Non-split almost complex and non-split Riemannian supermanifolds

0.13 MB·English

by  Matthias Kalus

Non-split almost complex and
non-split Riemannian supermanifolds
Matthias Kalus
5
1 Fakult¨at fu¨r Mathematik
0
2 Ruhr-Universita¨t Bochum

n D-44780 Bochum, Germany
a
J

8
2 Keywords: supermanifold; almost complex structure; Riemannian metric; non-split

] MSC2010: 32Q60, 53C20, 58A50
G
D Abstract
.
h Non-split almost complex supermanifolds and non-split Riemannian supermanifolds
t
a are studied. The first obstacle for a splitting is parametrized by group orbits on
m
an infinite dimensional vector space. Further it is shown that non-split structures
[ appear in the first case as deformations of a split reduction and in the second case

1 as the deformation of an underlying metric. In contrast to non-split deformations of
v complex supermanifolds, these deformations can be restricted by cut-off functions to
7
local deformations. A class of examples of nowhere split structures constructed from
1
1 almost complex manifolds of dimension 6 and higher, is provided for both cases.
7
0
Even almost complex structures and Riemannian metrics define global tensor fields on real
.
1 supermanifolds. Denoting a real supermanifold by M = (M,C∞) and the global super
0 M
5 vector fields on M by VM, the tensors lie in End(VM)¯0, resp. Hom(VM,VM∗ )¯0. Fixing a
1 Batchelor model M → (M,Γ∞ ), the Z-degree zero part J of an even almost complex
: ΛE∗ R
v structure J ∈ End(V ) is again an almost complex structure on M. This raises the
i M ¯0
X question, whether there is a Batchelor model, such that J equals its reduction J . Or
R
r equivalently, denoting by N the nilpotent superfunctions: if the exact sequence
a
0 → N2 → C∞ → C∞ ⊕Γ∞ → 0
M M E∗
is split with an α : C∞ ⊕Γ∞ → C∞ such that the induced automorphism αˆ : C∞ → C∞
M E∗ M M M
transforms J to J. In the case of a positive answer we call the tensor split. The analogue
R
question canbe formulatedforeven Riemannianmetrics where thereduction g ofa metric
R
g ∈ Hom(V ,V∗ ) is given by the Riemannian metric g +g .
M M ¯0 0 2
For complex structures being integrable almost complex structures, existence of a splitting
was studied in [2], [5] and [7]: there exist non-split complex supermanifolds, all of them
1
being deformations of split complex supermanifolds. The parameter spaces of deforma-
tions are given by orbits of the automorphism group of the associated Batchelor bundle
on a certain non-abelian first cohomology. Here the existence of local complex coordinates
makes the splitting problem a problem of global cohomology. The splitting question for
even symplectic supermanifolds was answered in [6] by identifying the symplectic super-
manifold with an underlying symplectic manifold and a Batchelor bundle with metric and
connection. It is shown that all terms of degree higher than 2 in a symplectic form can be
erased by the choice of a Batchelor model. Hence all symplectic supermanifolds are split
in the above sense.
In this paper the existence of a splitting for even almost complex structures as well as
even Riemannian metrics is studied. It is shown that all almost complex structures appear
as deformations of split structures and all Riemannian metrics appear as deformations of
underlying metrics. In both cases but in contrast to the complex case, these deformations
can be restricted by smooth cut-off functions to local deformations. For almost complex
structures the splitting problem stated above can be expressed as: what is the obstacle for
having local coordinates near any point such that the almost complex structure is repre-
sented by a purely numerical matrix. In the Riemannian case (similar to the symplectic
case in [6]), the reduction is asked to be a purely numerical matrix on V⊗2 and to have
M,−1
matrixentriesofdegreelessorequalto2onthethreeremainingblocksof(V ⊕V )⊗2.
M,0 M,−1
The first obstacle for a splitting is described for both problems. Finally explicit examples
of non-split almost complex structures, resp. Riemannian metrics are given. The results
and the applied methods are summarized in the following.
