A moment map for twisted-Hamiltonian vector fields on locally conformally K\"ahler manifolds

We extend the classical Donaldson-Fujiki interpretation of the scalar curvature as moment map in K\"ahler Geometry to the wider framework of locally conformally K\"ahler Geometry.


Introduction
By the foundational work of Fujiki [11] and Donaldson [8], it is known that the scalar curvature of Kähler metrics arise as a moment map for an infinite-dimensional Hamiltonian action.More precisely, given a compact symplectic manifold (M, ω o ), the space J alm (ω o ) of compatible almost complex structures can be endowed with a natural structure of infinite-dimensional Kähler manifold (J, Ω) that is invariant under the action of the automorphism group Aut(M, ω o ) by pullback.Here, an almost complex structure J is said to be compatible if ω o ( , J ) is a J-almost Hermitian (and hence almost Kähler) metric.This action preserves the analytic subset J (ω o ) ⊂ J alm (ω o ) of integrable almost complex structures and the map scal : J alm (ω o ) → C ∞ (M, R), that assigns to any J ∈ J alm (ω o ) the scalar curvature of the metric ω o ( , J ), is smooth, Aut(M, ω o )-equivariant and it verifies Here, ham(M, ω o ) denotes the Lie subalgebra of Hamiltonian vector fields, i.e. those infinitesimal automorphisms X ∈ aut(M, ω o ) such that X ω o = dh X for some h X ∈ C ∞ (M, R), uniquely determined up to a constant, and X * denotes the fundamental vector field associated to X.For this reason, we say that scal is a moment map for the action of Ham(M, ω o ) on J (ω o ), where Ham(M, ω o ) denotes the Hamiltonian diffeomorphism group.This fact has deep consequences in Kähler Geometry.In particular, the constant scalar curvature Kähler metrics arise as zeroes of a moment map equation.More precisely, the problem of finding constant scalar curvature Kähler metrics in a fixed Kähler class is reduced to finding a zero for the moment map in the orbit of the complex structure under the complexified action.Inspired by the Hilbert-Mumford criterion in Geometric Invariant Theory in finite-dimension, this leads to the notion of K-stability, see e.g.[32].
This paper is an attempt to generalize this moment map framework to the non-Kähler setting.More precisely, we deal with a compact almost symplectic manifold (M, ω) with dω = 0, admitting compatible complex structures.We notice here that, in the literature, there appeared other works concerning this problem, see e.g.[3,12], which are distinct from our approach.For other appearance of moment maps for group actions on locally conformally symplectic or Vaisman manifolds, see [20,17,31].However, we stress that the group action that we consider lives in an infinite-dimensional Kähler setting.Recently, García-Prada and Salamon [13] described a setting where the moment map is given by the Ricci form and studied moment map interpretations of the Kähler-Einstein condition.As the anonymous referee suggested, it would be interesting to try to extend this construction to our locally conformally Kähler setting.
One of the first natural non-Kähler cases of study [19] is given by the so-called locally conformally symplectic condition, i.e. we ask that dω = θ ∧ ω for a given closed 1-form θ.Notice that, by introducing the twisted exterior differential operator d θ := d − θ∧, the previous condition can be written as d θ ω = 0.This nomenclature is due to the following characterization: ω is locally conformally symplectic if and only if, for any point x ∈ M , ω is locally conformal to a symplectic form in a neighborhood of x.This shows clearly that the locally conformally symplectic condition is actually a property for the whole conformal class [ω] of ω.Moreover, there is a further relation with the symplectic framework: indeed, (M, ω) admits a unique minimal covering map π : M → M such that π * ω is globally conformal to a symplectic form ω o on M and the deck transformation group acts by homotheties on ( M , ω o ) (see e.g.[18, Section 2] and [6, Section 2.1]).Moreover, a vector field X ∈ X(M ) lifts through π to an infinitesimal automorphism π * X ∈ aut( M , ω o ) if and only if d θ (X ω) = 0 [6, Proposition 3.3] and π * X ∈ ham( M , ω o ) if and only if X ω is d θ -exact [6,Corollary 3.10].Therefore, it is natural to look at the action of the generalized Lie transformation group Aut (M, [ω]) generated by the Lie algebra of special conformal vector fields of (M, ω) (see Section 2.1 and Section 2.2) on the space J alm (ω) of compatible almost complex structures on (M, ω).As in the Kähler setting, one may expect the existence of a moment map, related to the scalar curvature and the 1-form θ, when one restricts aut (M, [ω]) to the Lie subalgebra of twisted-Hamiltonian vector fields and J alm (ω) to the analytic subset J (ω) of integrable almost complex structures.However, two main difficulties occur in this picture.The first one is that the group Aut (M, [ω]) preserves just the conformal class [ω], and so the map scal : J alm (ω) → C ∞ (M, R) that associates to any J ∈ J alm (ω) the Riemannian scalar curvature of ω( , J ) turns out to be non-Aut (M, [ω])-equivariant.The second one is that, during the linearization procedure of the map scal, the term D(θ ) appears, where D denotes the Levi-Civita connection and the metric duality, and it seems to us that it cannot be handled via an absorption scheme by adding extra terms depending on θ itself.
