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arXiv:1606.07701v1 [math.DG] 24 Jun 2016
Holonomy classification of Lorentz-K¨ahler manifolds
Anton S. Galaev
Abstract. The classification problem for holonomy of pseudo-Riemannian manifolds is actual
and open. In the present paper, holonomy algebras of Lorentz-K¨ahler manifolds are classified. A
simple construction of a metric for each holonomy algebra is given. Complex Walker coordinates
are introduced. Complex pp-waves are characterized in terms of the curvature, holonomy and
the potential. Classification of Lorentz-K¨ahler symmetric spaces is reviewed.
Keywords: Lorentz-K¨ahler manifold; holonomy group; complex Walker coordinates; complex
pp-wave; symmetric space; space of oriented lines
2010 Mathematical Subject Codes: 53C29, 53C25, 53C35, 53C50, 53C55
Contents
1. Introduction 1
2. Preliminaries 3
3. The Classification Theorem 4
4. Weakly irreducible subalgebras in u(1, n + 1) 5
5. Algebraic curvature tensors and Berger algebras 8
6. Lorentz-K¨ahler metrics 11
7. Example: the space of oriented lines in R318
8. Complex pp-waves 18
9. Lorentz-K¨ahler symmetric spaces 19
References 20
1. Introduction
The holonomy group of a pseudo-Riemannian manifold (M, g) is an important invariant that
gives rich information about the geometry of (M, g). This motivates the classification problem
for holonomy groups of pseudo-Riemannian manifolds. The problem is solved only for connected
holonomy groups of Riemannian and Lorentzian manifolds. The case of Riemannian manifolds
is classical and it has many applications in differential geometry and theoretical physics, see e.g.
[9, 14, 33]. The case of Lorentzian manifolds took the attention of geometers and theoretical
physicists during the last two decades, see the reviews [3, 24, 25] and the references therein.
In the other signatures, only partial results are known [12, 13, 7, 23, 27, 28, 34]. The main
different between holonomy groups of Riemannian and proper pseudo-Riemannian manifolds is
that for Riemannian manifolds all considerations may be reduced to the case of irreducible holo-
nomy groups, while in the case of proper pseudo-Riemannian manifolds one should consider also
holonomy groups preserving degenerate subspaces. Berger [10] classified connected irreducible
holonomy groups of pseudo-Riemannian manifolds of arbitrary signature. Note that there is a
more general classification of connected irreducible holonomy groups of torsion-free affine connec-
tions [39].
1
2 ANTON S. GALAEV
The general case of the signature (2, N ), i.e. the next signature after the Lorentzian one,
is quite complicated, this show the works [34, 27]. In the present paper we give a complete
classification of the connected holonomy groups (equivalently, of holonomy algebras) for pseudo-
K¨ahler manifolds of complex index one; such manifolds are called Lorentz-K¨ahler and they have
holonomy algebras contained in u(1, n + 1) ⊂so(2,2n+ 2) (n+ 1 is the complex dimension of the
manifold). These results were obtained in much more complicated way in the thesis [27].
We do the same three steps that were previously done for the holonomy of Lorentzian manifolds
[8, 36, 26]. The Wu theorem allows to assume that the holonomy algebra is weakly irreducible
(this means that it does not preserve any proper non-degenerate subspace of the tangent space).
Since the case of irreducible holonomy algebras is solved by Berger, we assume that the holonomy
algebra is not irreducible, and then (with one exception in dimension four) it preserves a complex
isotropic line. In Section 4 we classify weakly irreducible subalgebras in u(1, n + 1). For that we
consider the following geometrical approach: we consider the induced action of the corresponding
connected Lie subgroup G⊂U(1, n + 1) on the boundary of the complex hyperbolic space, which
may be identified with the sphere. This action induces an action of Gon the Heisenberg space
and on the Hermitian space Cn. Then Gpreserves a real affine subspace L⊂Cn, where it acts
transitively as the group of similarity transformations. This allows to find all groups G. Using this,
in Section 5 we classify weakly irreducible Berger subalgebras in u(1, n +1), i.e. weakly irreducible
subalgebras in u(1, n + 1) having the same algebraic properties as the holonomy algebras. Berger
algebras play the role of the candidates to the holonomy algebras.
Then appears the most important problem to check, which of these candidates are really
holonomy algebras. Note that in the case of Riemannian manifolds the same problem was open
for more than thirty years, and finally it was solved by Bryant [15]. In Section 6 we show that
some of the obtained by us Berger algebras can not appear as the holonomy algebras (note that
all previously known Berger algebras were always holonomy algebras), and we construct metrics
realizing all the other Berger algebras as holonomy algebras. We obtain the metrics by writing
down explicitly their potentials, which are relatively simple. In [27] were written down very
complicated metrics in real coordinates with the same holonomy algebras as here.
In Section 6 we also introduce the complex version of the Walker coordinates, which are
frequently used in Lorentzian geometry and Relativity [16].
Then we consider several examples. In Section 7 we find the holonomy algebra of the space of
oriented lines in R3, which admits the structure of the Lorentz-K¨ahler manifold of real dimension
four; this space is used in geometric optics, see the survey [30] and the references therein. There
are generalizations of this space giving farther examples of Lorentz-K¨ahler metrics [30, 31, 32].
The pp-wave metrics are very useful in theoretical physics. In Section 8 we consider their
generalization to the complex case. We give equivalent conditions in terms of the curvature,
holonomy and the potential for a complex Walker metric to be a complex pp-waves. Complex
pp-waves were studied in [38], where appear also examples of Lorentz-K¨ahler metrics and their
holonomy. Complex pp-waves are used also in the physical literature, e.g. [20, 41].
Simply connected pseudo-Riemannian symmetric spaces of index 2 were classified in [35], and
the case of Lorentz-K¨ahler manifolds was considered separately. In Section 9 we reformulate the
last result it terms of the curvature and holonomy.
Let us mention the works and situations, where Lorentz-K¨ahler manifolds and their holonomy
algebras appear. There are various examples of invariant Lorentz-K¨ahler metrics on Lie groups
and homogeneous spaces, e.g. [6, 17, 18, 40, 43]. In some cases the holonomy groups are com-
puted. Lorentz-K¨ahler metrics and their holonomy are related to conformal geometry, conformal
holonomy, and CR-structures through the ambient metric construction, see e.g. [5, 19, 22, 37],
in particular, to the conformal analog of Calabi-Yau manifolds [4].
Acknowledgements. The author is grateful to Helga Baum and Dmitri V. Alekseevsky for
useful discussions and support.
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 3
2. Preliminaries
The theory of the holonomy groups of pseudo-Riemannian manifolds can be found e.g. in
[9, 33, 23]. Let (M , g) be a pseudo-Riemannian manifold of signature (r, s). The holonomy
group Gxof (M, g) at a point x∈Mis the Lie group that consists of the pseudo-orthogonal
transformations of the tangent space given by the parallel transports along all piecewise smooth
loops at the point x. This group may be identified with a Lie subgroup of the pseudo-orthogonal
group O(r, s). The corresponding subalgebra of the pseudo-orthogonal Lie algebra so(r, s) is called
the holonomy algebra and it determines the holonomy group if the manifold is simply connected.
The Ambrose-Singer Theorem states that the holonomy algebra is spanned by the endomorphisms
τ−1
γ◦Ry(τγX, τγY)◦τγof TxM, where γis a piecewise smooth curve starting at the point x
with an end-point y∈M, and X, Y ∈TxM. This implies that the holonomy algebra satisfies a
strong algebraic condition being spanned by images of the algebraic curvature tensors; algebras
satisfying this property are called Berger algebras (see below), and they are candidates to the
holonomy algebras.
The fundamental principle for holonomy groups states that there exists a one-to-one corre-
spondents between parallel tensor fields Ton (M , g) and tensors T0of the same type at xpreserved
by the tensor extension of the representation of the holonomy group.
A subgroup (resp. subalgebra) of the pseudo-orthogonal Lie group (resp. algebra) is called
weakly irreducible if it does not preserve any non-degenerate proper subspace of the pseudo-
Euclidean space). The Wu decomposition theorem implies that if the holonomy algebra of a
pseudo-Riemannian manifold is not weakly irreducible, then the manifold is locally decomposable,
and the holonomy algebra is the direct sum of weakly irreducible holonomy algebras. This allows
us to assume that the holonomy algebra is weakly irreducible. Irreducible holonomy algebras of
pseudo-Riemannian manifolds were classified by Berger [10], so we are left with weakly irreducible
not irreducible holonomy algebras.