Contents. In the first section an almost complex structure is decomposed via the finite
log series into its reduction (the degree zero term) and its degree increasing term. With
respect to these components the lowest degree obstacle for isomorphy of almost complex
supermanifolds is deduced. Fixing the reduction, these obstacles are parametrized by the
orbits of a quotient of the group of transformations that are almost holomorphic with
respect to the reduction up to a certain degree, acting on a quotient of tensor spaces.
The second section deals with Riemannian metrics in an analogue way producing results
analogous to those in the almost complex case. Here the isometries of the reduction play
the role of the almost holomorphic transformations. However the more complicated action
oftheautomorphismgroupofthesupermanifoldonametricandthefactthatthereduction
has no pure degree, require an adjustment of the techniques.
Finally the third section contains a class of non-split examples for almost complex struc-
tures and Riemannian metrics. These are constructed on the supermanifold of differential
forms on an arbitrary almost complex manifold of dimension higher than 4. Some basic
factsofalmost complexgeometryandamethodtoconstruct super vector fields areapplied.
In the almost complex and in the Riemannian case, the constructed non-split tensors are
nowhere split, i.e. at no point of the manifold the matrix elements of the respective tensors
satisfy the split property mentioned above.
2
1 Non-split almost complex supermanifolds
Let (M,J) be an almost complex supermanifold with sheaf of superfunctions C∞. Denote
M
the C∞(M)-module of global superderivations of C∞ by V = V ⊕V . Furthermore
M M M M,¯0 M,¯1
fix a Batchelor model M → (M,Γ∞ ) yielding Z-gradings (denoted by lower indexes)
ΛE∗
and filtrations (denoted by upper indexes in brackets) on C∞, V and End(V ), the last
M M M
denoting C∞(M)-linear maps. The even automorphism of C∞(M)-modules J ∈ End(V )
M M M ¯0
can be uniquely decomposed into J = J (Id+J ) with invertible J = J and nilpotent
R N R 0
J . The finite exp and log series yield a unique representation Id + J = exp(Y) with
N N
Y ∈ End(2)(V ) .
M ¯0
Lemma 1.1. The tensor J is an almost complex structure if and only if J is an almost
R
complex structure and YJ +J Y = 0.
R R
Proof. From J2 = −Id we obtain J2 = −Id and exp(Y)J exp(Y) = J . For reasons
R R R
of degree Y J + J Y = 0. Assume that Y J + J Y = 0 holds for all k < n. Set
2 R R 2 2k R R 2k
k
Y := Y . It is exp(Y) = exp(Y ) + Y up to terms of degree > 2n. Hence
[2k] Pj=1 2j [2n−2] 2n
exp(Y)J exp(Y) = exp(Y )exp(−Y )J + Y J +J Y up to terms of degree
R [2n−2] [2n−2] R 2n R R 2n
> 2n. This completes the induction. The converse implication follows directly.
We call J the reduction of J, deforming J by t 7→ J exp(tY). In particular J yields
R R R R
an almost complex structure on M and an almost complex structure on the vector bundle
E → M. Hence even and odd dimension of M are even. Further topological conditions
on M and E for the existence of an almost complex structure can be obtained from [3]
and e.g. [1]. Adapted to our considerations the almost complex supermanifold (M,J) is
split if there is a Batchelor model, such that the almost complex structure J has nilpotent
component Y = 0. Note that this problem is completely local since Lemma 1.1 allows
cutting off the nilpotent Y in J = J exp(Y).
R
Let Φ = (ϕ,ϕ∗) be an automorphism of the supermanifold M. The global even isomor-
phism of superalgebras ϕ∗ ∈ Aut(C∞(M)) over ϕ is decomposable into ϕ∗ = exp(ζ)ϕ∗
M ¯0 0
withζ ∈ V(2) andϕ∗ preserving theZ-degree induced by theBatchelor model (see e.g. [5]).