In order to overcome these issues, we impose a further symmetry.More precisely, we consider the symplectic dual of θ, i.e. the unique vector field V ∈ X(M ) such that V ω = θ, and we assume that there exists a compact, connected Lie group K acting effectively on M such that its action preserves ω, is of Lee type, i.e. the flow of V is a 1-parameter subgroup contained in the center of K, and is twisted-Hamiltonian, i.e. the Lie algebra k := Lie(K) is contained in ham(M, [ω]) (see Section 2.3).Then, it turns out that the group Aut (M, [ω]) K := Aut (M, [ω]) ∩ Diff(M ) K preserves the 2-form ω (see (4.5)) and that, for any J ∈ J (ω) K := J (ω)∩J alm (ω) K , the endomorphism D(θ ) is orthogonal to the tangent space T J J alm (ω) K (see (3.18), (3.19) and (3.24)).Here, we denoted by Diff(M ) K the subgroup of diffeomorphisms that commute with K and by J alm (ω) K the subspaces of K-invariant, compatible almost complex structures on (M, ω).Therefore, if we denote by Ham(M, [ω]) the generalized Lie transformation group generated by ham(M, [ω]) (see Section 2.1 and Section 2.2), set Ham(M, [ω]) K := Ham(M, [ω]) ∩ Diff(M ) K , and let C ∞ (M ; R) K be the space of K-invariant smooth functions, the previous arguments allow us to prove the main result of this paper, that is Theorem A (see Proposition 4.1, Proposition 4.2 and Theorem 4.3).Let (M 2n , ω) be a compact, connected, 2n-dimensional, smooth manifold endowed with a locally conformally symplectic structure with non-exact Lee form θ. Assume that there exists a compact, connected Lie group K acting effectively on M such that its action preserves ω (k1), is of Lee type (k2) and is twisted-Hamiltonian (k3).Then, the map Here, we denoted by scal Ch the map that assigns to any J ∈ J (ω) K the Chern-scalar curvature of the locally conformally Kähler metric ω( , J ) (see Appendix A.1).As expected, the moment map framework developed in Theorem A leads to the existence of a Futaki-type invariant.More precisely, if we denote by κ : ham(M, [ω]) K → C ∞ (M, R) K the linear isomorphism that verifies X ω = d θ (κ(X)) for any X ∈ ham(M, [ω]) K (see (2.9) and Proposition 4.1), and by z(k) the center of the Lie algebra k, then the following corollary holds true.
Corollary B (see Corollary 4.4).Under the same hypotheses of Theorem A, the value and the map are independent of J, in the connected components of J (ω) K .
We expect that our results could possibly lead to new notions of canonical metrics and stability conditions in the context of locally conformally Kähler Geometry, playing an analogue role to the constant scalar curvature Kähler metrics and K-stability in Kähler Geometry.Note that the equation µ(J) = µ arising from Theorem A and B does not reduce to the constant scalar curvature Kähler equation on the minimal symplectic covering M (see Remark 4.5).However, in the special case when J is Vaisman, i.e.D(θ ) = 0 at J, then d * θ = 0 at J and so we recover the constant (Chern) scalar curvature equation on M .
The paper is organized as follows.In Section 1, we summarize some basic facts on locally conformally symplectic and locally conformally Kähler geometry.In Section 2, we recall some notions about generalized Lie transformation groups and torus actions.In Section 3, we study the properties of the space J alm (ω) K and we compute some linearization formulas that play a role in the proof of Theorem A. In Section 4, we prove our main results.Finally, in Appendix A we collect some facts and detailed computations concerning the Chern connection and the Weyl connection of a locally conformally Kähler manifold.