A pseudo-K¨ahler manifold is a pseudo-Riemannian manifold (M, g) with a g-orthogonal com-
plex structure Jthat is parallel. By the fundamental principle, the equivalent condition is that
the holonomy group of (M , g) is contained in the pseudo-unitary group U(r
2,s
2) (the numbers r
and smust be even). Together with J,Mbecomes a complex manifold; the tensor fields gand
Jdefine a pseudo-K¨ahler metric hon Mof complex signature ( r
2,s
2). We are interested in holo-
nomy groups of pseudo-K¨ahler manifold of real signature (2,2N), i.e. of complex signature (1, N ).
Such manifolds are called Lorentz-K¨ahler manifolds. Let N=n+ 1, n≥0. The tangent space
of a Lorentz-K¨ahler manifold may be identified with the pseudo-Hermitian space C1,n+1 with a
pseudo-Hermition metric h. From the Berger classification it follows that the only irreducible
holonomy algebras of Lorentz-K¨ahler manifolds are u(1, n + 1) and su(1, n + 1). In fact, these Lie
algebras exhaust irreducible subalgebras in u(1, n + 1) [21].
Fix a Witt basis p, e1,...,en, q of C1,n+1, i.e. the non-zero values of the metric hare the
following: h(p, q) = h(ej, ej) = 1. In particular, the vectors pand qare isotropic, and the vectors
e1,...,enform a basis of the Hermitian space, which we denote by Cn. We will be interested in
the subalgebra u(1, n + 1)Cpof u(1, n + 1) preserving the complex isotropic line Cp. In the matrix
form we have
u(1, n + 1)Cp=
a−¯
Ztic
0A Z
0 0 −¯a
a∈C, A ∈u(n),
Z∈Cn
c∈R
.
We will denote the above matrix by the 4-tuple (a, A, Z, c). For the Lie brackets we have
[(a, A, 0,0),(b, B, Z, c)] = (0,[A, B]u(n),¯aZ+AZ, 2ic Re a),[(0,0, Z, 0),(0,0, V , c)] = (0,0,0,2iIm h(Z, V )).
We obtain the decomposition
u(1, n + 1)Cp= (C⊕u(n)) ⋉(Cn⋉iR).
Note that the subalgebra Cn⋉iR⊂u(1, n + 1)Cpis isomorphic to the Heisenberg Lie algebra,
and the subalgebra su(1, n + 1)Cp⊂u(1, n + 1)Cpis isomorphic to the Lie algebra of the similarity
transformations of the Heisenberg space. We will denote the complex structure on C1,n+1 by J.
4 ANTON S. GALAEV
We will consider real vector subspaces of Cnof the form
L=Cm⊕L0,
where 0 ≤m≤n, we fix an h-orthogonal decomposition Cn=Cm⊕Cn−m, and L0⊂Cn−mis a
real form, i.e. iL0∩L0= 0, L0⊕iL0=Cn−m. An example of L0is
Rn−m= spanR{em+1,...,en}.
Note that in general L0is different from this one, since L0may not contain an h-orthonormal
basis of Cn−m.
For the case n= 0 let us consider the subalgebra g0⊂su(1,1) defined in the following way.
Consider the Witt basis p1, p2, q1, q2in R2,2and define the pseudo-Hermitian structure Jsuch
that Jp1=q2,Jp2=−q1. Let
g0= A0
0−At
A∈sl(2,R)⊂so(2,2).
The Lie algebra g0commutes with J, hence it is contained in u(1,1). Moreover, g0⊂su(1,1).
The Lie algebra g0preserves the vector subspace spanR{p1, p2}, which is not J-invariant.
3. The Classification Theorem
Theorem 3.1.1) A subalgebra g⊂u(1,1) is the weakly irreducible not irreducible holonomy
algebra of a Lorentz-K¨ahler manifold of complex dimension 2 if and only if gis conjugated to g0,
or gis conjugated to one of the following subalgebras of u(1,1)Cp:
•g1=u(1,1)Cp;
•g2= a0
0−¯a
a∈C;
•gγ
3= rγ ic
0−r¯γ
r, c ∈R,where γ∈Cis a fixed number.
2) Let n≥1. Then a subalgebra g⊂u(1, n + 1) is the weakly irreducible not irreducible
holonomy algebra of a Lorentz-K¨ahler manifold of complex dimension n+ 2 if and only if gis
conjugated to one the following subalgebras of u(1, n + 1)Cp:
•gk=k⋉(Cn⋉iR)
=
a−¯
Ztic
0A Z
0 0 −¯a
a+A∈k,
Z∈Cn
c∈R
,
where
k⊂C⊕u(n)
is an arbitrary subalgebra;
•gk,J,L =k⋉(L⋉iR)
=
a2i−¯
Zt−¯
Xtic
0A0Z
0 0 a2iEn−mX
0 0 0 a2i
a2(i+iidCn−m) + A∈k,
Z∈Cm, X ∈Rn−m,
c∈R
,
where L=Cm⊕Rn−m,0≤m < n,
k⊂RJ⊕u(m)
is an arbitrary subalgebra not contained in u(m);
•gk,L =k⋉(L⋉iR)
=
0−¯
Zt−¯
Xtic
0A0Z
0 0 0 X
0 0 0 0
A∈k,
Z∈Cm, X ∈L0,
c∈R
,
where L=Cm⊕L0,0≤m < n,L0⊂Cn−mis a real form,
k⊂u(m)
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 5
is an arbitrary subalgebra;
•gk0,ψ =k0⊕ {(0, ψ(X), X, 0)|X∈Cm−r⊕L0}⋉(Cr⋉iR)
=
0−¯
Zt−¯
Xtic
0A+ψ(X) 0 Z
0 0 0 X
0 0 0 0
A∈k0,
Z∈Cr, X ∈Cm−r⊕L0,
c∈R
,
where 1≤r≤m≤n,L=Cm⊕L0,L0⊂Cn−mis a real form, we consider the
decomposition
Cm=Cr⊕Cm−r,
k0⊂u(r)is an arbitrary subalgebra,
ψ:Cm−r⊕L0→u(r)
is a non-zero linear map such that the image ψ(Cm−r⊕L0)⊂u(r)is commutative, it
commutes with k0and has trivial intersection with k0.
Recall that, by definition, a special pseudo-K¨ahler manifold (or Calabi-Yau manifold of in-
definite signature) is just a Ricci-flat pseudo-K¨ahler manifold. Special pseudo-K¨ahler manifolds
are precisely pseudo-K¨ahler manifold with holonomy algebras contained in the special unitary Lie
algebra. We imminently get the following corollary.
Corollary 3.1.Weakly irreducible not irreducible holonomy algebras of Ricci-flat Lorentz-
K¨ahler manifolds are exhausted by the following algebras:
g0;
gγ
3with γ∈R;
gkwith k⊂R⊕R(ni −2 idCn)⊕su(n),
gk,J,L with k⊂R(mi −(2 + n−m)iidCm+mi idCn−m)⊕su(m);
gk,L with k⊂su(m);
gk0,ψ with k0⊕ψ(Cm−r⊕L0)⊂su(r).
Proof of Theorem 3.1.
Suppose that g⊂u(1, n + 1) is the weakly irreducible not irreducible holonomy algebra
of a Lorentz-K¨ahler manifold. Then gpreserves a proper real subspace W⊂R2,2n+2 =C1,n+1 .
Consequently gpreserves the orthogonal complement W⊥⊂R2,2n+2 and the intersection W∩W⊥,
which is isotropic and must be of real dimension 1 or 2. Let W1= (W∩W⊥)⊥.
Suppose that n= 0. If W1∩JW16= 0, then gis conjugated to a subalgebra of u(1,1)Cp. It is
not hard to write down all subalgebras of u(1,1)Cpand decide which of them are weakly irreducible
Berger subalgebras in u(1, n + 1)Cp, i.e. weakly irreducible subalgebras in u(1, n + 1) having the
same algebraic properties as the holonomy algebras. In Section 6 we construct metrics with each
of the obtained algebra being the holonomy algebra. If W1∩J W1= 0,then it is not hard to see
that gis contained in g0. Holonomy algebras of pseudo-Riemannian manifolds of signature (2,2)
are well studied [7, 28], by this reason we do not pay much attention to the case n= 0.
Suppose that n > 0. Then W2=W1∩J W1is non-trivial, g- and J-invariant, and degenerate.
Thus, gpreserves the complex isotropic line W2∩W⊥
2. Hence gis conjugated to a subalgebra of
u(1, n + 1)Cp.
The next three sections will be dedicated to the rest of the proof of Theorem 3.1. In Section
4 we will describe weakly irreducible subalgebras in u(1, n + 1)Cp. Using this, in Section 5 we
classify weakly irreducible Berger subalgebras in u(1, n + 1)Cp. In Section 6 we show that some of
the obtained Berger algebras can not appear as the holonomy algebras, and we construct metrics
realizing all the other Berger algebras as holonomy algebras. This will prove Theorem 3.1.