M,¯0 0
Denote by Aut(E∗) the bundle automorphisms over arbitrary diffeomorphisms of M, then
ϕ∗ is induced by an element ϕ ∈ Aut(E∗) over ϕ. The automorphism ϕ∗ transforms J into
0 0
ϕ∗.J given by (ϕ∗.J)(χ) := ϕ∗(J((ϕ∗)−1χϕ∗))(ϕ∗)−1. Denoting ad(ζ) := [ζ,·], assuming
ζ ∈ V(2k) andapplying ϕ∗ = (Id+ζ)ϕ∗ uptotermsinV(4k), it isϕ∗.J = ϕ∗.J+[ad(ζ),ϕ∗.J]
M,¯0 0 M,¯0 0 0
up to terms in End(4k)(V ) . Comparing both sides with respect to the degree yields:
M ¯0
Proposition 1.2. The almost complex supermanifolds (M,J) and (M,J′) with structures
J = J exp(Y), J′ = J′ exp(Y′), Y,Y′ ∈ End(2k)(V ) are isomorphic up to error terms
R R M ¯0
in End(4k)(V ) via an automorphism ϕ∗ with ϕ∗(ϕ∗)−1 ∈ exp(V(2k)) if and only if there
M ¯0 0 M,¯0
exist ϕ ∈ Aut(E∗) and ζ ∈ V(2k) such that J′ = ϕ∗.J and:
0 M,¯0 R 0 R
Y′ = ϕ∗.Y −ad(ζ )−J′ ad(ζ )J′ , k ≤ j < 2k
2j 0 2j 2j R 2j R
3
From now on we fix the reduction J and hence assume that for an automorphism ψ∗ of
R
M, the map ψ∗ is pseudo-holomorphic with respect to J , denoted ψ∗ ∈ Hol(M,J ). Let
0 R 0 R
Hol(M,J ,2k) be the automorphisms ψ∗ = exp(ξ)ψ∗ of M such that J = ψ∗.J up to
R 0 R R
terms in End(2k)(V ) . Note that ψ∗ ∈ Hol(M,J ,2k) includes ψ∗ ∈ Hol(M,J ) and
M ¯0 R 0 R
(2k)
that exp(V ) ⊂ Hol(M,J ,2k) is a normal subgroup.
M,¯0 R
Define on the endomorphisms of real vector spaces EndR(VM) the CM∞(M)-linear Z-degree
preserving map:
FJR : EndR(VM) → EndR(VM), FJR(γ) := γ +JRγJR
The set F (End(2k)(V )) is by Lemma 1.1 exactly the nilpotent parts Y of almost com-
JR M
plex structures J = J exp(Y) deforming J in degree 2k and higher. Note further that
R R
F (ad(V )) ⊂ End(V ) and more precisely F (ad(V(2k))) ⊂ End(2k)(V ) .
JR M M JR M,¯0 M ¯0
Definition 1.3. Let the upper index 2k ∈ 2N in curly brackets denote the sum of terms
of Z-degree 2k up to 4k − 2. For J = J exp(Y), Y ∈ End(2k)(V ) we call the class
R M ¯0
[Y{2k}] in the quotient of vector spaces F (End{2k}(V ) )/F (ad(V{2k})) the 2k-th split
JR M ¯0 JR M,¯0
obstruction class of J.
The Hol(M,J ,2k)-action on F (End(2k)(V ) ) is given up to terms in End(4k)(V )
R JR M ¯0 M ¯0
by (ψ∗,Y) 7→ J (J −ψ∗.J )+ψ∗.Y. Since ψ∗.F (ad(V{2k})) ⊂ F (ad(V{2k})), it is well-
R R R JR M,¯0 JR M,¯0
defined on F (End{2k}(V ) )/F (ad(V{2k})). By Proposition 1.2 it induces an action of
JR M ¯0 JR M,¯0
PHol(M,J ,2k) := Hol(M,J ,2k)/exp(V(2k)) on F (End{2k}(V ) )/F (ad(V{2k})).
