was proved in [BGMR,
theorem 1.7], a complex structure which is Hermitian with respect to a symmetric
metric is necessarily parallel. Therefore J is a compatible parallel complex structure
on G2 (Cm ). But this is impossible by theorem 2.4. ¤
Remark 4.4. If ν < 0 but M is non-compact the conclusion is not true: it is sufficient
to think of the hyperbolic quaternionic space HHn [P2]. M
The following two Propositions hold on any (not necessarily complete) quaternionKähler manifold.
Proposition 4.5. Let (M 4n , g, Q) be a quaternion-Kähler manifold with a compatible complex structure J and Lee form θ. Then the following identities hold for any
vector fields X, Y on M 4n :
(∇X θ)(Y ) + (∇J2 Y θ)(J2 X) =
1
1
θ(JX)θ(JY ) + θ(J3 X)θ(J3 Y ) − 2νg(X, Y ) (4.4)
2
2
(∇X θ)(Y ) + (∇J3 Y θ)(J3 X) =
1
1
θ(JX)θ(JY ) + θ(J2 X)θ(J2 Y ) − 2νg(X, Y ) (4.5)
2
2
1
1
(∇X θ)(Y ) + θ(X)θ(Y ) = (∇JX θ)(JY ) + θ(JX)θ(JY )
2
2
(4.6)
16
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Proof. From lemma 4.1 we have θ = 2ω2 ◦ J2
For α = 2 the identity (2.2) gives
,
ω2 = − 21 θ ◦ J2
,
ω3 = − 21 θ ◦ J3 .
(∇X ω2 )(Y ) − (∇Y ω2 )(X) + ω3 (X)ω1 (Y ) − ω3 (Y )ω1 (X) = νg(X, J2 Y )
that is
1
1
1
1
− (∇X θ)(J2 Y ) − θ((∇X J2 )Y ) + (∇Y θ)(J2 X) + θ((∇Y J2 )X)
2
2
2
2
1
1
− θ(J3 X)ω1 (Y ) + θ(J3 Y )ω1 (X)
2
2
= νg(X, J2 Y )
that is
1
1
1
− (∇X θ)(J2 Y ) − θ(ω1 (X)J3 Y − ω3 (X)JY ) + (∇Y θ)(J2 X)
2
2
2
1
1
1
+ θ(ω1 (Y )J3 X − ω3 (Y )JX) − θ(J3 X)ω1 (Y ) + θ(J3 Y )ω1 (X) = νg(X, J2 Y )
2
2
2
that is, after suitable cancellations,
1
1
1
1
− (∇X θ)(J2 Y ) + (∇Y θ)(J2 X) + ω3 (X)θ(JY ) − ω3 (Y )θ(JX) = νg(X, J2 Y )
2
2
2
2
By changing Y with J2 Y one finds
1
1
1
1
(∇X θ)(Y ) + (∇J2 Y θ)(J2 X) + ω3 (X)θ(J3 Y ) − ω3 (J2 Y )θ(JX) = −νg(X, Y )
2
2
2
2
and (4.4) follows by expressing ω3 by θ. For α = 3 we get
(∇X ω3 )(Y ) − (∇Y ω3 )(X) + ω1 (X)ω2 (Y ) − ω1 (Y )ω2 (X) = νg(X, J3 Y )
that is equivalent to
− (∇X θ)(J3 Y ) − θ(ω2 (X)JY − ω1 (X)J2 Y ) + (∇Y θ)(J3 X)
− θ(ω2 (Y )JX − ω1 (Y )J2 X) − ω1 (X)θ(J2 Y ) + ω1 (Y )θ(J2 X) = 2νg(X, J3 Y )
and, after cancellations,
1
1
2νg(X, J3 Y ) = −(∇X θ)(J3 Y ) + (∇Y θ)(J3 X) + θ(J2 X)θ(JY ) − θ(J2 Y )θ(JY )
2
2
After substitution of Y with J3 Y we get (4.5). The (4.6) follows by substraction of
(4.5) to (4.4) and substitution of X, Y with J2 X, J2 Y . ¤
For any point p ∈ M 4n let Π be the orthogonal projection from the space of
bilinear forms defined on Tp M 4n onto the subspace of Q-Hermitian bilinear forms,
that is for any ω ∈ Bilp one has
X
1
Πω = [ω +
ω( · , · )]
4
α
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
17
Proposition 4.6. For a compatible complex structure J with associated Lee form θ
on a quaternion-Kähler manifold (M 4n , g, Q) one has:
1) The bilinear form ∇θ + 21 θ ⊗ θ is J-Hermitian.
2) The Q-Hermitian part ∇+ θ = Π∇θ and the skew-Q-Hermitian part ∇− θ =
(1 − Π)∇θ of the covariant derivative ∇θ are given by
1
1
(4.7)
∇+ θ = Π(θ ⊗ θ) − νg + dθ
2
2
1
∇− θ = (∇θ)s − Π(θ ⊗ θ) + νg
(4.8)
2
s
where (∇θ) is the symmetric part of ∇θ.
3) The 2-form dθ is Q-Hermitian and harmonic.
Proof. Statement 1) is equivalent to the identity (4.6). To prove identity (4.7) , we
calculate
X
(∇Jα X θ)(Jα Y )
4∇+ θ(X, Y ) ≡ 4Π∇θ(X, Y ) = (∇X θ)(Y ) +
α
using identities (4.4), (4.5), (4.6) to epress (∇Jα X θ)(Jα Y ), α = 1, 2, 3.