4. Weakly irreducible subalgebras in u(1, n + 1)
Let U(1, n + 1)Cpbe the Lie subgroup of U(1, n + 1) preserving the line Cp. This group is
generated by the elements of the form:
6 ANTON S. GALAEV
ea0 0
0A0
0 0 e−¯a
,
1−¯
Zt−1
2¯
ZtZ+ic
0EnZ
0 0 1
,
where a∈C,A∈U(n), Z∈Cn, and c∈R.
Consider the group
Sim Cn= (R∗·U(n)) ⋌Cn
of similarity transformations of the Hermitian space Cnand the groups homomorphism
Γ : U(1, n + 1)Cp→Sim Cn
that sends the above elements of U(1, n + 1)Cpto e¯a·A∈R∗·U(n), and Z∈Cn, respectively.
The homomorphism Γ can be constructed in the following geometrical way. The group U(1, n +
1) acts in the natural way on the boundary ∂Hn+1 of the complex hyperbolic space, which consists
of complex isotropic lines in C1,n+1 [29]. The boundary may be identified with the sphere S2n+1.
Since the group U(1, n + 1)Cppreserves the point {Cp} ∈ ∂Hn+1 , it acts on the sphere with one
removed point; this space may be identified with the Heisenberg space
Cn⊕R,
and the action there is given by the Heisenberg similarity transformations. Then we may consider
the induced action on Cn, which will give us the homomorphism Γ.
The differential
Γ′:u(1, n + 1)Cp→simCn= (R⊕u(n)) ⋉ Cn
is given by
(a, A, Z, c)7→ (Re a, −iIm aidCn+A, Z).
The kernel of Γ′equals to RJ⊕iR⊂u(1, n + 1)Cp.
Suppose that G⊂U(1, n + 1)Cpis a weakly irreducible Lie subgroup in the sense that it does
not preserve any non-degenerate complex subspace in C1,n+1.
Proposition 4.1.The subgroup Γ(G)⊂Sim Cndoes not preserve any proper complex affine
subspace of Cn. Consequently, if Γ(G)⊂Sim Cnpreserves a proper real affine subspace L⊂Cn,
then the minimal complex affine subspace of Cncontaining Lis Cn.
Proof. First we prove that Γ(G)⊂Sim Cndoes not preserve any proper complex vector
subspace of Cn. Suppose that Γ(G) preserves a proper complex vector subspace L⊂Cn. Then,
Γ(G)⊂(R∗·U(L)·U(L⊥)) ⋌L,
where L⊥is the orthogonal complement to Lin Cn. We see that the group Gpreserves the proper
non-degenerate vector subspace L⊥⊂C1,n+1 .
Suppose that Γ(G)⊂Sim Cnpreserves a proper complex affine subspace L⊂Cn. The
subgroup group G⊂U(1, n + 1)Cpis conjugated to a subgroup G1⊂U(1, n + 1)Cpsuch that
Γ(G1)⊂Sim Cnpreserves a proper complex vector subspace corresponding to the affine subspace
L⊂Cn. This gives a contradiction, since it is clear that G1⊂U(1, n + 1)Cpis weakly irreducible.
This proves the proposition.
Let g⊂u(1, n + 1)Cpbe a weakly irreducible subalgebra. Let G⊂U(1, n + 1)Cpbe the
corresponding connected Lie subgroup. Suppose that Γ(G) preserves a real affine subspace L⊂Cn.
Considering, if necessary, another group G1conjugated to G, we may assume that L⊂Cnis a
real vector subspace. We may assume also that Lis minimal in the sense that Γ(G) does not
preserve any proper affine subspace in L. The induced action of Γ(G) on the Euclidean space L
is by similarity transformations. Since Γ(G) does not preserve any proper affine subspace of L, it
acts transitively on L[2]. Thus,
Γ(G)⊂(O(L⊥)·Sim L)∩Sim Cn= (R∗·((O(L⊥)·O(L)) ∩U(n))) ⋌L,
where the orthogonal complement is taken with respect to the Euclidean metric Re h, and
Sim L= (R∗·O(L)) ⋌L
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 7
is the Lie group of similarity transformations of the Euclidean space L. Moreover, the projection
of Γ(G) on Sim Lis a transitive group of similarity transformations. The Lie algebras of the
transitive Lie groups of similarity transformations of a Euclidean space Eare exhausted by the
following [2]:
•f⋉L,
•(f0⊕ {ψ(X) + X|X∈U})⋉W,
where f⊂R⊕so(L) is a subalgebra; for the second algebra, there exists an orthogonal decompo-
sition L=W⊕U,f0⊂so(W) is a subalgebra, ψ:U→so(W) is a non-zero linear map such that
f0∩ψ(U) = 0 and ψ(U) is commutative and it commutes with f0.
Let us describe the subspace L⊂Cn. It is clear that iL ∩L⊂Cnis a Hermitian subspace,
let us denote it by Cm. Let Cn−mdenote the orthogonal complement to Cm. We obtain the
decomposition
Cn=Cm⊕Cn−m.
Let L0⊂Lbe the orthogonal complement to Cm⊂Lwith respect to Re h. It is clear that
iL0∩L0= 0, i.e. L0⊂Cn−mis a real form. We obtain the decomposition
L=Cm⊕L0.
Let e1, ..., embe an h-orthonormal basis of Cm. Consider a basis fm+1,...,fnof the real vector
space L0. We may assume that this basis is orthonormal with respect to Re h. There exists a
skew-symmetric real matrix ω= (ωjk )n
j,k=m+1 such that
h(fj, fk) = δjk +iωj k, j, k =m+ 1,...,n.
It is known, that the basis fm+1,...,fncan be chosen in such a way that
ω= diag 0−λ1
λ10,...,0−λs
λs0,0,...,0
for some real numbers λk. Since his positive definite, it holds |λk|<1.
Let τ:Cn−m→Cn−mbe the anti-linear involution defining the real form L0⊂Cn−m. It
holds
τ(fk) = fk, τ(ifk) = −ifk, k =m+ 1,...,n.
Define the endomorphism θof L0by the equation
Reh(θX, Y ) = Imh(X, Y ), X, Y ∈L0
and extend it to a C-linear endomorphism of Cn−m. Clearly, θ∈u(n−m). Note that θ= 0 if
and only if L0contains an h-orthonormal basis of Cn−m.
For k= 1,...,s, let
em+2k−1=√2
2√1−λk
(fm+2k−1+ifm+2k), em+2k=√2
2√1 + λk
(fm+2k+ifm+2k−1).
Then it holds
θ(em+2k−1) = −iλkem+2k−1, θ(em+2k) = iλkem+2k
and
τ(em+2k−1) = −i√1 + λk
√1−λk
em+2k, τ(em+2k) = −i√1−λk
√1 + λk
em+2k−1.
Note that
(so(L⊥)⊕so(L)) ∩u(n) = u(m)⊕zu(n−m)τ,
where zu(n−m)τdenotes the ideal in u(n−m) consisting of elements commuting with τ.
Suppose that the pro jection of Γ′(g) to sim(L) is of the form (f0⊕ {ψ(X) + X|X∈U})⋉W.
Since f=f0⊕ψ(U) annihilates U, and it is contained in u(n), it annihilates also iU . Consequently,
fis contained in (U+iU)⊥. We denote this space by Cr. It is clear that Cr⊂L. This implies
that Cr⊂Cm. Let Cm−rbe the orthogonal complement to Crin Cm. We get the decomposition
L=Cr⊕Cm−r⊕L0
and we extend the map ψto Cm−r⊕L0. Thus we may assume that W=Cr, and U=Cm−r⊕L0.
8 ANTON S. GALAEV
We conclude that if g⊂u(1, n + 1)Cpis a weakly irreducible subalgebra with the associated
subspace L=Cm⊕L0⊂Cn, then Γ′(g) is one of the following:
•f⋉L,
•(f0⊕ {ψ(X) + X|X∈Cm−r⊕L0})⋉ Cr,
where f⊂R⊕u(m)⊕zu(n−m)τis a subalgebra; f0⊂u(r), ψ:Cm−r⊕L0→u(r) is a non-zero
linear map such that f0∩ψ(Cm−r⊕L0) = 0 and ψ(Cm−r⊕L0) is commutative and it commutes
with f0.
