Killing ﬁelds on compact pseudo-K¨ahler manifolds

. We show that a Killing ﬁeld on a compact pseudo-K¨ahler ddbar manifold is necessarily (real) holomorphic. Our argument works without the ddbar assumption in real dimension four. The claim about holomorphicity of Killing ﬁelds on compact pseudo-K¨ahler manifolds appears in a 2012 paper by Yamada, and in an appendix we provide a detailed explanation of why we believe that Yamada’s argument is incomplete.


Introduction
By a pseudo-Kähler manifold we mean a pseudo-Riemannian manifold (M, g) endowed with a ∇-parallel almost-complex structure J, for the Levi-Civita connection ∇ of g, such that the operator J x : T x M → T x M is a linear g x -isometry (or is, equivalently, g x -skew-adjoint) at every point x ∈ M.This implies integrability of J (see the comment preceding Lemma 3.1).We then call (M, g) a pseudo-Kähler ∂∂ manifold if, in addition, the underlying complex manifold M has the following ∂∂ property, also referred to as the ∂∂ lemma: (0.1) every closed ∂ exact or ∂ exact (p, q) form equals ∂∂λ for some (p − 1, q − 1) form λ.
It is well known that the ∂∂ property follows if M is compact and admits a Riemannian Kähler metric [5,Prop. 6.17 on p. 144].
Theorem A. Every Killing vector field on a compact pseudo-Kähler ∂∂ manifold is real holomorphic.
We provide two proofs of Theorem A, in Sections 2 and 3.The former is derived directly from the ∂∂ condition; the latter, shorter, relies on the Hodge decomposition, which is equivalent to the ∂∂ property [2, p. 269, subsect. (5.21)].
The Riemannian-Kähler case of Theorem A is well known, and straightforward [1, the lines following Remark 4.83 on pp.60-61].See also Remark 1.2.

Proof of Theorem B
All manifolds, mappings, tensor fields and connections are assumed smooth.
Lemma 1.1.Given a connection ∇ on a manifold M, let a vector field v on M be affine in the sense that its local flow preserves ∇.Then, for any ∇-parallel tensor field Θ on M, of any type, the Lie derivative Proof.Clearly, −£ v Θ is the derivative with respect to the real variable t, at t = 0, of the push-forwards Let (M, g) now be a fixed pseudo-Kähler manifold.If v is any vector field on M then, with J and ∇v treated as bundle morphisms TM → TM, (1.1) for B = ∇v and A = £ v J one has A = [J, B] and JA = −AJ , which is immediate from the Leibniz rule.For the Kähler form ω = g(J • , • ) of (M, g) and any g-Killing vector field v, it follows from (1.1) and Lemma 1.1 that (1.2) i) A = £ v J and α = £ v ω are related by α = g(A• , • ), while ii) A * = −A, JA = −AJ , ∇A = 0, ∇α = 0, and α is exact.
Given an exact p-form α on a compact pseudo-Riemannian manifold (M, g), (1.3) α is L 2 orthogonal to all parallel p times covariant tensor fields θ on M.
Remark 1.2.By (1.2-ii) and (1.3), for a Killing field v on a compact Riemannian Kähler manifold, £ v ω is L 2 -orthogonal to itself, and so, as a consequence of (1.2-i), v must be real holomorphic.
Let (M, g) be, again, a pseudo-Kähler manifold.The vector bundle morphisms C : TM → TM having C * = −C (that is, g x -skew-adjoint at every point x ∈ M ) constitute the sections of (1.4) the vector subbundle so(TM ) of End IR (TM ) = Hom IR (TM, TM ).
We denote by E the vector subbundle of so(TM ), the sections C of which are also complex-antilinear (so that JC = −CJ, in addition to C * = −C).Then Proof of Theorem B. By (1.5), with m = 2, the pseudo-Hermitian fibre metric in the line bundle E must be positive or negative definite.Hence so is its ginduced real part.For any Killing field v, (1.2-ii) implies that A = £ v J is a section of E which, due to (1.2) -(1.3), is L 2 -orthogonal to itself, and so £ v J = 0.
The above proof does not extend to compact pseudo-Kähler manifolds (M, g) of complex dimensions m > 2 with indefinite metrics.Namely, if the pair (j, k) represents the metric signature of g, with j minuses and k pluses (both j, k even, j + k = 2m), then the analogous signature of the real part (induced by g) of the pseudo-Hermitian fibre metric in E is (jk/2, [j 2 + k 2 − 2(j + k)]/4), with both components (indices) positive unless jk = 0 or j = k = 2.
One easily verifies this last claim, about the signature, by using a J x -invariant timelike-spacelike orthogonal decomposition of T x M, at any x ∈ M, to obtain obvious three-summand orthogonal decompositions of both so(TM ) and u(TM ) at x, two summands being spacelike, and one timelike.

Proof of Theorem A
We denote by Ω p,q M the space of complex-valued differential (p, q) forms on a complex manifold M. On such M, as ∂ ζ = 0 whenever dζ = 0, ( Since many expositions do not state what happens when, in the ∂∂ property (0.1), p or q equals 0, we note that, as Fangyang Zheng pointed out to us, (0.1) for (p, 0) forms easily follows from the case where p and q are positive.
Lemma 2.4.For a Killing vector field v on a pseudo-Kähler manifold (M, g), using the notation of (2.5), we have Proof.First, JBJ − B, as well as A = [J, B] and AJ, are g x -skew-adjoint at every point x ∈ M, since so is B = ∇v, and A anticommutes with J, cf.(1.1).Thus, ξ, ζ and γ = i(JBJ − B) are indeed differential forms of degrees 1, 2, 2.

