Compact lcK manifolds with parallel vector fields

Abstract We show that for n > 2 a compact locally conformally Kähler manifold (M2n , g, J) carrying a nontrivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by a compact Kähler manifold of dimension 2n − 2 and a real function.


Introduction
A locally conformally Kähler (lcK) manifold is a Hermitian manifold (M, g, J) of real dimension 2n ≥ 4 such that around each point, g is conformal to a Kähler metric relative to J, cf. [3].
The differentials of the (logarithms of the) conformal factors glue up to a well-defined closed 1-form on M -called the Lee form of the lcK structure -which is exact if and only if (M, g, J) is globally conformally Kähler.
Many complex manifolds which for topological reasons do no carry any Kähler metric, have compatible lcK metrics. For example the product metric on the Hopf manifold S 1 × S 2n−1 (with odd first Betti number) is lcK with respect to the complex structure induced from the identification The Lee form of this structure is easily computed to be the length element of S 1 , and is therefore parallel. Compact lcK manifolds with parallel Lee form are called Vaisman [7], and their structure is well-understood: they are mapping tori of automorphisms of Sasakian manifolds cf. [6]. Moreover, it was recently proved that every compact homogeneous lcK manifold is Vaisman [4].
In real dimension 4 it is well known that a compact complex manifold carries a compatible Kähler metric if and only if its first Betti number is even [2], [5]. It was generally believed that every complex surface with odd first Betti number would carry a compatible lcK structure, until Belgun has shown that some Inoue surfaces do not carry any lcK structure [1]. He also showed that every Hopf surface admits a compatible lcK metric, and classified all Vaisman complex surfaces.
In this paper we address the following question: Are there non-Vaisman compact lcK manifolds which carry a non-trivial parallel 1-form? It turns out that the answer to this question is positive, and moreover, one can describe the lcK structure of such manifolds in a very explicit way in all dimensions greater than 4 (cf. Theorem 3.5 below). These manifolds are globally conformally Kähler, but the metric is not Kähler in general. In dimension 4 this construction still gives examples of non-Vaisman lcK manifolds carrying a parallel 1-form, but we do not know whether these are the only examples.
A more general problem, which however will not be considered here, would be to describe all compact lcK manifolds with special holonomy (e.g. with reducible holonomy, or whose holonomy group belongs to the Berger list). Note that unlike Kähler manifolds, the Riemannian product of lcK manifolds is no longer lcK (at least not in a canonical way). This somehow indicates that the holonomy reduction of a lcK metric is a strong condition, which might lead in general to classification results in the vein of Theorem 3.5.

Some preliminaries on lcK manifolds
As explained in the introduction, a lcK manifold is a Hermitian manifold (M, g, J) of real dimension 2n ≥ 4 carrying an open cover U α and real maps f α : U α → R such that (U α , e −fα g, J) are Kähler manifolds. Denoting Ω(·, ·) := g(J·, ·) the fundamental form of M, the above condition yields Since the linear map Λ 1 M → Λ 3 M defined by σ → σ ∧ Ω is injective, (2.1) shows that df α = df β on U α ∩ U β , so the 1-forms df α glue together to a closed form θ on M -called the Lee form -such that θ| Uα = df α for all α. The Levi-Civita covariant derivatives ∇ and ∇ α of the conformal metrics g and e −fα g on U α are related by the well known formula where θ ♯ is the vector field dual to θ via the metric g. Using the fact that ∇ α J = 0 on U α , we thus obtain: Identifying 1-forms with vectors using the metric g =: ·, · , this relation can be equivalently written as If e i denotes a local orthonormal basis of TM we have Ω = 1 2 i e i ∧ Je i , so by (2.3) we immediately get which also follows from (2.1).