Let g0be the projection of gto su(1, n + 1). Note that if m > 0, then from the structure of
the Lie brackets in u(1, n + 1)Cpit follows that g0contains the ideal iR⊂u(1, n + 1)Cp. Even
more, below we will see that all holonomy algebras g⊂u(1, n + 1)Cpcontain this ideal, thus it is
enough to consider only such subalgebras. Then, since the kernel of Γ′|su(1,n+1)Cpcoincides with
iR, we immediately find that
g0= (Γ′|su(1,n+1)Cp)−1(Γ′(g)).
Note that
(Γ′|su(1,n+1)Cp)−1(iidCm) = i
n+ 2
−m0 0 0
0n+ 2 −m0 0
0 0 −m0
0 0 0 −m
+iR.
Now, for the Lie algebra gwe have the following there possibilities:
g=g0,g=g0⊕RJ, g={A+ζ(A)J|A∈g0},
where ζ:g0→Ris a linear map zero on the commutant g′
0= [g0,g0].
Thus we conclude that if g⊂u(1, n + 1)Cpis a weakly irreducible subalgebra containing the
ideal iR⊂u(1, n + 1)Cp, then gis one of the following:
•f⋉(L⋉iR),
•(f0⊕ {ψ(X) + X|X∈Cm−r⊕L0})⋉(Cr⋉iR),
where f⊂R⊕RJ⊕u(m)⊕zu(n−m)τis a subalgebra; f0⊂u(r)⊕RJ,ψ:Cm−r⊕L0→u(r)⊕RJ
is a non-zero linear map such that f0∩ψ(Cm−r⊕L0) = 0 and ψ(Cm−r⊕L0) is commutative and
it commutes with f0.
5. Algebraic curvature tensors and Berger algebras
Let (M, g , J) be a Lorentz-K¨ahler manifold of complex dimension n+ 2 ≥2, i.e. gis a
pseudo-Riemannian metric of signature (2,2n+ 2) on M, and Jis a parallel g-orthogonal complex
structure. Consider the corresponding Lorentz-K¨ahler metric
h=g+ig ◦J.
Fix a point x∈M. Let g⊂u(TxM, hx) be the holonomy algebra of the manifold (M, g ). We
identify the tangent space TxMwith the Lorentz-Hermitian space C1,n+1 .
Consider the complexified tangent bundle TCMand the standard decomposition
TCM=T1,0M⊕T0,1M
into the direct sum of eigenspaces of the C-linear extension of Jcorresponding to the eigenvalues
±i. Recall that sections of T1,0Mand T0,1Mare of the form
X−iJX, X +iJX,
where Xis a vector field on M. In particular, the bundle T1,0Mis identified with T M . The
metric gallows to identify the bundle T0,1Mwith the dual bundle to T1,0M. Moreover, let X, Y
denote both vector fields in T M and the corresponding vector fields in T1,0M, then it holds
(1) g(X, ¯
Y) = 4h(X, Y ),
where gdenotes the C-bilinear extension to TCMof the initial metric g.
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 9
Fix a basis p, e1,...,en, q in TxMas above. Denote by the same symbols p, e1,...,en, q the
basis 1
2(p−iJp),1
2(e1−iJe1),..., 1
2(en−iJen),1
2(q−iJq)∈T1,0
xM.
Then the vectors ¯p, ¯e1,...,¯en,¯qform a basis of T0,1
xM. Suppose that g⊂u(1, n + 1)Cp. The
holonomy algebra of the complex connection ∇coincides with the complexification gC=g⊗Cof
g. The complexification of u(1, n + 1)Cpconsists of the matrices of the form
T10
0T2,
where
T1=
a¯
Wtc
0A Z
0 0 b
, T2=−
b¯
Ztc
0AtW
0 0 a
with a, b, c ∈C,Z, W ∈Cnand A∈gl(n, C). Denote by g1,0the projection of gCto gl(T1,0
xM).
It is clear that this Lie algebra determines gC. This Lie algebra allows us also to determine g,
namely, consider the following anti-linear involution of g1,0:
σ:
a¯
Wtc
0A Z
0 0 b
7→ −
¯
b¯
Zt¯c
0¯
AtW
0 0 ¯a
.
Then gcoincides with the set of fixed points of σ.
The symbol ∇will denote both the Levi-Civita connection on (M , g) and the corresponding
complex connection on TCM. We use the same convection for R. It is well-known that Rsatisfies
the conditions
R(J X, JY ) = R(X, Y ), R(J X, Y ) + R(X, J Y ) = 0
for all vector fields on M. This implies
Lemma 5.1.For all vector fields Z, W ∈T1,0Mit holds
R(Z, W ) = R(¯
Z, ¯
W) = 0, R(Z, ¯
W) = R(¯
Z, W ),
where the conjugation is taken with respect to u(TCM)⊂gl(TCM). In particular,
Rx(Z, ¯
Z)∈ig.
It holds
R(X, Y ) = R(X, ¯
Y)−R(Y, ¯
X),
where on the left hand side Ris the curvature tensor of the connection in T M and X, Y are vector
fields on X, and on the right hand side Ris the curvature tensor of the complexified connection
and X, Y are vector fields from T1,0Mcorresponding to the initial X , Y .
Let now g⊂u(1, n + 1)Cpbe an arbitrary subalgebra. We may carry all the above notations
to this case. Denote C1,n+1 simply by V, and as above we write
VC=V⊗C=V1,0⊕V0,1=V⊕¯
V .
For the Lie algebra gC=g⊗Cconsider the following space of algebraic curvature tensors:
R(gC) = R: Λ2VC→gC
R(X, Y )Z+R(Y , Z)X+R(Z, X )Y= 0,∀X, Y, Z ∈VC
R(X, ¯
Y) = σ(R(¯
X, Y )),∀X, Y ∈V.
We say that g⊂u(1, n+1)Cpis a Berger algebra if gCis generated by the images of the elements R:
Λ2VC→gCfrom the space R(gC). From the Ambrose-Singer theorem and the above consideration
it follows that if g⊂u(1, n + 1)Cpis the holonomy algebra of a Lorentz-K¨ahler manifold, then it
is a Berger algebra.
The proof of the following lemma is direct.
Lemma 5.2.A linear map R: Λ2VC→gCbelongs to R(gC)if and only if it satisfies
R(X, ¯
Y) = σ(R(¯
X, Y )), R(X, ¯
Y)Z=R(Z, ¯
Y)X, ∀X, Y , Z ∈V.
10 ANTON S. GALAEV
For an element ξ∈gCwe denote by ξ1,0its projection onto gl(V1,0) = gl(V). The following
lemma easily follows from the previous one. We denote by X, Y vectors from Cn.
Lemma 5.3.Each algebraic curvature tensor R∈ R(u(1, n + 1)Cp⊗C)is uniquely given by
the equalities
R1,0(p, ¯q) =
α Ntβ
0 0 0
0 0 0
, R1,0(X, ¯q) =
g(N, X )T(X)tg(¯
K, X )
0P(X)AX
0 0 0
,
R1,0(X, ¯
Y) =
0g(P(X)·,¯
Y)g(AX, ¯
Y)
0R0(X, ¯
Y)P(Y)tX
0 0 0
, R1,0(q, ¯q) =
β¯
Ktc
0A K
0 0 ¯
β
for arbitrary elements α, β, ∈C,c∈R,N∈Cn,K∈Cn,T∈ ⊙2Cn,R0∈ R(u(n)C)and
P∈gl(n, C)(1) =⊙2Cn⊗Cn.
Now, for an arbitrary subalgebra g⊂u(1, n + 1)Cpit holds
R(gC) = R(u(1, n + 1)Cp⊗C)∩(Λ2(VC)∗⊗gC).
Recall that we assign to each weakly irreducible subalgebra g⊂u(1, n +1)Cpa real vector subspace
L=Cm⊕L0⊂Cnsuch that L0⊂Cn−mis a real form.
Theorem 5.1.Let n≥1. If g⊂u(1, n + 1)Cpis a weakly irreducible Berger subalgebra then
it is one of the following algebras:
•gk=k⋉(L⋉iR)
=
a1+ia2−¯
Zt−¯
Xtic
0A0Z
0 0 a2(iEn−m+θ)X
0 0 0 −a1+ia2
a1+a2(i+iidCn−m+θ) + A∈k,
Z∈Cm, X ∈L0,
c∈R
,
where
k⊂R⊕R(i, i idCn−m+θ)⊕u(m)
is an arbitrary subalgebra;
•gk,ψ =k⊕ {(0, ψ(X), X, 0)|X∈Cm−r⊕L0}⋉(Cr⋉iR)
=
0−¯
Zt−¯
Xtic
0A+ψ(X) 0 Z
0 0 0 X
0 0 0 0
A∈k,
Z∈Cr, X ∈Cm−r⊕L0,
c∈R
,
where 1≤r≤m, we consider the decomposition
Cm=Cr⊕Cm−r,
k⊂u(r)is an arbitrary subalgebra,
ψ:Cm−r⊕L0→u(r)
is a non-zero linear map such that the image ψ(Cm−r⊕L0)⊂u(r)is commutative, it
commutes with k0and has trivial intersection with k0.