R R M,¯0 JR M ¯0 JR M,¯0
It follows that for an almost complex supermanifold that is split up to terms of degree 2k
and higher, the 2k-th split obstruction class is well-defined up to the PHol(M,J ,2k)-
R
action. Note that for a given almost complex structure J = J exp(Y) the obstructions
R
can be checked starting with j = 1 iteratively: if Y = ad(ζ ) + J ad(ζ )J can be
2j 2j R 2j R
solved for a ζ ∈ V then there is an automorphism of the supermanifold M such that
2j M,2j
J = J exp(Y′) with Y′ ∈ End(2(j+1))(V ) . In the non-split case this procedure ends
R M ¯0
with a well-defined 2k and associated orbit of 2k-th split obstruction classes. We note as
a special case:
Proposition 1.4. Let (M,J ) be a split almost complex supermanifold of odd dimension
R
2(2m+r), m ≥ 0, r ∈ {0,1}. The almost complex supermanifolds (M,J) with reduction
J that are split up to terms of degree (2m+ r)+ 1 and higher, correspond bijectively to
R
the PHol(M,J ,2(m+1))-orbits on F (End(2(m+1))(V ) )/F (ad(V(2(m+1)))).
R JR M ¯0 JR M,¯0
As a technical tool we note an identification for the quotient appearing in the split ob-
struction classes. Denote by E1 = E1 ⊕E1 the global super-1-forms on M and by d
M M,¯0 M,¯1 M
the de Rham operator on the algebra EM of superforms. It is End(VM) = VM⊗CM∞(M)EM1 .
Proposition 1.5. For all k, the map
ΘJR : FJR(cid:0)VM ⊗CM∞(M) EM1 (cid:1)(¯02k) −→ FJR(End(2k)(VM)¯0)/FJR(ad(VM(2k,¯0)))
4
locally for homogeneous arguments defined by
F (χ⊗d f) 7−→ (−1)|f||χ|f ·F (ad(χ))+F (ad(V(2k)))
JR M JR JR M,¯0
is a well-defined, surjective morphism of Z-filtered super vector spaces. For any element
ψ∗ ∈ Hol(M,J ,2k) and [ψ∗] ∈ PHol(M,J ,2k) it is Θ (ψ∗.Z) = [ψ∗].(Θ (Z)).
R R JR JR
Proof. For homogeneous components of χ ⊗ dMf ∈ VM ⊗CM∞(M) EM1 the decomposition
χ⊗d f = (−1)|f||χ|(f ·ad(χ)−ad(fχ)) is well-defined up to terms in ad(V ).
M M
2 Non-Split Riemannian supermanifolds
Let(M,g)beaRiemanniansupermanifoldwithevennon-degeneratesupersymmetric form
g ∈ Hom(VM ⊗CM∞(M) VM,CM∞(M))¯0. Here we will mostly regard g as an isomorphism of
C∞-modules g ∈ Hom(V ,V∗ ) with g(X)(Y) = (−1)|X||Y|g(Y)(X) for homogeneous
M M M ¯0
arguments. The context will fix which point of view is used. For a given Batchelor model
M → (M,Γ∞ ) decompose g = g (Id+ g ) with invertible g = g +g and nilpotent
ΛE∗ R N R 0 2
g ∈ End(2)(V ) such that g g ∈ Hom(4)(V ,V∗ ) . With the finite log and exp
N M ¯0 0 N M M ¯0
(2) (2k)
series we write g = g exp(W) with W ∈ End (V ) , where End (V ) denotes those
R g0 M ¯0 g0 M ¯0
W ∈ End(2k)(V ) such that g W ∈ Hom(2k+2)(V ,V∗ ) .
M ¯0 0 M M ¯0
Lemma 2.1. If the tensor g = g exp(W), W ∈ End(2k)(V ) is a Riemannian metric
R g0 M ¯0
then g is a Riemannian metric and g (W(·),·) = g (·,W(·)) up to terms of degree 4k+2
R R R
and higher.
Proof. Due to supersymmetry g (exp(W)(·),·) = g (·,exp(W)(·)). The approximation
R R
exp(W) = 1+W holds up to terms of degree 4k with error term 1W2 in degree 4k. Since
2 2k
W ∈ End(2k)(V ) it is g W2 ∈ Hom(4k+2)(V ,V∗ ) .
g0 M ¯0 R 2k M M ¯0
We call g the reduction of g. Here the metric g appears as a deformation of the underlying
R
Riemannian metric g on M via t 7→ (g +t·g )exp( ∞ tjW ). Note that g also yields
0 0 2 Pj=1 2j R
a non-degenerate alternating form on the bundle E. So in contrast to the non-graded
case there is a true condition for the existence of a Riemannian metric: the existence of
a nowhere vanishing section of E ∧ E → M. In particular the odd dimension of M has
to be even. A Riemannian supermanifold (M,g) is split, if there is a Batchelor model,
such that the Riemannian metric g has nilpotent component W = 0. Again the appearing
deformations are essentially local via cutting off g exp(W) by (g +f·g )exp( ∞ fjW )
R 0 2 Pj=1 2j
with cut-off function f.