Parallel vector fields on lcK manifolds
Assume throughout this section that that the dimension of M is strictly larger than 4 and that V is a non-trivial parallel vector field on M. We can of course rescale V such that it has unit length. Consider the components of θ along V and JV : Since ∇V = 0 we have ∇ X (JV ) = (∇ X J)V , so using (2.3) we get In particular we have Taking the exterior product with V in this relation yields V ∧ da + 1 2 aθ ∧ Ω = 0, and since by assumption n > 2 we get V ∧ da + 1 2 aθ = 0, so there exists some function f on M such that (3.5) da Since θ is closed and V is parallel, the Kostant formula yields .6) and (3.7) we compute at x: This relation is tensorial in X, so it actually holds at every point of M.
Remark now that if A ∧ V + B ∧ JV = 0 for some vectors A and B, then both vectors belong to the plane generated by V and JV . The previous relation thus shows that there exist some 1-forms µ and ν such that We take the exterior product with X in this relation and sum over some local orthonormal basis X = e i . As dθ = 0, we get µ ∧ V + ν ∧ JV = 0, hence by the previous remark there exist smooth functions α, β, γ on M such that µ = αV − γJV and ν = γV + βJV . Taking X = V in Equation (3.9) and using (3.6) yields Equation (3.9) thus becomes Using this relation together with (2.2) we readily obtain In particular the exterior derivative of Jθ reads We now take the scalar product with V in (3.12) and obtain As n > 2, this shows that (3.18) 2df + f θ = aαV.
We now use (3.11) in order to express the differential of the square norm |θ| 2 . For every tangent vector X we have so from (3.10) we get d|θ| 2 = 2(f − α)θ − |θ| 2 θ + 2aαV + 2bβJV = −2f θ + 2aαV + 2bβJV, whence using (3.18): We are now ready to prove the key result of this section From now on M will be assumed compact.
Lemma 3.2. The following relations hold: Proof. Taking the covariant derivative in (3.20) with respect to some arbitrary vector X and using (3.2), (3.5) and (3.15) yields: Comparing with (3.11) we thus get: 2 X, aV + bJV (aV + bJV ) + α X, V V + β X, JV JV and identifying the corresponding terms yields the result.
Using Lemma 3.2 we now get from (3.5):  which by symmetrization gives: We first note that for each s, t the metric h(s, t) is Kähler. Indeed, J defines by restriction to D an integrable complex structure on each local leaf R n−2 , whose Kähler form Ω(s, t) is just the restriction of Ω. Consequently, dΩ(s, t) is the restriction to the leaves of dΩ = θ ∧ Ω, which vanishes since θ| D = 0. The fundamental group of M induces a co-compact group of isometries of the globally conformally Kähler manifold (M,g) := (R 2 × N, ds 2 + dt 2 + e 2c(t) g N ). Our aim is to show that the Lee form of M is exact. Note that the Kähler form ofM isΩ = ds ∧ dt + e 2c(t) Ω N , which satisfies showing that the Lee form ofM is 2dc. It suffices to check that the function c is Γ-invariant. This follows from a more general statement: Lemma 3.4. Assume that (N d , g N ) is a complete simply connected Riemannian manifold of dimension d ≥ 1, c : R → R is a smooth function and Γ is a co-compact group acting totally discontinuously by isometries on the Riemannian manifold (R 2 × N, ds 2 + dt 2 + e 2c(t) g N ). Assume moreover that Γ preserves the vector fields ∂ s and ∂ t . Then the function c is invariant by Γ.
Proof. The last assumption shows that every element γ ∈ Γ has the form γ(s, t, x) = (s + s γ , t + t γ , ψ γ (x)), where s γ and t γ are real numbers and ψ γ is a diffeomorphism of N. The condition that γ is an isometry of the metric ds 2 + dt 2 + e 2c(t) g N reads Thus ψ γ is a homothety of (N, g N ) with ratio (3.23) ρ γ := e c(t)−c(t+tγ ) (note that, in particular, this expression does not depend on t).
Assume, for a contradiction, that c is not Γ-invariant. By (3.23), there exists γ 0 ∈ Γ such that ρ γ 0 < 1. The map ψ γ 0 is a contraction of the complete metric space (N, d N ), where d N is the distance induced by g N . By the Banach fixed point theorem, ψ γ 0 has a unique fixed point x 0 ∈ N and Let γ be any element of Γ. For every integer k ∈ N we have , so by (3.24), the sequence {y k } converges to (s γ , t γ , x 0 ) =: y 0 . Since the action of Γ is totally discontinuous, this implies that y k = y 0 for k sufficiently large, whence ψ γ (x 0 ) = x 0 for every γ ∈ Γ.
Consider now the continuous map f : R 2 × N → R + defined by f (s, t, x) := e c(t) d N (x, x 0 ). Using (3.23) an immediate induction shows that thus showing that c is onto on R.
In particular, f is onto on R + .
For every γ ∈ Γ we have using (3.23): ( Thus f is Γ-invariant and induces a continuous mapf : Γ\(R 2 × N) → R. Since f is onto,f is also onto, contradicting the fact that the action of Γ on R 2 × N is co-compact.
Summarizing, we have proved: Theorem 3.5. Let (M, g, J, θ) be a compact lcK manifold of complex dimension n > 2 admitting a non-trivial parallel vector field V . Then the following (exclusive) possibilities occur: (i) The Lee form θ is a (non-zero) constant multiple of V ♭ , so M is a Vaisman lcK manifold. (ii) (M, g, Ω, θ) is globally conformally Kähler and there exists a complete simply connected Kähler manifold (N, g N , Ω N ) of real dimension 2n − 2, a smooth real function c : R → R and a discrete co-compact group Γ acting freely and totally discontinuously on R 2 × N, preserving the metric ds 2 + dt 2 + e 2c(t) g N , the Hermitian 2-form ds ∧ dt + e 2c(t) Ω N and the vector fields ∂ s and ∂ t , such that M is diffeomorphic to Γ\(R 2 × N), and the structure (g, Ω, θ) corresponds to (ds 2 + dt 2 + e 2c(t) g N , ds ∧ dt + e 2c(t) Ω N , dc) through this diffeomorphism.