5.1. Proof of Theorem 5.1. We consider the Lie algebras obtained in Section 4 and verify,
which of these Lie algebras are Berger algebras.
Lemma 5.4.If g⊂u(1, n + 1)Cpis a weakly irreducible Berger subalgebra contained in
R⊕R(i+iidCn−m)⊕u(m)⊕zu(n−m)τ⋉(L⋉iR),
then gis contained in
R⊕R(i+iidCn−m+θ)⊕u(m)⋉(L⋉iR).
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 11
Proof. Let R∈ R(gC) be as in Lemma 5.3. It is clear that both R0and Ptake their values
in u(m). Consider the tensor A, which should be of the form A=iA1+iA2+a2idCn−m, where
A1∈u(m), A2∈zu(n−m)τ, and a2∈R. Note that the complexification of the vector subspace
L⊂ghas the form
(2) L⊗C=
0¯
Wt−τ(X)t0
0 0 0 Z
0 0 0 X
0 0 0 0
W, Z ∈Cm, X ∈Cn−m
.
Let X∈iL0, then this and the condition R(X, ¯q)∈gCimply
T(Y, X ) = −g(Y, τ(iA2X+a2X))
for all Y∈Cn. Since X∈iL0and A2preserves iL, it holds τ(iA2X+a2X) = iA2X−a2X.
Using (1) and the fact that Tis symmetric, we obtain
h(Y, iA2X−a2X) = h(X, iA2Y−a2Y)
for all Y∈iL0. This implies
Reh(X, A2Y) = a2Imh(X, Y )
for all X, Y ∈L0, i.e. A2=−a2θ. This proves the lemma.
Lemma 5.5.If g⊂u(1, n + 1)Cpis a weakly irreducible Berger subalgebra, then iR⊂g.
Proof. Let g⊂u(1, n+1)Cpbe a weakly irreducible Berger subalgebra and suppose that iR6⊂ g.
Then it is clear that m= 0, i.e. L=L0. By the previous lemma, prC⊕u(n)g⊂R(i+iidCn−m+θ).
Let R∈ R(gC) be as in Lemma 5.3. From the condition that Rtakes values in gCit follows
immediately that R= 0.
Lemma 5.6.If g⊂u(1, n + 1)Cpis a weakly irreducible Berger subalgebra of the form (f0⊕
{ψ(X) + X|X∈Cm−r⊕L0})⋉(Cr⋉iR), then f0⊕ψ(Cm−r⊕L0)⊂u(r).
Proof. Suppose that prC⊕u(n)gcontains an element of the form i+iidCn−m+A, where A∈
u(r). Since gis a Berger algebra, there exists R∈ R(gC) such that
R1,0(q, ¯q) =
1∗ ∗
0idCn−m−iA ∗
0 0 1
.
Let X∈Cm−r⊕L0. It holds
R1,0(X, ¯q) =
0∗prCm−r⊕L0T(X)t∗
0 0 0 ∗
0 0 0 X
0 0 0 0
.
This implies that X−prCm−r⊕L0T(X)∈g. Considering iX, we get iX +iprCm−r⊕L0T(X)∈g.
The vectors of the last two types span Cm−r⊕L0. We conclude that ψ= 0. This gives a
contradiction.
The lemmas show that if g⊂u(1, n + 1)Cpis a weakly irreducible Berger subalgebra, then g
must be one of the algebras from the statement of the theorem. Conversely it is easy to see that
all algebras from the statement of the theorem are weakly irreducible Berger algebras.
6. Lorentz-K¨ahler metrics
6.1. Complex Walker metrics. Suppose that (M, g ) is a Lorentzian manifold such that its
holonomy group preserves an isotropic line, then (M, g) admits a parallel distribution ℓof isotropic
lines. Locally there exist so called Walker coordinates v, x1, ..., xn, u such that the metric ghas
the form
g= 2dvdu +h+ 2Adu +H(du)2,
12 ANTON S. GALAEV
where h=hij (x1, ..., xn, u)dxidxjis an u-dependent family of Riemannian metrics, A=Ai(x1,...,xn, u)dxi
is an u-dependent family of one-forms, and H=H(v, x1, ..., xn, u) is a local function on M[16].
Let us show that Walker coordinates may be generalized to the case of a Lorentz-K¨ahler
manifold. Suppose that (M , h) is a Lorentzian-K¨ahler manifold such that its holonomy group
preserves a complex isotropic line in the tangent space. Then (M, h) admits a parallel distribution
ℓ⊂T M of complex isotropic lines. It is clear that the corresponding line bundles ℓ⊂T1,0M
and ¯
ℓ⊂T0,1Mare parallel. Moreover, the distribution ℓ⊥⊂T M of complex codimension 1 is
parallel, and the subbundles ℓ⊥⊂T1,0Mand ℓ⊥⊂T0,1Mare parallel as well. It is clear that all
parallel distributions are involutive. From the holomorphic version of the Frobenius Theorem it
follows that locally on Mthere exist local holomorphic coordinates v, z1,...,zn, u such that the
vector field ∂vgenerates the subbundle ℓ⊂T1,0M, and the vector fields ∂v, ∂z1,...,∂zngenerate
the subbundle ℓ⊥⊂T1,0M. Consequently, the metric hcan be written in the following way:
h=h¯uv d¯udv +h¯vu d¯vdu +h¯
jk d¯zjdzk+h¯uk d¯udzk+h¯
kud¯zkdu +h¯uud¯udu.
Since the metric his pseudo-K¨ahler, it holds
(3) ∂ah¯
bc =∂ch¯
ba,
Hence the coefficients of the metric depend on the coordinates in the following way:
h¯vu =h¯vu (¯v, ¯zl, u, ¯u), h¯
jk=h¯
jk(zl,¯zl, u, ¯u),
h¯
ku =h¯
ku(¯v, z l,¯zl, u, ¯u), h¯uu =h¯uu(v, ¯v, zl,¯zl, u, ¯u).
It is easy the check that the inverse matrix, defined by the equality
h¯ab h¯ac =δb
c,
is given by
h¯vv =1
|h¯uv |2h¯
luh¯
lj h¯uj −h¯uu , h¯vu =1
h¯vu
, h¯
jk=˜
h¯
jk, h¯vk =−1
h¯vu
h¯
ju h¯
jk, h¯uk =h¯uu = 0,
where ˜
h¯
jk is the inverse matrix to h¯
jk.
For the obtained metric and a6=vit holds
Γa
vb =h¯ca∂bh¯cv =h¯ua ∂bh¯uv = 0,
i.e. the vector field ∂vis isotropic and recurrent, consequently it defines a local parallel distribution
of complex isotropic lines ℓ⊂T1,0M.
We will consider the frame p, e1,...en, q, ¯p, ¯e1,...¯en,¯q, where
p=1
h¯vu
∂v, ej=Ck
j∂zk−h¯uk
h¯uv
∂v, q =∂u−h¯uu
h¯uv
∂v.
Here Ck
jis a matrix such that ¯
Ck
jh¯
klCl
s=δjl . It is clear that such matrix exists.
6.2. Non-existence results. Here we will show that some of the Berger algebras obtained
above cannot appear as the holonomy algebras of Lorentz-K¨ahler metrics.
Theorem 6.1.Let gkbe the Berger algebra as in Theorem 5.1. If m < n and gkis the
holonomy algebra of a Lorentz-K¨ahler manifold, then k⊂R(i+iidCn−m)⊕u(m). Consequently,
if k6⊂ ⊕u(m), then θ= 0, and L0=Rn−m.