The (4.8) and the first statement of 3) follow from (4.7). To prove the last statement in 3) we can argue as in [MS] (where Q-Hermitian 2-forms are called self-dual
and it was proved that such a 2-form is closed if and only if it is harmonic): for a
Q-Hermitian 2-form ω one has ?ω = c2 ω ∧ µn−1 where c2 = −1/(2n − 1)! (see [GP]).
Hence δdθ = − ? d ? dθ = −c2 ? d(dθ ∧ µn−1 ) = 0 and 4dθ = 0. ¤
Remark 4.7.
(1) When J is a compatible complex structure more can be said about the 2-form
ΩJ considered in the previous section §2:
1
(4.9)
ΩJ = − (θ ◦ J2 ) ∧ (θ ◦ J3 ) − νF
4
and
1
ΩJ (·, J·) = − [(θ ◦ J2 ) ⊗ (θ ◦ J2 ) + (θ ◦ J3 ) ⊗ (θ ◦ J3 )] − νg
(4.10)
4
Hence, if ν > 0 then −ΩJ tames J , that is
1
(4.11)
g 0 := [(θ ◦ J2 ) ⊗ (θ ◦ J2 ) + (θ ◦ J3 ) ⊗ (θ ◦ J3 )] + νg
4
is a Riemannian metric which is almost-Kähler with respect to J and in fact,
0
by a classical result, g 0 is Kähler and J is parallel with respect to ∇g .
(2) On a non-compact complete quaternion-Kähler manifold (M 4n , g, Q) , a complex structure compatible with Q may exist. For example, any quaternionKähler manifold (M 4n , g, Q) which admits a simply transitive solvable group
G of isometries and different from quaternion hyperbolic space has a Ginvariant compatible complex structure, see [A].
M
18
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Conformally balanced complex structures, Dirac and Twistor operators.
Referring to the fundamental work of Salamon [S1], we would now like to establish a
relation between compatible almost complex structures on a quaternion-Kähler manifold (M 4n , g, Q) and two first order differential operators acting on sections of Q
which we will call Dirac and Twistor opertors by analogy with the four-dimensional
case; as usual they are defined by first taking covariant derivative with respect to the
Levi-Civita connection and then projecting onto the two irreducible Sp(n) · Sp(1)submodules of the target space.
Our notations are taken from the four-dimensional case and are different from
those used by Salamon who denotes the complexified decomposition (4.12) by S 2 H ⊗
E ⊗ H = (E ⊗ H) ⊕ (E ⊗ S 3 H); what we called Twistor operator is denoted by
Salamon by D and our Dirac operator by δ.
Salamon also introduced complex analytic methods in quaternionic geometry by
defining the twistor fibration t : Z → M . The total space Z is a (2n + 1)-dimensional
complex manifold equipped with a holomorphic contact structure and the fibers
t−1 (p) are holomorphically imbedded CP1 ’s
The general principle of Penrose is that holomorphic properties of Z reflect geometric properties of (M 4n , g, Q) . With respect to this, Salamon shows that the
kernel of the twistor operator is isomorphic to the real part of the space of holomorphic sections of the contact line bundle O(2) over Z [S1, Lemma 6.4]. He also proves
that in the compact non-Ricci-flat case the Dirac operator maps the kernel of the
twistor operator isomorphically onto the space of Killing vector fields of (M, g).
In what follows we will let V denote the tangent bundle T M . Then using notations
of [AM2] we have:
∗
⊕ (V ∗ ⊗ Q)0
V ∗ ⊗ Q = V(1)
where
∗
V(1)
= {Aξ =
∗
X
α
(4.12)
(ξ ◦ Jα ) ⊗ Jα , ξ ∈ V ∗ } ∼
=V∗
and (V ⊗ Q)0 are the trace-free tensors.
P
More precisely, given a tensor T ∈ V ∗ ⊗ Q we can write T = α ξα ⊗ Jα and set
ξ=−
1X
(ξρ ◦ Jρ )
3 ρ
and
T0 =
X
α
∗
(ξ ◦ Jα ) ⊗ Jα ∈ V(1)
Then the above direct sum decomposition is T = T 0 + (T − T 0 ).
Finally if p1 and p2 denote the projections onto the irreducible components we
will call
∗
D : Γ(Q) → Γ(V(1)
)
the Dirac operator given by covariant differentiation ∇ followed by p1 and
D̄ : Γ(Q) → Γ((V ∗ ⊗ Q)0 )
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
19
the Twistor operator p2 ◦ ∇.
A non zero section σ of Q , restricted to the open submanifold M̂ = {σ 6= 0}
can always be written as σ = f J for some smooth function f and compatible almost
complex structure J on M . To compute the covariant derivative of σ we choose a
local admissible basis (Jα ) of Q such that J = J1 . We have
∇σ = ∇f J = f (ω3 ⊗ J2 − ω2 ⊗ J3 ) + df ⊗ J
and
3Dσ = [f (ω3 ◦ J3 + ω2 ◦ J2 ) + df ] ⊗ J1
+ [f (ω3 − ω2 ◦ J1 ) − df ◦ J3 ] ⊗ J2
+ [f (−ω3 ◦ J1 − ω2 ) + df ◦ J2 ] ⊗ J3
X
=−
[f θ + df ] ◦ J ◦ Jα ⊗ Jα
(4.13)
α
Which can also be rewritten as
X
X
δσ ◦ Jα ⊗ Jα
δ(f F ) ◦ Jα ⊗ Jα = −
3Dσ = −
α
(4.14)
α
so that D can be identified with the codifferential δ : Γ(Q) −→ T ∗ M .