Preliminaries and notation
Let M 2n be a compact, connected, orientable, smooth manifold of real dimension 2n.For any vector bundle E → M over M , we denote by C ∞ (M ; E) the space of smooth sections of E. For the sake of notation, we set X(M ) := C ∞ (M ; T M ) and, for any X ∈ X(M ), Θ X t : M → M will be the flow of X, t ∈ R.
We fix an almost symplectic structure on M , i.e. a smooth 2-form ω such that ω x is non degenerate for any x ∈ M , that corresponds to an Sp(2n, R)-reduction of the frame bundle of M .We say that ω is locally conformally symplectic if, for any point x ∈ M , there exist a neighborhood U ⊂ M of x and a smooth function f : U x → R such that e −f ω| U is closed.By the Poincaré Lemma, the local conformal changes are organized in a 1-form θ, called the Lee form of (M, ω), that verifies the condition dω = θ ∧ ω , dθ = 0 . (1.1) By introducing the twisted exterior differential operator from (1.1) it follows that d θ ω = 0.Moreover, as θ is closed, it is immediate to check that d θ • d θ = 0 and so d θ defines a cohomology H * d θ (M, R) called Morse-Novikov cohomology.For later use, we prove the following Lemma 1.1.For any vector fields X, Y ∈ X(M ), the following equalities hold true: where denotes the interior product X ω := ω(X, ).
Proof.Equation (1.3) is a direct application of the Cartan formula and the locally conformally symplectic condition d θ ω = 0.Moreover, by [22,Proposition I.3.10(b)] and (1.3) we get which concludes the proof.
Symplectic structures are recovered for θ = 0.More generally, if θ is exact, then ω admits a global conformal change to a symplectic structure: in this case, we say that ω is globally conformally symplectic.In this paper, we will focus on stricly locally conformally symplectic structures, namely, the case when θ is non-exact.We refer to e.g.[9,5,6,30] and the references therein for an up-to-date account on locally conformally symplectic geometry.In the stricly locally conformally symplectic case, by [35,Proposition 2.1] we have the following We recall (see e.g.[18, Section 2] and [6, Section 2.1]) that the locally conformally symplectic manifold (M, ω) admits a unique, up to equivalence, minimal symplectic covering, that is the data of: • a covering π : M → M with deck transformation group Γ; • an injective group homomorphism ρ : Γ → (R, +) and a smooth function In the following, we will denote the minimal symplectic covering of (M, ω) simply by ( M , ω o ).
We recall that an almost complex structure on M is a section J ∈ C ∞ (M ; End(T M )) such that J 2 = −Id.Notice that J acts on T * M by Jϕ := ϕ • J −1 and therefore it extends to the whole tensor bundle over M .By the Newlander-Nirenberg Theorem, it is known that J is integrable, i.e.M 2n admits a complex manifold structure that induces J, if and only if the Nijenhuis tensor of J vanishes, i.e.
For the sake of notation, any integrable almost complex structure will just be called complex structure.An (almost) complex structure J on (M, ω) is said to be compatible with ω if ω(J , J ) = ω and ω( , J ) x > 0 for any x ∈ M . ( Any such (almost) complex structure induces a Riemannian metric g J := ω( , J ) which is said to be locally conformally (almost) Kähler.Indeed, the pulled back metric π * g J to the minimal symplectic covering of (M, ω) is globally conformal to the (almost) Kähler metric ω o ( , π * J ).We also remark that the Riemannian volume form on M induced by g J coincides with the top exterior power ω n .

Locally conformally Kähler metrics.
Let us fix a compatible complex structure J on (M, ω) and denote by g := g J the corresponding locally conformally Kähler metric.In the following, we will denote by and the musical isomorphisms induced by g, e.g.