Proof. Consider the Berger algebra gkas in Theorem 5.1. Suppose that m < n and suppose
that gk⊗Cis the complexification of the holonomy algebra of a Lorentz-K¨ahler manifold (M, h)
at a point x∈M. Suppose that k⊗Ccontains an element
ξ=ia1+iA +a2(−id +iθ),
where A∈u(m). Suppose that a16= 0, or a2θ6= 0. Consider the coordinates and a local frame
as in the previous section. Note that the holonomy algebra of the induced connection on the
bundle ℓ⊥/ℓ coincides with pru(n)k; this subbundle and its connection may be identified with
the subbundle of T1,0Mgenerated by the vector fields e1,...,enand the connection ∇restricted
and then projected to that subbundle. Let λbe one of the non-zero eigenvalues of iθ. Let
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 13
Eλx ⊂spanC{em+1,...,en}be the eigenspace corresponding to the eigenvalue −a2(1 −λ) of the
operator ξrestricted to spanC{em+1 ,...,en}. The corresponding subspace in ℓ⊥
x/ℓxis clearly
holonomy-invariant, and we obtain a parallel subbundle Eλ⊂ℓ⊥/ℓ. Denote by the same symbol
Eλthe corresponding subbundle of T1,0M, which is parallel modulo p. The holonomy algebra
gk⊗Cpreserves the space of tensors of the form
¯px∧Z−px∧κ(Z),
where Z∈Eλx and κ(Z) may be found from the condition
h(τ(X), Z) + h(κ(Z), X ) = 0,∀X∈spanC{e1x,...enx}.
The element ξacts in this subspace as the multiplication by a1i+a2λ, and its orthogonal comple-
ment in the holonomy algebra acts in this space trivially. Consequently we get a parallel subbundle
Fλof the bundle of bivectors. Let y∈M, and γbe any curve from xto y. Then Fλy =τγFλx, and
the element τγ◦ξ◦τ−1
γof the holonomy algebra at the point yacts on Fλy as the multiplication
by a1i+a2λ. The same element acts on Cpy∧¯pyas the multiplication by 2a1i. Consequently, Fλ
has trivial projection to the bundle < p ∧¯p > generated by p∧¯p, and Fλconsists of sections of
the form
¯p∧Z−p∧¯
W ,
where Z∈Γ(Eγ) and Wis uniquely defined by Z. For each such section and each vector field V
from TCMmust hold
pr<p∧¯p> ∇V( ¯p∧Z−p∧¯
W) = 0.
Hence,
¯p∧pr<p> ∇VZ−p∧pr<¯p> ∇V¯
W= 0,
i.e.
h(∇VZ, ¯q) + h(∇V¯
W , q) = 0.
Let V=∂za. Since ¯q=∂¯u−h¯uu
h¯v u ∂¯v, it holds
h(∇∂zaZ, ¯q) = −h(Z, ∇∂za¯q) = 0.
Let V=∂¯za. By the similar argument it holds h(∇∂¯za¯
W , q) = 0. Consequently,
h(∇∂¯zaZ, ¯q) = 0.
This shows that the subbundle Eλ⊂T1,0Mis parallel. The corresponding distribution Eλ⊂T M
is also parallel, and it is non-degenerate. This gives a contradiction, since the holonomy algebra
is weakly irreducible. Thus, a1= 0, and a2θ= 0. This proves the theorem.
6.3. Construction of the metrics with all possible holonomy algebras. 1) First con-
sider the complex dimension 2. Metrics with the holonomy algebra g0may be found in [7, 28].
Let a, b ∈Cbe some numbers. We assume that if a= 0, then b= 0. If a=b= 0, we set
fC= ¯vu + ¯uv;
if a6= 0, b = 0, we set
fC(v, ¯v, u, ¯u) = (Re −v
iau e−ia|u|2−1,if u6= 0;
0,if u= 0;
if a, b 6= 0, then let
fC(v, ¯v, u, ¯u) = (Re −v
iau ea2
b√πv
√bu erf a
√b+√b|u|2
2−erf a
√b,if u6= 0;
0,if u= 0;
Recall that the Error function erf satisfies
derfx
dx =2
√πe−x2.
This function appears here since we solve a differential equation for the potential fof the Lorentz-
K¨ahler metric hobtained from a condition on the curvature tensor Rof h. This condition reads
h¯uv =e−ia|u|2−ib
4|u|4.
14 ANTON S. GALAEV
We could avoid the Error function and other differentiable functions defined by conditions, if we
write down the metric without giving its potential, but in this case we would have to carry about
the condition (3).
Let gdenotes the holonomy algebra of a metric given by some potential f
Similarly as in the proof of Theorem 6.2 the following may be shown:
if we chose f=fCwith a=i,b= 1, then g=g1;
if we choose f=fCwith a=γ6= 0, b= 0, then g=gγ
3;
if we choose f= ¯uv + ¯vu +|u|4, then g=g0
3;
finally, the holonomy algebra of the metric h=e¯uv d¯udv +e¯v ud¯vdu coincides with g2.
2) Let n≥1. Denote by v, z1, ..., z n, u the coordinates on Cn+2. Let k⊂C⊕u(n) be a
subalgebra. Let us choose the following elements spanning k:
a+A1, b +A2, A3, ..., AN,
where a, b ∈C,A1, ..., AN∈u(n). We may assume the following: if a= 0, then b= 0; if a6= 0
(resp. b6= 0), then A1(resp. A2) belongs to the center of pru(n)k.
Consider the following Hermitian matrices:
Bα=−1
(α!)2iAα, α = 1, ..., N
and
G=
N
X
α=1
Bα|u|2α.
Define the function
fu(n)=¯
ZTeGZ,
where eGis the matrix exponential and Z=
z1
.
.
.
zn
.
We may suppose that the basis e1,...,enis chosen in such a way that, for some 0 ≤n0≤
m,A1restricted to Cn0= spanC{e1,...,en0}is an isomorphism of Cn0, and it annihilates
spanC{en0,...,en}. Define the following function:
fCm=1
4Re i¯u2
m
X
k=n0+1
(zk)2!.
Define the function
f1=fC+fu(n)+fCn.
Consider the Lie algebra gk,J,L. We assume that a=i,b= 0,
A1=˜
A1+iidCn−m,
where ˜
A1∈u(m), and A2,...,AN∈u(m). Let n0be the number defined by ˜
A1in the same way
as it was defined above by A1. Let
fRn−m(zl,¯zl, u, ¯u) = (−1
2Re Pn
j=m+1 1
¯u21−e|u|2+u
¯ue|u|2,if u6= 0;
0,if u= 0;
Define the function
f2=fC+fu(n)+fCm+fRn−m.
Consider the Lie algebra gk,L. We assume that a=b= 0, A1,...,AN∈u(m). Using the real
form L0⊂Cn−m, we define the matrix
B= (Bkj )n
k,j=m+1 = diag √2
2√1−λ1√1 + λ1
−i√1 + λ1−i√1−λ1,...,√2
2√1−λs√1 + λs
−i√1 + λs−i√1−λs,1,...,1!.
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 15
Let
fL0=−Re
n
X
j=m+1
n−m
X
α=1
iBj m+α¯zj
((N+α)!)2|u|2(N+α)u
N+α+ 1
,
and
f3=fC+fu(n)+fCm+fL0.
Let us turn to the algebra gk0,ψ . Let k=k0⊕ψ(Cm−r⊕L0). Consider the unitary matrices
A1=ψ(er+1), ..., Am−r=ψ(em), Am−r+1 =ψ(ier+1), ..., A2m−2r=ψ(iem),
A2m−2r+1 =ψ(fm+1), ..., An+m−2r=ψ(fn),
where, as above, fm+1,...,fnis a basis of the real vector space L0. Let An+m−2r+2 , ..., ANbe a
basis of k0. Let
D=iEm−r−Em−r0
0 0 iB .
Using this matrix, we define the function
fψ=−1
2Re
n
X
j=r+1
n+m−2r
X
α=1
Dk α(¯zk)2
((α+ 2)!)2|u|2(α+2) u
α+ 3
.
Finally consider the function
f4=fC+fu(n)+fCr+fψ.
Theorem 6.2.The holonomy algebras of the the Lorentz-K¨ahler metrics with the potentials
f1,f2,f3,f4coincide respectively with the Lie algebras gk,gk,J,L,gk,L,gk0,ψ.
6.4. Proof of Theorem 6.2. Let (M, h) be a Lorentz-K¨ahler manifold, let g⊂u(TxM) be
the holonomy algebra of the Levi-Civita connection on (M , h). Recall that the complexification
g⊗C⊂gl(TC
xM) coincides with holonomy algebra of the induced complex connection on TCM,
and it is determined by the projection of g⊗Cto gl(T1,0
xM). The holonomy algebra g⊗Cis
generated by the following elements:
∇Zr···∇Z1Rx(X, ¯
Y), r ≥0, X, Y ∈T1,0
xM, Z1,...Zr∈TC
xM.
Let zabe local complex coordinates on M. We denote by R1,0(∂za, ∂¯zb) the matrix of the field of
the endomorphisms R(∂za, ∂¯zb) restricted to T1,0M. Recall that the Christoffel symbols ΓA
BC are
given by the equality
∇∂zB∂zA= ΓA
CB ∂zC,
where we assume that A, B, C take all values aand ¯a. The possibly non-zero symbols are Γa
bc and
Γ¯a
¯
b¯c=Γa
bc. It holds
Γa
bc =h¯
da∂ch¯
db.