We now compute the twistor operator:
D̄σ = f ω3 ⊗ J2 − f ω2 ⊗ J3 + df ⊗ J − Dσ
(4.15)
We will now use the above facts to prove the following result which holds whether
M is compact or not, and generalizes [Theorem 2.1, P1] to higher dimensions.
Definition 4.8. A Hermitian metric (g, J) is said to be balanced if (J is integrable
and) the Lee form θ = 0; conformally balanced if θ is exact.
For the sake of simplicity, when g is fixed we also say that J is balanced or,
respectively, conformally balanced.
Of course when n = 1 one has θ = 0 if and only if F is symplectic. For n > 1,
by first of (4.1) it follows immediately that a compatible complex structure J on the
quaternion-Kähler manifold (M 4n , g, Q) is balanced if and only if it is parallel.
Theorem 4.9. Let (M 4n , g, Q) be quaternion-Kähler if n ≥ 2 or anti-self-dual if
n = 1 with twistor space Z. Suppose J is a compatible almost complex structure and
let X denote the image of the sections ±J : M → Z in the twistor space. Then,
X is associated to the holomorphic contact line bundle O(2) if and only if (g, J) is
conformally balanced.
Proof. The smooth submanifold X given by the (disjoint) union of J(M ) and −J(M )
is a divisor of O(2) if and only if there is a holomorphic section s ∈ H 0 (Z, O(2)) vanishing exactly on X. By Salamon correspondence [S1, Lemma 6.4] this is equivalent
20
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
to having a smooth section φ(s) ∈ Γ(Q) satisfying the Twistor equation D̄φ(s) = 0 ∈
Γ(V ∗ ⊗ Q)0 . Now, Salamon correspondence tells us that φ(s) = f J where J is the
given almost complex structure and f is a smooth real function on M which never
vanishes because X contains no twistor fibers, for example. By equations (4.13) and
(4.15) we easily see that D̄f J = 0 is equivalent to ω2 ◦ J2 = ω3 ◦ J3 - i.e. J is an
integrable complex structure - and furthermore d log f 2 = ω2 ◦ J2 + ω3 ◦ J3 - where
θ = ω2 ◦ J2 + ω3 ◦ J3 is the Lee form of (g, J). It is now easy to check that this is
equivalent to the fact that the Hermitian metric (|f |g, J) is conformally balanced. ¤
Remark 4.10. We can now give a more self-contained proof of the statement that
there is no compatible complex structure on a complete quaternion-Kähler manifold
(M 4n , g, Q) of positive scalar curvature; see theorem 4.3.
Recall that M is necessarily compact and simply connected [S1]. By contradiction,
let θ be the Lee form of a compatible complex structure J then dθ is harmonic by
proposition 4.6 3), and since M is compact dθ = 0. In fact, since M is simply
connected we conclude that the Lee form is exact: θ = dh for some smooth function
h which is globally defined on M . In this situation (4.2) becomes
kdhk2 + 2∆h = 8nν
showing that the function h cannot have a minimum; this is a contradiction because
M is compact. M
Proposition 4.11. Assume that the compatible almost complex structure J on M
is globally conformally balanced - i.e. J is integrable and θ = dlogf 2 is exact. Then
df ◦ J ≡ 12 f θ ◦ J ≡ 21 f δF is a Killing 1-form (that is the vector field ξ dual to df ◦ J
is a Killing vector field). Furthermore, if ξ = 0 then the scalar curvature ν = 0.
Proof. We must show that under the present hypothesis the covariant derivative of
1-form df ◦ J is skew-symmetric. For any vectors X, Y , by taking into account that
df ◦ J ≡ 21 f θ ◦ J, one has
∇X (df ◦ J)(Y ) + ∇Y (df ◦ J)(X)
1
1
= df (X)θ(JY ) + f (∇X θ)(JY ) + df (ω3 (X)J2 Y − ω2 (X)J3 Y )
2
2
1
1
+ df (Y )θ(JX) + f (∇Y θ)(JX) + df (ω3 (Y )J2 X − ω2 (Y )J3 X)
2
2
1 1
1
= f [ θ(X)θ(JY ) + θ(Y )θ(JX) + (∇X θ)(JY ) + (∇Y θ)(JX)]
2 2
2
(Last equality being deduced from identities df = 12 f θ and ω2 = − 21 θ ◦ J2 , ω3 =
− 12 θ ◦ J3 ). By (4.6) one gets ∇X (df ◦ J)(Y ) + ∇Y (df ◦ J)(X) = 0. Finally, ξ = 0 if
and only if θ = 0: in this case (g, J) is Kähler by (3.5) and therefore the holonomy
of g is in Sp(n) · Sp(1) ∩ U (2n) ⊂ Sp(n). ¤
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
21
Conformally balanced hypercomplex structures. It is well known that the
quaternionic projective space HPn is locally hypercomplex: local admissible hypercomplex bases H = (Jα ) on HPn are obtained by considering systems of quaternionic
projective coordinates. But Theorem 4.3 excludes the existence of global hypercomplex bases on HPn (In fact a classical result of W.S. Massey states that there does’nt
exist any almost complex structure on HPn ). On the other hand, as recalled by Remark 4.4, the hyperbolic quaternionic space HHn admits global hypercomplex bases.
In fact we wonder if for ν < 0 this is essentially the only possibility of complete
quaternion-Kähler manifolds having this property, up to a Riemannian covering. A
first result in this direction is the following one.