Moreover, given two vector bundles of tensors E, F → M and a linear differential operator P : induced by g.We denote by D the Levi-Civita connection of (M, g) and by Rm(X, the Riemannian curvature tensor, the Riemannian Ricci tensor and the Riemannian scalar curvature of (M, g), respectively.Furthermore, we denote by δ the Riemannian divergence acting on symmetric endomorphism fields of (M, g), that can be locally written as [7, Section 1.59] where (ẽ α ) α∈{1,...,2n} is any local g-orthonormal frame on M .The Levi-Civita connections of the local Kähler metrics induced by g glue together to a globally defined connection ∇, called the Weyl connection of (M, J, g) [33,34].By [34,Equation (2.4)], it has the following expression: By the very definition and direct computations, it satisfies [34, Theorem 2.2] where denotes the torsion tensor of ∇.For later use, we prove the following where d θ is the twisted differential operator defined in (1.2).
Proof.By (1.2) and (1.8), we get and so (1.9) follows.Fix now a local g-orthonormal frame (ẽ α ) α∈{1,...,2n} on M .By [7, Lemma 1.60] (see also [7, Errata at page 514]), we get ) and so, by (1.2) and (1.7), we get We also denote by ∇ Ch and scal Ch the Chern connection and the Chern-scalar curvature of (M, J, g), respectively.For the convenience of the reader, we collected in Appendix A some properties of both the connections ∇ and ∇ Ch that will play a role afterwards.
Finally, we recall that a (real) smooth vector field X ∈ X(M ) is said to be J-holomorphic if L X J = 0. Notice that, as J is integrable, by (1.8) we directly get Lemma 1.4.For any X ∈ X(M ) we have In particular, the following three conditions are equivalent:

Twisted Hamiltonian diffeomorphisms and torus actions
Let (M 2n , ω) be a compact, connected, smooth, locally conformally symplectic manifold of real dimension 2n with non-exact Lee form θ. We denote by ( M , ω o ) its minimal symplectic cover and by Γ the deck transformation group of the projection π : M → M .For the sake of notation, we denote by X( M ) Γ the closed subspace of Γ-invariant, smooth vector fields on the total space M and by the Lie algebra of Γ-invariant infinitesimal automorphisms of ( M , ω o ) and the Lie algebra of Γ-invariant Hamiltonian vector fields of ( M , ω o ), respectively.

A remark on infinite dimensional transformation groups.
We its exponential map and by its adjoint representation.Here and in the following, we will not go into details of the theory of infinitedimensional Lie groups.As a good reference, we recommend [27,28].
We also recall that an isotopy to the identity is a smooth map φ : [0, 1] × M → M such that φ t := φ(t, ) ∈ Diff(M ) for any t ∈ [0, 1] , φ 0 = Id and that ϕ ∈ Diff(M ) is said to be isotopic to the identity if there exists an isotopy to the identity (φ t ) t∈[0,1] such that φ 1 = ϕ.Since Diff(M ) is locally connected by smooth arcs, it follows that its identity component Diff(M ) 0 coincides with the subset of diffeomorphisms of M that are isotopic to the identity (see e.g.[4, Section I.1]).We also recall that there exists a bijective correspondence between the set of all the isotopies to the identity and the set of time-dependent vector fields on M , i.e. the smooth maps Z : x ∈ M (see e.g.[24, Theorem 9.48, Exercise 9-20, Exercise 9-21]).This relation is explicitly given by the following system of ODE's: Let now g ⊂ X(M ) be a closed Lie subalgebra of vector fields on M .In general, it is not possible to find a strong ILH-Lie subgroup in the sense of Omori whose Lie algebra is g (see [28,Section III.5]).However, this problem of integrability can be overcome by considering the weaker notion of generalized Lie group in the sense of Omori [28, Definition I.3.1,Definition I.3.2] in the following way.First of all, we call g-isotopy to the identity any isotopy to the identity (φ t ) t∈[0,1] such that the corresponding time-dependent vector field (Z t ) t∈[0,1] verifies Z t ∈ g for any t ∈ [0, 1].Then, the following result holds true.Proposition 2.1.Let g ⊂ X(M ) be a closed Lie subalgebra of vector fields on M and assume that Ad(φ t )X ⊂ g for any g-isotopy to the identity (φ t ) t∈[0,1] , for any X ∈ g .
( ) Then, the subset G := {ϕ ∈ Diff(M ) : there exists a g-isotopy to the identity is a generalized Lie group in the sense of Omori, with Lie algebra g, whose exponential map and adjoint representation are given by the restrictions of (2.2) and (2.3), respectively.