AsbeforeletΦ = (ϕ,ϕ∗), ϕ∗ = exp(ζ)ϕ∗ beanautomorphismofthesupermanifoldM. We
0
obtain ϕ∗.g given by (ϕ∗.g)(χ)(χ′) = ϕ∗ g((ϕ∗)−1χϕ∗,(ϕ∗)−1χ′ϕ∗) . Assuming ζ ∈ V(2k)
(cid:0) (cid:1) M,¯0
this yields:
ϕ∗.g = ϕ∗.g −(ϕ∗.g)(ad(ζ)⊗Id+Id⊗ad(ζ))+ζ(ϕ∗.g) (1)
0 0 0
5
in Hom(V ,V∗ ) up to terms in Hom(4k)(V ,V∗ ) . Note that for the term ζ(ϕ∗.g), the
M M ¯0 M M ¯0 0
metric is regarded as an element in Hom(VM ⊗CM∞(M) VM,CM∞(M))¯0. Define VM(2k,g)0,¯0 to be
theelements inζ ∈ V(2k) satisfyingg (ad(ζ)⊗Id+Id⊗ad(ζ))+ζg ∈ Hom(2k+2)(V ,V∗ ) .
M,¯0 0 0 M M ¯0
Comparing the terms in (1) with respect to the degree yields:
Proposition 2.2. The Riemannian supermanifolds (M,g) and (M,g′) with Riemannian
metrics g = g exp(W), g′ = g′ exp(W′), W ∈ End(2k)(V ) , W′ ∈ End(2k)(V )
R R g0 M ¯0 g0′ M ¯0
are isomorphic up to error terms in Hom(4k)(V ,V∗ ) via an automorphism ϕ∗ with
M M ¯0
ϕ∗(ϕ∗)−1 ∈ exp(V(2k) ) if and only if there exist ϕ ∈ Aut(E∗) and ζ ∈ V(2k) such that
0 M,g0′,¯0 0 M,g0′,¯0
g′ = ϕ∗.g and:
R 0 R
W′ = ϕ∗.W −ad(ζ )− (g′ )−1(ad∗(ζ)−ζ)g′ , k ≤ j < 2k
2j 0 2j 2j (cid:16) R R(cid:17)
2j
Here ϕ∗.W is defined by (ϕ∗.W)(χ) := ϕ∗(W((ϕ∗)−1χϕ∗))(ϕ∗)−1 and the homomorphism
0 0 0 0 0 0
ad∗ : VM → EndR(EndR(VM,CM∞(M))) denotes the representation dual to ad.
Fix g from now on and denote by Iso(M,g ,2k +2) the automorphisms ψ∗ = exp(ξ)ψ∗
R R 0
of M such that g = ψ∗.g up to a term S := g −ψ∗.g ∈ Hom(2k+2)(V ,V∗ ) . Note
R R R R M M ¯0
that this forces g g−1S ∈ Hom(2k+2)(V ,V∗ ) . Further exp(V(2k) ) ⊂ Iso(M,g ,2k+2)
0 R M M ¯0 M,g0,¯0 R
is a normal subgroup.