Proposition 4.12. Let (M 4n , g, Q) be a complete quaternion-Kähler manifold. Assume that there exists a global admissible basis H = (Jα ) of Q, consisting of integrable
complex structures. Then
1) ν ≤ 0 and ν < 0 only if (M 4n , g) is non-compact.
Moreover
2) if ν < 0 and one of the complex structures Jα (α = 1, 2, 3) is conformally
balanced then there exists an (eventually singular) integrable distribution D on M 4n
which is Q-invariant and whose regular orbit is a (non trivial) totally geodesic quaternionic submanifold with constant quaternionic curvature, that is locally isometric to
HHk , 1 ≤ k ≤ n, with a standard metric.
Proof. Let H = (Jα ) be a global admissible basis which is hypercomplex. Then
by theorems 4.3 and 2.4 we exclude that ν 6= 0 if M 4n is compact. It remains to
prove the second statement. Hence let assume ν < 0, M 4n non-compact and, say J1
conformally balanced. Then formula (2.9) and similar formulas for J2 , J3 together
with Lemma 4.1 imply that the three complex structures Jα have equal Lee forms :
θ = θ 1 = θ2 = θ3
and, moreover, θ does not vanishes.
By 1) of Proposition 4.6 the 2-tensor ∇θ + 21 θ ⊗ θ is Q-Hermitian. In fact, by
summing up the (4.6) for J = Jα , α = 1, 2, 3 respectively, one gets
1
1
∇− θ = − θ ⊗ θ + Π(θ ⊗ θ)
2
2
and by adding to (4.7) it results, since dθ = 0,
1
∇θ = − θ ⊗ θ + Π(θ ⊗ θ) − νg
2
(4.16)
1
By assumption, θ = dh for some function h. We define the 1-form η := e 2 h θ. Then
1
1
∇η = e 2 h [∇θ + Π(θ ⊗ θ)]
2
and hence ∇η is a symmetric Q-Hermitian 2-tensor on M , that is g −1 η is a quaternionic non-isometric infinitesimal transformation on M 4n (see [AM1]). By result of
[AM3, Proposition 6] we can conclude. ¤
22
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
5. Laplacians of self-dual forms
4n
Let (M , g, Q) be a quaternion-Kähler manifold if n > 1 and a self-dual Einstein
4-manifold if n = 1. Then the following formulas hold.
Proposition 5.1. Let H = (Jα ) be an admissible basis of Q. Then
4F1 = (4ν + kω2 k2 + kω3 k2 )F1
+ (δω3 − < ω1 , ω2 >)F2 − (δω2 + < ω1 , ω3 >)F3
and
∇∗ ∇F1 = (kω2 k2 + kω3 k2 )F1
+ (δω3 − < ω1 , ω2 >)F2 − (δω2 + < ω1 , ω3 >)F3
(5.1)
(5.2)
(Note that if ν > 0 then 4F1 6= 0 everywhere).
Moreover, for F = F1 one has
4F − ∇∗ ∇F = 4νF
(5.3)
Proof. Let recall the expression of δF given by (2.12). Now we compute dδF . We
have
dδF (X, Y ) = [∇X (ω3 ◦ J2 − ω2 ◦ J3 )](Y ) − [∇Y (ω3 ◦ J2 − ω2 ◦ J3 )](X)
= (∇X ω3 )(J2 Y ) − (∇Y ω3 )(J2 X) − (∇X ω2 )(J3 Y ) + (∇Y ω2 )(J3 X)
+ ω3 [(ω1 (X)J3 − ω3 (X)J1 )(Y )] − ω2 [(ω2 (X)J1 − ω1 (X)J2 )(Y )]
− ω3 [(ω1 (Y )J3 − ω3 (Y )J1 )(X)] + ω2 [(ω2 (Y )J1 − ω1 (Y )J2 )(X)]
Hence
dδF (X, Y ) = (∇X ω3 )(J2 Y ) − (∇Y ω3 )(J2 X) − (∇X ω2 )(J3 Y ) + (∇Y ω2 )(J3 X)
− [ω2 (X)ω2 (J1 Y ) + ω3 (X)ω3 (J1 Y ) − ω2 (Y )ω2 (J1 X) − ω3 (Y )ω3 (J1 X)]
+ ω1 (X)ω2 (J2 Y ) − ω1 (Y )ω2 (J2 X) + ω1 (X)ω3 (J3 Y ) − ω1 (Y )ω3 (J3 X)