Proof.Let ϕ (1) , ϕ (2) ∈ G, take two g-isotopy to the identity (φ and denote by (Z (i) ) t∈[0,1] the corresponding time-dependent vector field, with i = 1, 2. Then, it is immediate to observe that (φ t ) t∈[0,1] , with φ t := φ (1) t for any t ∈ [0, 1], is an isotopy to the identity and that the corresponding time-dependent vector field (Z t ) t∈[0,1] is given by (2.6) By ( ) and (2.6), it follows that (φ t ) t∈[0,1] is a g-isotopy to the identity and so, since φ 1 = ϕ (1) • ϕ (2) , it follows that G is a subgroup of Diff(M ) 0 .It is straightforward now to check that G verifies all the properties listed in [28, Definition I.According to Proposition 2.1, we refer to the group G defined in (2.5) as the generalized Lie transformation group generated by g.

The twisted-Hamiltonian diffeomorphism group.
Following [25,6], we consider the set of special conformal vector fields of (M, ω) and the subset of twisted-Hamiltonian vector fields of (M, ω) By (1.4), it is easy to observe that both (2.7) and (2.8) are closed Lie subalgebras of X(M ) and that Since ω is non degenerate, from Lemma 1.2 it follows that there exists a unique linear isomorphism Moreover, by (1.4), the isomorphism κ verifies It is remarkable to notice that, by [6], the Lie algebras (2.7) and (2.
where Γ is the deck transformation group of π.
As a direct corollary of Proposition 2.2, by using a lifting argument, one can prove the following According to Proposition 2.1 and Corollary 2.3, we denote by Aut (M, [ω]) and Ham(M, [ω]) the generalized Lie transformation groups generated by aut (M, [ω]) and ham(M, [ω]), respectively.We call them special conformal automorphism group and twisted-Hamiltonian diffeomorphism group of (M, ω), respectively.

A note on twisted-Hamiltonian group actions.
We denote by V ∈ X(M ) the symplectic dual of θ, i.e. the unique vector field on M defined by the condition V ω = θ.Notice that from (1.2) and the very definition, we get and therefore, by (1.3), it follows that L V ω = 0 .
(2.11) Let now K be a compact, connected Lie group that acts effectively on M and k := Lie(K) ⊂ X(M ) its Lie algebra.By compactness, K can be written as the product K = Z(K) • [K, K], where Z(K) and [K, K] denote the center and the commutator of K, respectively.At the Lie algebra level, this decomposition corresponds to k = z(k) ⊕ [k, k].We make the following assumptions on the group K: (k1) the K-action preserves ω; (k2) the K-action is of Lee type, i.e.V ∈ z(k); (k3) the K-action is twisted-Hamiltonian, i.e.K ⊂ Ham(M, [ω]).
Let us notice that condition (k1) is not too restrictive, in view of the following Lemma 2.4.Let K be a compact, connected Lie group acting effectively on M .If M admits a locally conformally symplectic form, then it also admits a K-invariant locally conformally symplectic form.
Proof.Fix K a locally conformally symplectic form on M with Lee form θ. Since the cohomology of M is Kinvariant, see e.g.[10, Proposition 1.28], we can perform a conformal change in order to make θ K-invariant.
Then, the statement follows by standard averaging methods.
Notice that, by condition (k2), the group K cannot be semisimple.For example, the standard actions of SU(2) on the Hirzebruch surfaces are not of Lee-type.A less trivial example of a non-Lee-type action is given by the homogenous action of U(2) on the linear Hopf manifold.Indeed one can directly check that the vector V turns out to be tangent to the semisimple part of U(2), see [2, Prop.3.17].
Remark 2.5.Let L be the flow of V , i.e. the 1-parameter subgroup of Diff(M ) defined as (2.12) We notice that L ⊂ Ham(M ) thanks to (2.10).Consider now its closure L in Diff(M ) with respect to the compact-open topology.Notice that, if there exists an almost complex structure J compatible with ω such that L V J = 0 , (2.13) then, by (2.11), V is a Killing vector field for the Riemannian metric g J = ω( , J ).Therefore, by the Myers-Steenrod Theorem, it follows that L = T is a torus, whose action on M clearly satisfies conditions (k1) and (k2).We stress that, if (M, ω) admits a compatible complex structure Ĵ whose associated Riemannian metric is Vaisman, then condition (2.13) is automatically satisfied and thus it admits a torus action of Lee type.On the other hand, in [26], the authors construct examples of compact locally conformally Kähler manifolds satisfying (2.13) which are not Vaisman.