Another proof of Theorem A
On a compact complex manifold M with the ∂∂ property, every cohomology space H k (M, C) has the Hodge decomposition [2, p. 269, subsect.(5.21)]: with each H p,q M consisting of cohomology classes of closed (p, q) forms.The complex conjugation of differential forms descends to a real-linear involution of H k (M, C), the fixed points of which obviously are the real cohomology classes (those containing real closed differential forms).In terms of the decomposition (3.1), a complex cohomology class (3.2) is real if and only if, for all p and q, its H q,p com ponent equals the conjugate of its H p,q component.
for the Nijenhuis tensor N of an almost-complex structure J on a manifold M and any vector fields u, v, clearly becomes when one uses any fixed torsionfree connection ∇ on M. We call ∇ a Kähler connection for the given almost-complex structure J if it is torsionfree and ∇J = 0.By (3.3), J then must be integrable.This implies integrability of J in any pseudo-Kähler manifold, as one then has ∇J = 0 for the Levi-Civita connection ∇.
Lemma 3.1.For any ∇-parallel real 2-form α on a complex manifold M with a Kähler connection ∇, such that α(J • , J •) = −α, the complex-valued 2- We do not know whether -aside from Theorem B and the Riemannian case -Theorem A remains valid without the ∂∂ hypothesis.For possible future reference, let us note that, as shown above, one has the following conclusions about a Killing field v on a compact pseudo-Kähler manifold, whether or not the ∂∂ property is assumed.First, for α = £ v ω, the complex-valued 2-form ζ = α − iα(J • , • ) is parallel and holomorphic (see the preceding proof and Lemma 3.1).Also, by (1.2), α is exact, while A = £ v J : TM → TM is parallel and complex-antilinear, as well as nilpotent at every point.This last conclusion follows since the constant function tr IR A k , with any integer k ≥ 1, has zero integral as a consequence of (1.3) applied to α = g(A• , • ) and θ = g(A k−1 • , • ).

Appendix: Yamada's argument
Yamada's claim [7, Proposition 3.1] that on a compact pseudo-Kähler manifold, Killing fields are real holomorphic, has a proof which reads, verbatim, Let X be a Killing vector field.From Propositions 1.2 and 2.12, Z = X − √ −1 JX is holomorphic.Because the real part of a holomorphic vector field is an infinitesimal auto morphism of the complex structure, we have our proposition.Proposition 1.2 of [7], cited from Kobayashi's book [3], amounts to the well-known harmonic-flow condition satisfied by Killing fields v on pseudo-Riemannian manifolds.Thus, 2.12 in (A.1) should read 2.14, since Propositions 1.2 and 2.14 refer to the Ricci tensor quite prominently, while 2.12 does not mention it at all; also, Proposition 2.14 contains, in its second part, a holomorphicity conclusion.
In the ninth line of the proof of the second part of Proposition 2.14, it is established -correctly -that, for every (1, 0) vector field Y , and Z in (A.1), ∇ ′′ Z is L 2 -orthogonal to ∇ ′′ Y .Then an attempt is made to conclude that ∇ ′′ Z = 0, arguing by contradiction: if ∇ ′′ Z = 0 at some point z 0 , one can -again correctly -find Y having g(∇ ′′ Z, ∇ ′′ Y ) = 0 everywhere in some neighborhood of z 0 .As a next step, it is claimed that a contradiction arises: cited verbatim, By considering a cut off function, we see that there exists a complex vector field Y such that M g(∇ ′′ Z,∇ ′′ Y ) dv = 0.It is here that the argument seems incomplete: such a cut-off function ϕ equals 1 on some small "open ball" B centered at z 0 , and vanishes outside a larger "concentric ball" B ′ , and after the original choice of Y has been replaced by ϕY , there is no way to control the integral of g(∇ ′′ Z, ∇ ′′ (ϕY )) over B ′ B (while the integrals over B and M B ′ have fixed values).More precisely, the sum of the three integrals must be zero, ∇ ′′ Z being L 2 -orthogonal to all ∇ ′′ Y .

(1. 5 )
E is a complex vector bundle of rank m(m − 1)/2, where m = dim C M, with a pseudo Hermitian fibre metric having the real part induced by g.In fact, C → JC provides the complex structure for E. Nondegeneracy of g restricted to E follows from g-orthogonality of the decomposition End IR (TM ) = End C (TM ) ⊕ E ⊕ D, the sections C of the subbundle D being characterized by JC = −CJ and C * = C, with End C (TM ) orthogonal to E ⊕ D since any antilinear morphism C : TM → TM is conjugate, via J, to −C, and so tr IR C = 0.The pseudo-Hermitian fibre metric in E arises by restricting •, • − i J •, • to E, for the pseudo-Riemannian fibre metric •, • in End IR (TM ) induced by g.The rank m(m − 1)/2 follows since so(TM ) = u(TM ) ⊕ E, with u(TM ) ⊆ so(TM ) characterized by having sections C : TM → TM that commute with J (which, due to their g-skew-adjointness, makes them also g c -skew-adjoint, for g c = g − iω): so(TM ) and u(TM ) have the real ranks m(2m − 1) and m 2 .