Let recall the expression of dF given by (2.1.2). Now we compute δdF . We have
X
δdF (X, Y ) = −
(∇Ei dF )(Ei , X, Y )
i
=−
X
[(∇Ei ω3 )(Ei )F2 (X, Y ) + ω3 (Ei )g((∇Ei J2 )X, Y )
i
− (∇Ei ω2 )(Ei )F3 (X, Y ) − ω2 (Ei )g((∇Ei J3 )X, Y )]
− ω3 (X)ω1 (J3 Y ) + ω3 (X)ω3 (J1 Y ) − (∇J2 Y ω3 )(X)
+ ω2 (X)ω2 (J1 Y ) − ω2 (X)ω1 (J2 Y ) + (∇J3 Y ω3 )(X)
X
[g((∇Ei J2 )Ei , X)ω3 (Y ) + F2 (Ei , X)(∇Ei ω3 )(Y )
−
i
− g((∇Ei J3 )Ei , X)ω2 (Y ) − F3 (Ei , X)(∇Ei ω2 )(Y )]
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
23
Hence
δdF (X, Y ) = −
X
i
[(∇Ei ω3 )(Ei )F2 (X, Y ) − (∇Ei ω2 )(Ei )F3 (X, Y )
+ ω3 (Ei )g(ω1 (Ei )J3 X − ω3 (Ei )J1 X, Y )
− ω2 (Ei )g(ω2 (Ei )J1 X − ω1 (Ei )J2 X, Y )
+ ω3 (Y )g(ω1 (Ei )J3 Ei − ω3 (Ei )J1 Ei , X)
− ω2 (Y )g(ω2 (Ei )J1 Ei − ω1 (Ei )J2 Ei , X)]
+ (∇J2 X ω3 )(Y ) − (∇J3 X ω2 )(Y )
− ω3 (X)ω1 (J3 Y ) + ω3 (X)ω3 (J1 Y ) − (∇J2 Y ω3 )(X)
+ ω2 (X)ω2 (J1 Y ) − ω2 (X)ω1 (J2 Y ) + (∇J3 Y ω3 )(X)
That is
δdF (X, Y ) = δω3 F2 (X, Y ) − δω2 F3 (X, Y )
− < ω3 , ω1 > F3 (X, Y ) + kω3 k2 F1 (X, Y )
+ kω2 k2 F1 (X, Y )− < ω1 , ω2 > F2 (X, Y )
+ ω1 (J3 X)ω3 (Y ) − ω3 (J1 X)ω3 (Y ) − ω2 (J1 X)ω2 (Y ) + ω1 (J2 X)ω2 (Y )
+ (∇J2 X ω3 )(Y ) − (∇J3 X ω2 )(Y )
− ω3 (X)ω1 (J3 Y ) + ω3 (X)ω3 (J1 Y ) − (∇J2 Y ω3 )(X)
+ ω2 (X)ω2 (J1 Y ) − ω2 (X)ω1 (J2 Y ) + (∇J3 Y ω3 )(X)
By previous formulas one has
4F1 (X, Y ) = (dδ + δd)F1 (X, Y )
= (∇X ω3 )(J2 Y ) − (∇Y ω3 )(J2 X) − (∇X ω2 )(J3 Y ) + (∇Y ω2 )(J3 X)
+ (∇J2 X ω3 )(Y ) − (∇J3 X ω2 )(Y ) − (∇J2 Y ω3 )(X) + (∇J2 Y ω3 )(X)
+ ω1 (J3 X)ω3 (Y ) − ω3 (J1 X)ω3 (Y ) − ω2 (J1 X)ω2 (Y ) + ω1 (J2 X)ω2 (Y )
− ω3 (X)ω1 (J3 Y ) + ω3 (X)ω3 (J1 Y ) + ω2 (X)ω2 (J1 Y ) − ω2 (X)ω1 (J2 Y )
− ω2 (X)ω2 (J1 Y ) − ω3 (X)ω3 (J1 Y ) + ω2 (Y )ω2 (J1 X) + ω3 (Y )ω3 (J1 X)
+ ω1 (X)ω2 (J2 Y ) − ω1 (Y )ω2 (J2 X) + ω1 (X)ω3 (J3 Y ) − ω1 (Y )ω3 (J3 X)
+ (kω2 k2 + kω3 k2 )F1 (X, Y )
+ (δω3 − < ω1 , ω2 >)F2 (X, Y ) − (δω2 + < ω1 , ω3 >)F3 (X, Y )
24
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
that is
4F1 (X, Y ) = dω3 (X, J2 Y ) + dω3 (J2 X, Y ) − dω2 (J3 X, Y ) − dω2 (X, J3 Y )
+ ω1 ∧ ω2 (J2 X, Y ) + ω1 ∧ ω2 (X, J2 Y )
− ω3 ∧ ω1 (J3 X, Y ) − ω3 ∧ ω1 (X, J3 Y )
+ (kω2 k2 + kω3 k2 )F1 (X, Y )
+ (δω3 − < ω1 , ω2 >)F2 (X, Y ) − (δω2 + < ω1 , ω3 >)F3 (X, Y )
= νg(J2 X, J3 Y ) + νg(X, J3 J2 Y ) − νg(J3 X, J2 Y ) − νg(X, J2 J3 Y )
+ (kω2 k2 + kω3 k2 )F1 (X, Y )
+ (δω3 − < ω1 , ω2 >)F2 (X, Y ) − (δω2 + < ω1 , ω3 >)F3 (X, Y )
= 4νg(J1 X, Y )
+ (kω2 k2 + kω3 k2 )F1 (X, Y )
+ (δω3 − < ω1 , ω2 >)F2 (X, Y ) − (δω2 + < ω1 , ω3 >)F3 (X, Y )
and (5.1) follows immediately. Now we compute ∇∗ ∇F1 . By using (2.1.1) we have
(∇∗ ∇F1 )(X, Y ) = −
X
[(∇Ei ω3 )(Ei ))F2 (X, Y ) + ω3 (Ei )(∇Ei F2 )(X, Y )
i
− (∇Ei ω2 )(Ei ))F3 (X, Y ) − ω2 (Ei )(∇Ei F3 )(X, Y )]
= δω3 F2 (X, Y ) − δω2 F3 (X, Y )
− ω3 (Ei )g(ω1 (Ei )J3 X − ω3 (Ei )J1 X, Y )
+ ω2 (Ei )g(ω2 (Ei )J1 X − ω1 (Ei )J2 X, Y )
and (5.2) follows. By (5.1), (5.2) it is straightforward to deduce (5.3).