Concerning Remark 2.5 it is worth mentioning that, in general, even when T = L is compact, it is not obvious that condition (k3) is satisfied, i.e. that T is contained in Ham(M, [ω]).Indeed, by means of Proposition 2.2, it follows that the special conformal automorphism group Aut (M, [ω]) is closed in Diff(M ) and so, since d θ (V ω) = 0, we get t := Lie(T) ⊂ aut (M, [ω]).On the other hand, it is not known, to the best of our knowledge, whether the twisted-Hamiltonian diffeomorphism group Ham(M, [ω]) is closed or not, and therefore we do not know whether the inclusion t ⊂ ham(M, [ω]) holds in general or not.We remark that the corresponding problem in the classical setting has been solved in [29], where the author proved, as a corollary of his main theorem, that the Hamiltonian diffeomorphism group of a compact, connected symplectic manifold is closed in the diffeomorphism group with the C ∞ -topology.

K-invariant, compatible complex structures
Let (M 2n , ω) be a compact, connected, smooth, locally conformally symplectic manifold of real dimension 2n with non-exact Lee form θ. Assume that K is a compact, connected Lie group that acts effectively on M and verifies (k1), (k2), (k3) as in Section 2.3.In the following, for any vector bundle of tensors E → M over M , we denote by C ∞ (M ; E) K the closed subspace of smooth sections of E that are K-invariant.

The space of K-invariant, compatible almost complex structures.
Let us define J alm (ω) K := J almost complex structure on M compatible with ω as in (1.5) and K-invariant .(3.1) Following [11,Remark 4.3], the space J alm (ω) K has a natural structure of smooth ILH manifold in the sense of Omori [28, Definition I.1.9],since it can be regarded as the space of K-invariant smooth sections of a fibre bundle with typical fibre the Siegel upper half-space, see also [23,Theorem A].Note indeed that, in the classical picture for almost Kähler structures, one does only need ω to be non-degenerate.
Remark 3.1.By Remark 2.5, it follows that if the closure L of the 1-parameter group generated by V is non-compact, then the space of L-invariant almost complex structure on M compatible with ω is empty.
We recall now the characterization of the tangent space of J alm (ω) K .In the following, we denote by sp(T M, ω) ⊂ End(T M ) the subbundle defined fiber-wise by Lemma 3.2.The tangent space of J alm (ω) K at J is given by Proof.Let (J t ) t ⊂ J alm (ω) K be a curve starting at J = J 0 with initial tangent vector u = J 0 ∈ T J J alm (ω) K .In particular, J t satisfies J 2 t = −Id and ω(J t , J t ) = ω, and so 0 u ) , which proves the first inclusion Notice now that any u ∈ C ∞ (M ; sp(T M, ω)) satisfying Ju + uJ = 0 can be written as Finally, for any a ∈ C ∞ (M ; sp(T M, ω)) K , we define J t := exp(−ta)J exp(ta) and we observe that J t ∈ J alm (ω) K for any t ∈ R, J 0 = J and J 0 = [J, a] = âJ , which proves the remaining inclusion.
As a consequence of (3.2), any K-invariant tensor field a ∈ C ∞ (M ; sp(T M, ω)) K determines a vector field on J alm (ω Proof.Take a, b ∈ C ∞ (M ; sp(T M, ω)) K and notice that the flow of the basic vector field â is which proves (3.4).
Following [11, page 179] (see also [16, page 227]), we define the tautological complex structure J and the tautological symplectic structure Ω by setting ) Here, , L 2 (M,g J ) denotes the standard L 2 -pairing induced by g J .We refer to the triple (J, G, Ω) as the tautological Kähler structure of J alm (ω) K .This nomenclature is due to the following result (compare with [11,Theorem 4.2]).
Proposition 3.4.The tautological Kähler structure (J, G, Ω) defined in (3.5) is a Kähler structure on the ILH manifold J alm (ω) K with respect to which the basic vector fields (3.3) are holomorphic Killing.
Proof.By the very definitions, it is straightforward to realize that (J J , G J ) is a linear Hermitian structure on T J J alm (ω) K for any J ∈ J alm (ω) K .Since the flow Θ Jâ of the vector field Jâ is given by Θ Jâ t (J) = exp(−tJa)J exp(tJa) , a direct computation shows that for any and so, by (3.2), it follows that L âJ = 0 .