The components of the curvature tensor are defined by the equality
R(∂zC, ∂zD)∂zB=RA
BC D∂zA.
Only the following coefficients may be non-zero:
Ra
bc ¯
d=−Ra
b¯
dc,−R¯a
¯
bd¯c=R¯a
¯
b¯cd =Ra
bc ¯
d,
and it holds
Ra
bc ¯
d=−∂¯zdΓa
bc.
Let for each fixed c, Γcdenote the matrix (Γa
bc). Let ξbe the matrix of a one of the fields
∇∂zAr···∇∂zA1R1,0(∂za, ∂¯zb).
It holds
(4) ∇∂zcξ=∂zcξ+ [Γc, ξ],∇∂¯zcξ=∂¯zcξ.
Let now hbe the metric defined by a potential ffrom the theorem. It holds
h¯uv =e−ia|u|2−ib
4|u|4, h¯
jk =eG¯
jk.
16 ANTON S. GALAEV
We will consider the frame p, e1,...en, q, ¯p, ¯e1,...¯en,¯q, where
p=1
h¯uv
∂v, ej=e−1
2¯
Gkj ∂zk−h¯uk
h¯uv
∂v, q =∂u−h¯uu
h¯uv
∂v.
Using this frame, the curvature tensor of the metric hcan be expressed through some tensor fields
in the same way as in Section 5 (we will use the same notations). Note that
Γv
vv = Γv
vk = Γl
jk = 0.
Consequently, α= 0, N= 0, P= 0, and R0= 0.
Next,
Γk
uu =h¯
lkh¯v u∂uh¯
lu
h¯vu ,
and
K= pr<e1,...en>R(q, ¯q)q= pr<e1,...en>R(∂u, ∂¯u)∂u
= pr<e1,...en>Rk
uu¯u∂zk=
n
X
j=1 G1
2¯
G¯
jkRk
uu¯uej.
Consequently,
K=−
n
X
k,j=1 G1
2¯
G¯
jk ∂¯ue−G¯
lk h¯vu ∂uh¯
lu
h¯vu ej.
Similarly,
T(ej) = −
n
X
k,k1,j1=1 G−1
2¯
G¯
j1jG−1
2¯
G¯
k1kh¯uv ∂¯u1
h¯uv
∂zj1h¯uk1¯ek.
Let g⊂u(1, n + 1) be the holonomy algebra of the metric hat the point 0.
Lemma 6.1.It holds prC⊕u(n)g=k.
Proof.
It is easy to check that
Γv
vu =−ia¯u−ib
2|u|2¯u,
and
Rp
p(q, ¯q) = ia +ib|u|2.
Consequently, a non-trivial value at the point 0 at the position (v, v ) may have only the matrices
of the following covariant derivative of R:
(5) ∇∂u∇∂¯uRv
vu¯u(0) = ∇∂¯u∇∂uRv
vu¯u(0) = bi.
Consider the matrix ˜
Γu= (Γk
ju )n
j,k=1. Let ˜
R(X, ¯
Y) be the similar matrix obtained from R
and vector fields X, Y ∈T1,0M. Let X, Y ∈ {p, e1,...,en, q}, then the last matrix ˜
R(X, ¯
Y) is
non-zero only for X=Y=q. Moreover,
˜
R(q, ¯q) = ˜
R(∂u, ∂¯u).
It holds ˜
Γu=e−G∂ueG.
The formula for the derivation of the matrix exponential reads [42, Sec. 5, Th. 1.2]:
e−G∂ueG=∞
X
k=0
(−1)k
(k+ 1)!(adG)k∂uG=∂uG−1
2![G, ∂uG] + 1
3![G, [G, ∂uG]] − · ·· .
Next, ˜
R(q, ¯q) = −∂¯u˜
Γu.
Lemma 6.2.It holds
∇r
∂u∇r
∂¯u˜
R(∂u, ∂¯u)(0) = iAr, r = 0,...,N.
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 17
Proof. From (4) it follows that
∇r
∂¯u˜
R(∂u, ∂¯u) = −∂r+1
¯u˜
Γu=1
r!iArur−1
2!∂r+1
¯u[G, ∂uG] + 1
3! ∂r+1
¯u[G, [G, ∂uG]] − · · · .
Note that the covariant derivative with respect to ∂ucontains the Lie brackets with the matrix
˜
Γu. On the other hand, ˜
Γuis dividable by ¯u, consequently
∇r
∂u∇r
∂¯u˜
R(∂u, ∂¯u)(0) = ∂r
u∂r
¯u˜
R(∂u, ∂¯u)(0)
=iAr−1
2!∂r+1
¯u∂r
u[G, ∂uG](0) + 1
3!∂r+1
¯u∂r
u[G, [G, ∂uG]](0) − · ·· .
Note that since all terms in Gcontain the same powers of uand ¯u, it holds
∂r+1
¯u∂r
u[G, ∂uG](0) = X
r1+r2=r+1
r1≥0, r2≥1
[∂r1
¯u∂r1
uG, ∂r2
¯u∂r2
uG](0) = [G, ∂r+1
¯u∂r+1
uG](0) = 0.
Here we used the symmetry in r1,r2combined with the skew-symmetry of the Lie brackets, and
the fact that G(0) = 0. By the same reason ∂r+1
¯u∂r
u[G, [G, ∂uG]](0) as well as all terms containing
more Lie brackets with G, are zero.
Note that in particular,
˜
R(∂u, ∂¯u)(0) = iA1,∇∂u∇∂¯u˜
R(∂u, ∂¯u)(0) = ∇∂¯u∇∂u˜
R(∂u, ∂¯u)(0) = iA2.
The elements A1and A2are in the center of pru(n)k, consequently, by the construction, the other
covariant derivatives of ˜
Rat the point 0 take values in the subalgebra of prgl(n,C)korthogonal to
A1and A2. Using this and comparing the last equalities with (5), we conclude that k⊂prC⊕u(n)g.
The inverse inclusion easily follows from the the construction and the above formulas. Lemma 6.1
is proved.
Let now f=f1.
Lemma 6.3.It holds Cn⋉iR⊂g.
Proof. We know already that each holonomy algebra contains iR. It holds
R1,0(ej,¯q)(0) =
0T(ej)t∗
0 0 A1ej
0 0 0
∈g1,0.
Next, T(ej) = 0 for j= 1,...,n0, and T(ej) = −i¯ejfor j=n0+ 1,...,n. This shows that g
contains A1ej, iA1ej∈Cnfor j= 1,...n0. From the definition of n0it follows that gcontains
Cn0. For each j > n0,gcontains ejand iej. We conclude that gcontains Cn⋉iR.
From Lemmas 6.1 and 6.3 it follows that g=gk.
Suppose that f=f2.
Lemma 6.4.It holds prCn⋉iRg=L⋉iR=Cm⊕Rn−m⋉iR⊂g.
Proof. Similarly as in Lemma 6.3 it can be shown that Cm⋉iR⊂g. Let j≥m+ 1. It holds
R(ej,¯q) =
0∗¯et
j∗
0 0 0 ∗
0 0 0 −ej
0 0 0 0
.
Consequently, gcontains Rn−m. The covariant derivatives of this tensor give trivial projections
on Cn−m. Finally, K= 0, i.e.
R1,0(q, ¯q) = −idT1,0M+cp ∧¯p
for some function c. Hence, the covariant derivatives of R1,0(q, ¯q) give trivial projections on
Cn−m.
From Lemmas 6.1 and 6.4 it follows that g=gk,J,L.
18 ANTON S. GALAEV
Suppose that f=f3. In this case k⊂u(m). As above it can be shown that Cm⋉iR⊂g. It
holds
pr<em+1,...,en>K=
n−m
X
α=1
iBj m+α
((N+α−1)!)2|u|2(N+α−1)ej.
Together with the proof of Lemma 6.1 this shows that pru(n)g=k⊂g. The covariant derivatives
of Kwill give L0⊂g. Finally, pr<em+1,...,en>◦T= 0.Thus, g=gk,L.
The metric with the potential f4may be considered in the same way. We will get g=gk0,ψ.
The theorem is proved.
7. Example: the space of oriented lines in R3
The space of oriented lines in R3admits the following Lorentz-K¨ahler metric [30]:
h=1
(1 + |u|2)2d¯udv +d¯vdu +2i(¯vu + ¯uv)
(1 + |u|2)3d¯udu.
It is easy to show that it holds
R1,0(p, ¯q)(0) = 0 2
0 0, R1,0(q, ¯q)(0) = 2∗
0 2.