¤
Now we want to compute the Laplacian on self-dual 2-forms. Let σ be a section of
Λ2+ ≡ g◦Q, that is σ is a self-dual 2-form on M. Let write σ = f F where F ≡ F1 is the
Kähler form of a compatible almost complex structure J ≡ J1 and H = (J1 , J2 , J3 ) is
a local basis of Q. Now we calculate the Laplacian 4σ. Let (Ei )i=1,..,n be an (locally
defined) orthonormal frame and X a vector field on M . We have
δ(f F )(X) = −
=
X
X
i
i
df (Ei )F (Ei , X) − f
df (Ei )g(Ei , JX) − f
X
(∇Ei F )(Ei , X)
i
X
(∇Ei F )(Ei , X)
i
Hence
δσ ≡ δ(f F ) = (df ◦ J) + f δF
Remark 5.2. Note that δ(f F ) = 0 ⇐⇒ θJ = d(−logf ).
(5.4)
M
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
25
Moreover, for any two vector fields X, Y we have
dδ(f F )(X, Y ) = ∇X (df ◦J)(Y )−∇Y (df ◦J)(X)+df ∧δF (X, Y )+f dδF (X, Y ) (5.5)
that is
dδ(f F ) = d(df ◦ J) + df ∧ δF + f dδF
(5.6)
On the other hand, we have
dσ ≡ d(f F ) = df ∧ F + f dF
and hence
δd(f F )(X, Y ) = −
−
X
X
i
∇Ei (df ∧ F )(Ei , X, Y )
df (Ei )dF (Ei , X, Y ) + f δF (X, Y )
i
that is
δd(f F )(X, Y ) = −
−
X
X
i
i
∇Ei (df ∧ F )(Ei , X, Y )
df (Ei )(ω3 ∧ F2 − ω2 ∧ F3 )(Ei , X, Y ) + f δF (X, Y )
Now we have
X
i
(df ∧ F )(Ei , X, Y ) =
X
[df (Ei )F (X, Y ) + df (X)F (Y, Ei ) + df (Y )F (Ei , X)]
i
and
−
X
i
∇Ei (df ∧ F )(Ei , X, Y ) = (δdf )F (X, Y ) −
X
df (Ei )(∇Ei F )(X, Y )
i
− (∇Ei df )(X)F (Y, Ei ) − df (X)∇Ei F (Y, Ei )
− (∇Ei df )(Y )F (Ei , X) − df (Y )(∇Ei F )(Ei , X)
26
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
hence
−
X
i
∇Ei (df ∧ F )(Ei , X, Y )
= (δdf )F (X, Y ) −
X
i
µ
df (Ei ) ω3 (Ei )F2 (X, Y ) − ω2 (Ei )F3 (X, Y )
− (∇JY df )(X) − df (X)
+ (∇JX df )(Y ) − df (Y )
Xµ
i
Xµ
i
¶
ω3 (Ei )F2 (Y, Ei ) − ω2 (Ei )(F3 (Y, Ei )
¶
ω3 (Ei )F2 (Ei , X) − ω2 (Ei )F3 (Ei , X)
¶
= (δdf )F (X, Y ) − hdf, ω3 iF2 (X, Y ) + hdf, ω2 iF3 (X, Y )
+ (∇JX df )(Y ) − (∇JY df )(X) − df (X)ω3 (J2 Y ) − df (X)ω2 (J3 Y )
+ df (Y )ω3 (J2 X) − df (Y )ω2 (J3 X)
= (δdf )F (X, Y ) − hdf, ω3 iF2 (X, Y ) + hdf, ω2 iF3 (X, Y )
+ (∇JX df )(Y ) − (∇JY df )(X) − df (X)θ(JY ) + df (Y )θ(JX)
Moreover we have
−
X
i
X
i
df (Ei )(ω3 ∧ F2 − ω2 ∧ F3 )(Ei , X, Y )
[−df (Ei )ω3 (Ei )F2 (X, Y ) − df (Ei )ω3 (X)F2 (Y, Ei ) − df (Ei )ω3 (Y )F2 (Ei , X)
+ df (Ei )ω2 (Ei )F3 (X, Y ) + df (Ei )ω2 (X)F3 (Y, Ei ) + df (Ei )ω2 (Y )F3 (Ei , X)]
= −hdf, ω3 iF2 (X, Y ) − (df ◦ J2 )(Y )ω3 (X) + (df ◦ J2 )(X)ω3 (Y )
+ hdf, ω2 iF3 (X, Y ) + (df ◦ J3 )(Y )ω2 (X) − (df ◦ J3 )(X)ω2 (Y )
Hence
δd(f F )(X, Y ) = (δdf )F (X, Y ) − hdf, ω3 iF2 (X, Y ) + hdf, ω2 iF3 (X, Y )
+ (∇JX df )(Y ) − (∇JY df )(X) − df (X)(θ ◦ J)(Y ) + df (Y )(θ ◦ J)(X)
− hdf, ω3 iF2 (X, Y ) − (df ◦ J2 )(Y )ω3 (X) + (df ◦ J2 )(X)ω3 (Y )
+ hdf, ω2 iF3 (X, Y ) + (df ◦ J3 )(Y )ω2 (X) − (df ◦ J3 )(X)ω2 (Y )
+ f δdF (X, Y )
(5.7)
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
27
¿From which we deduce that
4σ(X, Y ) ≡ 4(f F )(X, Y )
= (∇X df )(JY ) − ∇Y (df )(JX) + (∇JX df )(Y ) − (∇JY df )(X)
µ
¶
µ
¶
+ df ◦ ω3 (X)J2 Y − ω2 (X)J3 Y − df ◦ ω3 (Y )J2 X − ω2 (Y )J3 X
− df (J2 Y )ω3 (X) + df (J2 X)ω3 (Y ) + df (J3 Y )ω2 (X) − df (J3 X)ω2 (Y )
− hdf, ω3 iF2 (X, Y ) + hdf, ω2 iF3 (X, Y )