Parallel to the analysis of the almost complex structures we define the maps
F : End(V ) → End(V ), F (γ) := γ +g−1γ∗g
gR M M gR R R
G : V → End(V ), G (ζ) := ad(ζ)+g−1(ad∗(ζ)−ζ)g
gR M M gR R R
denoting by γ∗ the induced element in End(V∗ ) and ad∗ as above. By Lemma 2.1 the
M
(2k)
elements in F (End (V ) ) are up to degree ≥ 4k+2 the appearing Ws in Riemannian
gR g0 M ¯0
(2k)
metrics g = g exp(W) that are split up to degree ≥ 2k. Further G (V ) lies in
R gR M,g0,¯0
(2k)
F (End (V ) ).
gR g0 M ¯0
Definition 2.3. For g = g exp(W) with W ∈ End(2k)(V ) we call the class [W{2k}]
R M ¯0
in the quotient of vector spaces F (End(2k)(V ) ){2k}/G (V(2k) ){2k} the 2k-th split ob-
gR g0 M ¯0 gR M,g0,¯0
struction class of g.
The Iso(M,g ,2k+2)-action on F (End(2k)(V ) ) is given up to terms in End(4k)(V )
R gR g0 M ¯0 M ¯0
by (ψ∗,W) 7→ g−1(ψ∗.g − g ) + ψ∗.W. It is ψ∗.G (V(2k) ) ⊂ G (V(2k) ) by di-
R R R gR M,g0,¯0 gR M,g0,¯0
rect calculation. Analog to the almost complex case using Proposition 2.2, the action of
(2k)
Iso(M,g ,2k+2)inducesaPIso(M,g ,2k+2) := Iso(M,g ,2k+2)/exp(V )-action
R R R M,g0,¯0
on the quotient F (End(2k)(V ) ){2k}/G (V(2k) ){2k}. Hence the 2k-th split obstruction
gR g0 M ¯0 gR M,g0,¯0
class is well-defined up to the PIso(M,g ,2k+2)-action for a Riemannian supermanifold
R
that is split up to terms of degree 2k+2 and higher. We have in particular analogously to
the almost complex case:
6
Proposition 2.4. Let (M,g ) be a split Riemannian supermanifold of odd dimension
R
2(2m+r), m ≥ 0, r ∈ {0,1}. The Riemannian supermanifolds (M,g) with reduction g
R
that are split up to terms of degree (2m+r)+3 and higher, correspond bijectively to the
Iso(M,g ,2(m+2))-orbits on F (End(2(m+1))(V ) )/G (V(2(m+1))).
R gR M ¯0 gR M,g0,¯0
Also an analogy to Proposition 1.5 holds:
Proposition 2.5. The map
Θ : F (End(2k)(V ) ) −→ F (End(2k)(V ) )/G (V(2k) )
gR gR g0 M ¯0 gR g0 M ¯0 gR M,g0,¯0
locally defined by
F (χ⊗d f) 7−→ (−1)|f||χ|f ·G (χ)+G (V(2k) )
gR M gR gR M,g0,¯0
is a well-defined surjective morphism of Z-filtered vector spaces. For any element ψ∗ in
Iso(M,g ,2k +2) and [ψ∗] ∈ PIso(M,g ,2k +2) it is Θ (ψ∗.Z) = [ψ∗].(Θ (Z)).
R R gR gR
Proof. Apply F to χ ⊗ d f = (−1)(|f||χ|(f · ad(χ) − ad(fχ)) and add fχ − fχ in the
gR M
bracket. This yields a map F (End(2k)(V ) ) → F (End(2k)(V ) )/G (V(2k)). Since
gR g0 M ¯0 gR M ¯0 gR M,¯0
χ⊗d f is in End(2k)(V ) , its degree 2k term is of the form f˜ ∂ ⊗d fˆ for an odd
M g0 M ¯0 P i∂ξi M i
(2k)
coordinate system (ξ ). This forces fχ ∈ V by direct calculation.
i M,g0,¯0
3 Examples of global nowhere split structures
Here explicit examples of non-split almost complex structures, resp. non-split Riemannian
metrics are given. The constructed tensors are nowhere split.
Let (M,J ) be an almost complex manifold of dimension 2n and let M be the superman-
M
ifold defined by differential forms, i.e. C∞ = E . The vector fields in V act on C∞ by
M M M M
Lie derivation. Let further π : V → V be the odd C∞-linear operator well-defined by
M M M
π2 = Id and π(χ)(ω) := ι ω for χ ∈ V ⊂ V and ω ∈ C∞.