This implies that the holonomy algebra of the metric hcoincides with u(1,1)Cp.
8. Complex pp-waves
Here we give the following characterization of the complex pp-waves.
Theorem 8.1.Let (M, h)be a Lorentz-K¨ahler manifold of complex dimension n+ 2 ≥2with
a parallel isotropic vector field p∈Γ(T M). Then the following conditions are equivalent:
1) The holonomy algebra gof (M, h)is contained in Cn⋉iR⊂u(1, n + 1)Cp.
2) The curvature tensor of the Levi-Civita connection satisfies R(p⊥, p⊥) = 0.
3) The curvature tensor of the extension of the Levi-Civita connection to TCMsatisfies
R(p⊥,¯p⊥) = 0.
4) Locally Madmits complex coordinates v, z1,...,zn, u such that
h=d¯udv +d¯vdu +
n
X
k=1
d¯zkdzk+h¯ukd¯udzk+h¯
kud¯zkdu +h¯uud¯udu,
and the coefficients of the metric depend on the coordinates in the following way:
h¯
ku =h¯
ku(¯zl, u, ¯u), h¯uu =h¯uu (zl,¯zl, u, ¯u), ∂zk∂¯zjh¯uu = 0.
5) Locally Madmits complex coordinates v, z1,...,zn, u such that the potential of his of
the form
f= ¯uv + ¯vu +
n
X
k=1 |zk|2+ Re(φ(zj, u, ¯u)).
Proof. From results of Section 5 it follows that the first three conditions are equivalent. It
is obvious that the last two conditions are equivalent. Using computations as in Section 6.4, it
is not hard to show that the fourth condition implies the third one. Let us show that the third
condition implies the fourth one.
Consider local coordinates as in Section 6.1. There exists a function αsuch that p=α∂v.
Since ∇p= 0, it holds ∂¯zaα= 0. Considering the coordinate transformation
˜v=F(v, zk, u),˜zk=zk,˜u=u,
where the function Fsatisfies ∂vF=1
α, we get p=∂˜v. This allows us to assume that p=∂v.
The condition ∇∂v= 0 implies
0 = Γv
va =h¯uv∂ah¯uv .
Consequently,
∂ah¯uv = 0.
HOLONOMY CLASSIFICATION OF LORENTZ-K¨
AHLER MANIFOLDS 19
from this and Section 6.1 it follows that h¯uv is a function of ¯u. Introducing the new coordinate ˜u
such that h¯uv d¯u=d¯
˜u, we get h¯
˜uv = 1, i.e. we may assume that h¯uv = 1.
It is clear that the induced connection in the bundle p⊥/ < p > (where pis considered as
a vector field in T1,0M) is flat. Consequently, there exist vector fields e1,...,en,¯e1,...,¯ensuch
that h(¯ek, ej) = δij ,
ej=Bk
j∂zkmodulo p;
moreover, these vector fields are parallel modulo p. From this condition it follows that
∂vBk
j=∂¯vBk
j=∂¯uBk
j=∂¯zlBk
j= 0.
Consider the family of K¨ahlerian metrics h0=h¯
kj d¯zkdzjdepending on the parameters u, ¯u.
The u-families of vector fields e1,...,en,¯e1,...,¯enare parallel with respect to the connections
defined the families of these metrics. Consequently,
[ej, ek] = [¯ej, ek] = [¯ej,¯ek] = 0,
and there exist coordinates ˜z1,...˜znsuch that ej=∂˜zjand
h0=
n
X
j=1
d¯zjdzj.
Since Bj
kare functions of the variables z1,...zn, u, the coordinates ˜z1,...,˜znare related to
z1,...,znby a holomorphic transformation depending holomorphically on the parameter u. By
this reason, we may consider the new coordinates v, ˜z1,...,˜zn, u. With respect to these coordinates
it holds h¯
kj = 0. The equality (3) implies the proof of the theorem.
9. Lorentz-K¨ahler symmetric spaces
Berger [11] classified indecomposable simply connected pseudo-Riemannian symmetric spaces
with simple groups of isometries. Classification of simply connected Lorentz-K¨ahler symmetric
spaces is obtained in [35]. Here we give an alternative formulation of this result in terms of the
curvature and holonomy.
Let (M, g ) be a simply connected pseudo-Riemannian symmetric space. Let Hbe the group
of transvections and G⊂Hbe the stabilizer of a point x∈M, then the holonomy group of
(M, g) coincides with the isotropy representation of G. The groups Hand G, and consequently
the manifold (M, g ), may me reconstructed from the holonomy algebra g⊂so(m) (m=TxM), of
(M, g) and the value Rxof curvature tensor of (M, g) at the point xin the following way, see e.g.
[1]. Define on the vector space
h=g⊕m
the structure of the Lie algebra in the following way:
[A, X] = AX, [A, B] = [A, B]g,[X, Y ] = −Rx(X, Y ),
where A, B ∈gand X, Y ∈m. Then Hmay be found as the simply connected Lie group with the
Lie algebra h, and G⊂His the connected Lie subgroup corresponding to the subalgebra g⊂h.
Note that it holds g=Rx(m,m).
Thus in order to classify indecomposable simply connected Lorentz-K¨ahler symmetric spaces
it is enough to find all weakly irreducible holonomy algebras g⊂u(m) (m=C1,n+1) admitting
g-invariant surjective maps R: Λ2m→gthat satisfy the Bianchi identity. It is convenient to
consider the complex extension of Rand use the results of Section 5.
For an indecomposable symmetric space (M , g) the following conditions are equivalent [1]: h
is simple; g⊂so(m) is totally reducible; the Ricci tensor of (M , g) is non-degenerate.
If g⊂u(1, n+1) is irreducible, then the above equivalent conditions imply that g=u(1, n +1),
and we obtain the complex de Sitter and anti de Sitter symmetric spaces:
dSn+2(C) = SU(1, n + 2)/U(1, n + 1),AdSn+2(C) = SU(2, n + 1)/U(1, n + 1).
The subalgebra g0⊂u(1,1) can not be the holonomy algebra of a symmetric space since it is
completely reducible and is contained in su(1,1) (i.e. the corresponding space would have to be
Ricci-flat).
20 ANTON S. GALAEV
Thus we may assume that g⊂u(1, n + 1) is weakly irreducible and it is contained in u(1, n +
1)Cp. Using the classification of holonomy algebras, results of Section 5 and solving a simple
exercise in linear algebra we arrive to the following theorem (we give the non-zero values of Ron
the basis vectors).
Theorem 9.1.If (M, g )is an indecomposable Lorentz-K¨ahler symmetric space with the holo-
nomy algebra g⊂u(1, n + 1)Cp, then (M, g )is given by exactly one of the following pairs (g, R):
a) g= 0ic
0 0
c∈R∈Cand R1,0(q, ¯q) = 0 1
0 0 ;
b) gand −R, where gand Rare from a);
c) g= a0
0−¯a
a∈Cand R1,0(p, ¯q) = 1 0
0 0 ;
d) n= 1,g=
0−x ic
0 0 x
0 0 0
x, c ∈R
and
R1,0(e1,¯q) =
0 0 −i
0 0 0
0 0 0
,R1,0(q, ¯q) =
0−i0
0 0 i
000
;
e) gand −R, where gand Rare from d);
f) n≥1,0≤m≤n,g=
2ai −¯
Zt−¯
Xtic
0aiEm0Z
0 0 2aiEn−mX
0 0 0 2ai
Z∈Cm, X ∈Rn−m,
a, c ∈R
,
R1,0(p, ¯q) = 2R1,0(ej,¯ej) = 1
2R1,0(ek,¯ek) =
001
000
000
,
R1,0(ej,¯q) = 1
2
0 0 0
0 0 ej
0 0 0
, R1,0(ek,¯q) =
0−¯et
k0
0 0 ek
0 0 0
,
R1,0(q, ¯q) = idC1,n+1 −1
2idCm,where 1≤j≤m,m+ 1 ≤k≤n.
In the case c), g⊂u(1,1) is completely reducible, and it corresponds to the symmetric space
SL(2,C)/C∗of simple isometry group. Symmetric spaces corresponding to other pairs have non-
simple isometry groups and they are found in [35]. Spaces corresponding to the cases a), b), d)
and e) are described also in [1].
Corollary 9.1.All indecomposable simply connected Calabi-Yau Lorentz-K¨ahler symmetric
spaces are exhausted by the cases a), b), d) and e) form the above theorem.
This result is affirmative with the result from [1], where it is shown that there are exactly
two (up to isometry) indecomposable simply connected Calabi-Yau pseudo-K¨ahler manifolds in
dimension 4, and the same holds for the dimension 6.
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