+ 4f F (X, Y ) + f 4 F (X, Y )
that is
4(f F )(X, Y ) = 4f F (X, Y ) + f 4 F (X, Y )
+ (∇X df )(JY ) − (∇Y df )(JX) + (∇JX df )(Y ) − (∇JY df )(X)
− hdf, ω3 iF2 (X, Y ) + hdf, ω2 iF3 (X, Y )
that is
4(f F ) = (4f )F + f 4 F − hdf, ω3 iF2 + hdf, ω2 iF3
(5.8)
and hence, by combining with formula (5.1), we have the following
Proposition 5.3. For any self-dual 2-form σ = f F where F is the locally defined
Kähler form of a compatible J one has
4(f F ) = [f (4ν + kω2 k2 + kω3 k2 ) + 4f ]F
[f (δω3 − hω1 , ω2 i) − hdf, ω3 i]F2
(5.9)
+ [−f (δω2 + hω1 , ω3 i) + hdf, ω2 i]F3
We conclude this paragraph by showing that for Laplacian of self-dual 2-forms
associated to integrable almost complex structures, on a not necessarily compact M ,
one has the following
Proposition 5.4. Let assume that the compatible almost complex structure J on M
is integrable. Then one has
1
4F = (4ν + kθk2 )F
2
and
∇∗ ∇F =
1
kθk2 F
2
(5.10)
(5.11)
Proof. It is an immediate consequence of previous identities (5.1) and (5.2) and of
the following Lemma. ¤
28
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Lemma 5.5. For J1 integrable the following identities hold:
δω2 + < ω1 , ω3 >= 0
(5.12)
δω3 − < ω1 , ω2 >= 0
(5.13)
Proof. We know that:
(dωα + ωβ ∧ ωγ )(·, Jα ·) = −νg
(α = 1, 2, 3)
Hence, in particular, the 2-forms at left member are Q-Hermitian and for α = 3 one
has the identities
(∇X ω3 )(J3 Y ) − (∇J3 Y ω3 )(X) + ω1 (X)ω2 (J3 Y ) − ω1 (J3 Y )ω2 (X)
= (∇Jρ X ω3 )(J3 Jρ Y ) − (∇J3 Jρ Y ω3 )(Jρ X)
(ρ = 1, 2, 3)
+ ω1 (Jρ X)ω2 (J3 Jρ Y ) − ω1 (J3 Jρ Y )ω2 (Jρ X)
and, by taking into account the identity
∇X ω3 = −(∇X ω2 ) ◦ J1 + ω2 (X)(ω2 ◦ J3 ) + ω2 (J1 X)(ω2 ◦ J2 )
(5.14)
which is obtained by differentiating the identity ω2 ◦ J2 = ω3 ◦ J3 , it results
(∇X ω2 )(J2 Y ) − ω2 (X)ω2 (Y ) + ω2 (J1 X)ω2 (J1 Y )
+ (∇J3 Y ω2 )(J1 X) − ω2 (J3 Y )ω2 (J3 X) + ω2 (J2 Y )ω2 (J2 X)
+ ω1 (X)ω2 (J3 Y ) − ω1 (J3 Y )ω2 (X)
= (∇Jρ X ω2 )(J2 Jρ Y ) − ω2 (Jρ X)ω2 (Jρ Y ) + ω2 (J1 Jρ X)ω2 (J1 Jρ Y )
+ (∇J3 Jρ Y ω2 )(J1 Jρ X) − ω2 (J3 Jρ Y )ω2 (J3 Jρ X) + ω2 (J2 Jρ Y )ω2 (J2 Jρ X)
+ ω1 (Jρ X)ω2 (J3 Jρ Y ) − ω1 (J3 Jρ Y )ω2 (Jρ X)
(ρ = 1, 2, 3)
By choosing ρ = 1 and by substituting J2 Y to Y we get the identity
− (∇X ω2 )(Y ) − (∇J1 Y ω2 )(J1 X) − (∇J1 X ω2 )(J1 Y ) − (∇Y ω2 )(X)
= ω2 (X)ω2 (J2 Y ) − ω2 (J1 X)ω2 (J3 Y ) + ω1 (X)ω2 (J1 Y ) − ω1 (J1 Y )ω2 (X)
− ω2 (J1 Y )ω2 (J3 X) + ω2 (Y )ω2 (J2 X)
− ω2 (J1 X)ω2 (J3 Y ) + ω2 (X)ω2 (J2 Y ) + ω2 (Y )ω2 (J2 X)
− ω2 (J1 Y )ω2 (J3 X) − ω1 (J1 X)ω2 (Y ) + ω1 (Y )ω2 (J1 X)
By contraction with respect to g one obtains
that is
4δω2 = − < ω1 , ω3 > − < ω1 , ω3 > − < ω1 , ω3 > − < ω1 , ω3 >
δω2 + < ω1 , ω3 >= 0
By interchanging J3 and J2 one gets also
δω3 − < ω1 , ω2 >= 0
(It is sufficient to repeat the computations for the admissible basis H 0 = (Jα 0 ) where
(J1 0 = J1 , J2 0 = J3 , J3 0 = −J2 ) and to take into account that ω1 0 = ω1 , ω2 0 = ω3 ,
ω3 0 = −ω2 . ¤
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
29
6. Action of the real and quaternionic
Weyl tensors on self-dual forms
Let (M 4n , g, Q), n > 1, be a quaternion-Kähler manifold with curvature tensor R.