χ M M,¯0 M
By [4, prop. 4.1] there exist non-degenerate 2-forms η ∈ C∞ compatible with J . We fix
M M
one and denote by g′ the J -invariant Riemannian metric η(·,J (·)) on M. Furthermore
M M
we embed
End(VM) ∼= (VM ⊗CM∞(M) EM1 ) ֒→ VM by χ⊗α 7→ ξχ⊗α := α·χ
and obtain ξ ,ξ ∈ V and π(ξ ),π(ξ ) ∈ V .
Id JM M,1 Id JM M,0
We define on M by C∞-linear continuation to End(V ), resp. Hom(V ,V∗ ):
M M M M
(i) the split almost complex structure JR by JM ⊕(π ◦JM ◦π) ∈ EndCM∞(VM ⊕π(VM))
7
(ii) the split Riemannian metric gR by g′+(η◦(π⊗π)) ∈ HomCM∞(VM⊗2 ⊕π(VM)⊗2,CM∞)
and g (V ⊗π(V )) = 0
R M M
Note the following technical lemma for later application:
Lemma 3.1. Let f,g ∈ End(V ), ω ∈ C∞ . Then:
M M,2
a) π(ξ )(ω) = 1(ω(f(·),·)+ω(·,f(·)))
f 2
b) [π(ξ ),π(ξ )] = −π(ξ )
f g [f,g]
c) J (ξ ) = −ξ and J (π(ξ )) = −π(ξ )
R JM Id R JM Id
Further fix in the almost complex, resp. Riemannian case the tensors:
(i) J = J exp(η·Y ) with Y ∈ End(2)(V ) by Y = F (π(ξ )⊗d η)
R η η M ¯0 η JR JM M
(ii) g = g exp(η ·W ) with W ∈ End(2)(V ) by W = F (π(ξ )⊗d η)
R η η M ¯0 η gR JM M
We prove:
Lemma 3.2. Assume that n > 1. The endomorphisms Y and W are nowhere vanishing.
η η
Proof. With Lemma 3.1 c) follows Y = π(ξ )⊗d η−π(ξ )⊗((d η)◦J ). Applying
η JM M Id M R
(Y (π(ξ )))(η)we obtain(π(ξ )(η))2+(π(ξ )(η))2. ByLemma 3.1 a)it isπ(ξ )(η) = 0
η JM JM Id JM
since η is compatible with J , and (Y (π(ξ )))(η) = η2. So Y is nowhere vanishing.
M η JM η
For the second statement note W = π(ξ ) ⊗ d η + g−1(d η) · g (π(ξ )). Further
η JM M R M R JM
(W (π(ξ )))(η) = (π(ξ )(η))2+(g−1(d η))(η)·η(ξ ,ξ ). Due to the compatibility of
η JM JM R M JM JM
η and J , it is η(ξ ,ξ ) = η and as before, π(ξ )(η) = 0. Further a calculation yields
M JM JM JM
(g−1(d η))(η) = g (g−1(d η),g−1(d η)) = η up to terms of degree 4 and higher. Hence
R M R R M R M
(W (π(ξ )))(η) = η2 up to terms of degree 6 and higher. So W is nowhere vanishing.
η JM η
Finally it follows:
Theorem 3.3. Assume that n > 2. The almost complex structure J = J exp(η ·Y ) and
R η
the Riemannian metric g = g exp(η ·W ) on M are nowhere split.
R η
Proof. For ψ∗ = exp(ξ)ψ∗ ∈ Hol(M,J ,4) we obtain [ad(ξ ),J ] = 0. With the iden-
0 R 2 R
tity exp(ξ ).J = exp(ad(ξ ))J exp(ad(ξ )) = exp([ad(ξ ),·])(J ) = J it follows that ψ∗
2 R 2 R 2 2 R R
maps Y ∈ F (End{4}(V ) ) to F (ad(ξ )) + ψ∗.Y up to terms of degree ≥ 6. Hence
JR M ¯0 JR 4 0
{4}
F (ad(V )) is a PHol(M,J ,4)-orbit up to terms of degree ≥ 6. Using Proposi-
JR M,¯0 R
tion 1.5 and ηY = F (ηπ(ξ ) ⊗ d η) it is sufficient to check that ηF (ad(ηπ(ξ )))
η JR JM M JR JM
does not vanish in degree 4. From the proof of Proposition 1.5 we know the identity
F (ad(ηπ(ξ ))) = η · F (ad(π(ξ ))) − Y . Now Lemma 3.1 b) and c) yields that
JR JM JR JM η
F (ad(π(ξ )))(π(ξ )) = 0. Hence it is F (ad(ηπ(ξ )))(π(ξ ))(η) = −Y (π(ξ ))(η)
JR JM JM JR JM JM η JM
which is η2 as it was shown in the proof of Lemma 3.2. This proves the first statement.
It is by direct calculation ηW ∈ End(4)(V ) . Let ψ∗ = exp(ξ)ψ∗ ∈ Iso(M,g ,6), then
η g0 M ¯0 0 R
the degree 4 term of ψ∗.g vanishes while the degree 6 term is ψ∗.(g W +g W )+(ψ∗.g ) .
0 2 4 0 6 R 6
8
Further (1+ξ )ψ∗ preserves g and so does exp(ξ )ψ∗. The term (exp(ξ +ξ ).g ) equals
2 0 R 2 0 4 6 R 6
(g (ad(ξ + ξ ) ⊗Id + Id⊗ ad(ξ +ξ ))− (ξ + ξ )g ) . So ψ∗ maps W ∈ End(4)(V )
R 4 6 4 6 4 6 R 6 g0 M ¯0
to G (ξ )+ ψ∗(W ) up to terms of degree ≥ 6. Since ψ∗ preserves the vanishing degree
gR 4 0 4
{4} {4}
four term of g , it follows that ξ ∈ V . Hence G (V ) is a PIso(M,g ,6)-
R 4 M,g0,¯0 gR M,g0,¯0 R
orbit up to terms of degree ≥ 6. Analogue to the almost complex case and follow-
ing Proposition 2.5 it is sufficient to show that ηG (ηπ(ξ )) is nowhere vanishing.
gR JM
We have the identity G (ηπ(ξ )) = η · G (π(ξ )) − W . Note that for α ∈ V∗
gR JM gR JM η M
it is (g−1(α))(η) = α(g−1(d η)) and g−1(d η) = π(ξ ) up to terms of degree two
R R M R M Id
and higher. Using these details, Lemma 3.1 b) and the definition of g one obtains
R
G (π(ξ ))(π(ξ ))(η) = −π(ξ )(η(π(ξ ),π(ξ ))) up to terms of degree four and
gR JM JM JM Id JM
higher. A direct calculation using the graded Leibniz rule and J -invariance of η shows
M
that G (π(ξ ))(π(ξ ))(η) vanishes up to terms of degree four and higher. Hence
gR JM JM
G (ηπ(ξ ))(π(ξ ))(η) = −W (π(ξ ))(η) up to terms of degree 6 and higher. In the
gR JM JM η JM
proof of Lemma 3.2 it was shown that the degree 4 term of this expression is η2. This
proves the second statement.
References
[1] E. Thomas, Complex structures on real vector bundles, Amer. J. Math. 89 (1967),
887-908.
[2] P. Green, On holomorphic graded manifolds, Proc. Amer. Math. Soc. 85 (1982), no.
4, 587-590
[3] W. S. Massey, Obstructions to the existence of almost complex structures, Bull. Amer.
Math. Soc. 67 (1961), 559-564
[4] D. McDuff, D. Salamon, Introduction to symplectic topology, Oxford Mathematical
Monographs. Oxford Science Publications. The Clarendon Press, Oxford University
Press, New York, 1995
[5] M. Rothstein, Deformations of complex supermanifolds, Proc. Amer. Math. Soc. 95
(1985), no. 2, 255-260
[6] M. Rothstein, The structure of supersymplectic supermanifolds, Differential geometric
methods in theoretical physics (Rapallo, 1990), 331-343, Lecture Notes in Phys., 375,
Springer, Berlin, 1991
[7] A. Yu. Vaintrob, Deformations of complex structures on supermanifolds, Functional
Anal. Appl. 18 (1984), no. 2, 135-136
9