We have the formulas (see for example [AM2])
(
K
R = νRHP + W Q
;
ν = 4n(n+2)
(6.1)
K
R = 4n(4n−1)
RS(1) + W R
where RS(1) is the curvature tensor of the 4n-dimensional sphere of constant curvature
1, RHP is the curvature tensor of the metric of quaternionic curvature 1 for the ndimensional quaternionic projective space HPn . We recall that for any vector fields
X, Y, Z, T ∈ M 4n
RS(1) (X, Y, Z, T ) = g(X, T )g(Y, Z) − g(X, Z)g(Y, T )
and
X
1
g(X, Jα Y )g(Jα Z, T )
RHP (X, Y, Z, T ) = [g(X, T )g(Y, Z) − g(X, Z)g(Y, T ) + 2
4
α
X
X
g(X, Jα Z)g(Jα Y, T )]
g(Y, Jα Z)g(Jα X, T ) +
−
α
α
(6.2)
W R is the Weyl curvature tensor, W Q is the quaternionic Weyl curvature tensor.
Hence
K
K
WR =
RHP + W Q −
RS(1)
(6.3)
4n(n + 2)
4n(4n − 1)
By defining the action of curvature tensor on a 2-form F by
R(F )hk = −
1X
R(Ei , Ej , Eh , Ek )F (Ei , Ej )
2 i,j
(6.4)
one has the following results.
Proposition 6.1. Let F be a self-dual 2-form on the quaternion-Kähler manifold
(M, g, Q), n > 1. Then one has
W Q (F ) = 0
,
RS(1) (F ) = F
,
RHP (F ) = nF
(6.5)
and
W R (F ) = cF
(6.6)
where c is constant,
c={
K (n − 1)(2n + 1)
[
]}
2n (n + 2)(4n − 1)
(6.7)
30
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Proof. Without any loss of generality we can assume that F is the Kähler form of a
(local) compatible almost complex structure J on (M, g, Q). We have
X
−2W Q (F )hk =
< W Q (Ei , Ej )Eh , Ek > g(IEi , Ej )
i,j
=
X
< W Q (Eh , Ek )Ei , Ej > g(IEi , Ej )
i,j
=
X
i
=−
< W Q (Eh , Ek )Ei , JEi >
X
< JW Q (Eh , Ek )Ei , Ei >
i
= T race[JW Q (Eh , Ek )] = 0
as it was proved in [AM2]. We have
X
−2RS(1) (F )hk =
[g(Ei , Ek )g(Ej , Eh ) − g(Ei , Eh )g(Ej , Ek )]g(JEi , Ej )
i,j
X
=
[g(Ei , Ek )g(JEi , Eh ) − g(Ei , Eh )g(JEi , Ek )]
i
= 2g(Eh , JEk )
= −2Fhk
We have
−2RHP (F )hk =
1
[−2RS(1) (F )hk ]
4
X
1 X
g(Ej , Jα Eh )g(Jα Ei , Ek )
+ [2
g(Ei , Jα Ej )g(Jα Eh , Ek ) −
4 i,j;α
i,j;α
X
+
g(Ei , Jα Eh )g(Jα Ej , Ek )]g(JEi , Ej )
i,j;α
X
1
= [−2F (Eh , Ek ) − 2
g(JEj , Jα Ej )g(Jα Eh , Ek )
4
j;α
X
X
g(JEj , Jα Eh )g(Jα Ej , Ek )]
g(JEi , Jα Eh )g(Jα Ei , Ek ) −
−
j;α
i;α
1
= [−2F (Eh , Ek ) − 8ng(JEh , Ek )
4
X
X
−
g(Jα Ek , JJα Eh ) −
g(Jα Ek , JJα Eh )]
α
α
1
= [−2F (Eh , Ek ) − 8nF (Eh , Ek ) + 2F (Eh , Ek )]
4
= −2nF (Eh , Ek )
ALMOST COMPLEX STRUCTURES ON QUATERNION-K ÄHLER MANIFOLDS
31
Hence,
K
K
· (n) + 0 −
· 1]F
4n(n + 2)
4n(4n − 1)
K (n − 1)(2n + 1)
[
]F
=
2n (n + 2)(4n − 1)
W R (F ) = [
¤
Remark 6.2. From (6.6) it is possible to show how the formulas (2.11), (4.2) and
(5.3) could be obtained respectively from formulas (12), (26) and (7) of [G1]. Our
formulas agree with Gauduchon formulas, by taking into account that for the scalar
product between 2-forms (·, ·) used in [G1] one has
(F, F ) =
1
kF k2
2
M
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[GP]
[Gr]
[H]
[K]
[LS]
[MS]
[M]
[P1]
[P2]
[P3]
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[S2]
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[T1]
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Gen. Antonova 2 - 99, 117279 Moscow, Russian Federation
E-mail address: [email protected]
Dipartimento di Matematica, Universit à di Roma “La Sapienza”, P.le A. Moro 2,
00185 Roma, Italy
E-mail address: [email protected]
Dipartimento di Matematica, Universit à di Roma Tre, Via C. Segre 2, 00146 Roma,
Italy
E-mail address: [email protected]

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