(3.6)An analogous computation shows that L Jâ J = 0 and so By [23,Proposition 1.4], the integrability of J is equivalent to the vanishing of the Nijenhuis tensor N J and so, by ( , it follows that J is integrable.
A straightforward computation shows that for any and so Notice that, in virtue of (3.2), this implies that and so, from (3.6) and (3.8) it follows that the basic vector fields are holomorphic Killing.Finally and this concludes the proof.Finally, we introduce the subset J (ω) K := J ∈ J alm (ω) K : J is integrable . (3.9) In the same spirit as [11,Theorem 4.2], one can show that J (ω) K is an analytic subset of J alm (ω) K .As already mentioned in Section 1, any element J ∈ J (ω) K yields a locally conformally Kähler structure (J, g J = ω( , J )) on M , with fundamental (1, 1)-form ω, such that K ⊂ Aut(M, J, g J ).

Linearization formulas.
The aim of this subsection is to compute the linearization of some geometric quantities related to the locally conformally Kähler structures induced by elements in J (ω) K .
Fix J ∈ J (ω) K and a direction âJ ∈ T J J alm (ω) K , for some a ∈ C ∞ (M ; sp(T M, ω)) K .For the sake of shortness, we set g := g J and å := −J âJ .An easy computation shows that the endomorphism å is g-symmetric, J-anti-invariant and trace-free, i.e. g(å(X), Y ) = g(X,å(Y )) , Jå + åJ = 0 , tr(å) = 0 . (3.10) In the following, for the sake of shortness, given any function F defined on J alm (ω) K , we will denote by F the differential of F at J in the direction of âJ .A straightforward computation proves the following Lemma 3.6.The following equalities hold true: Moreover, if we denote by Ric ∇ the Weyl-Ricci tensor of (M, J, g), then it follows that We recall that the endomorphisms (Ric ∇ ) is g-symmetric and J-invariant (see Corollary A.5 in Appendix A) and so, by (3.10), we get g(å, (Ric ∇ ) ) = 0 .

The moment map and the Futaki invariant
Let (M 2n , ω) be a compact, connected, smooth, locally conformally symplectic manifold of real dimension 2n, with non-exact Lee form θ, and assume that K is a compact, connected Lie group that acts effectively on M and verifies (k1), (k2), (k3) as in Section 2.3.
Let us consider the group of K-invariant, special conformal automorphisms of (M, ω) and the subgroup of K-invariant, twisted-Hamiltonian diffeomorphisms of (M, ω) with Diff(M ) K := {ϕ ∈ Diff(M ) : ϕ • ψ = ψ • ϕ for any ψ ∈ K}.Analogously, we consider the Lie algebra of K-invariant, special conformal vector fields of (M, ω) together with the Lie subalgebra of K-invariant, twisted-Hamiltonian vector fields of (M, ω) b) The map κ defined in (2.9) restricts to a linear isomorphism ham(M, Therefore, by Lemma 1.2, this means that θ(X) = 0 and so (4.5) follows by (1.3).Fix now X ∈ ham(M, [ω]).By definition, the action of K preserves ω, and hence it preserves θ as well.Therefore, for any ψ ∈ K, we get which implies, together with Lemma 1.2, that X is K-invariant if and only if κ(X) is K-invariant.
In the following proposition, we introduce a natural action of the group Aut (M, [ω]) K on the space J alm (ω) K of K-invariant, compatible, almost complex structures and we prove that it preserves the tautological symplectic structure Ω defined in (3.5) and the subset J (ω) K defined in (3.9).Proposition 4.2.The generalized Lie transformation group (M, [ω]) K acts on J alm (ω) K by pull-back and this action preserves both the subset J (ω) K ⊂ J alm (ω) K and the symplectic form Ω. Moreover, for any X ∈ aut (M, [ω]) K , the fundamental vector field X * ∈ C ∞ J alm (ω) K ; T J alm (ω) K associated to X is Proof.The action (4.6) is clearly well defined.Moreover, since the integrability of any J ∈ J alm (ω) K is equivalent to the vanishing of the Nijenhuis tensor N J , it follows that the subset J (ω) K is Aut (M, [ω]) Kinvariant.Formula (4.7) follows directly from the definition of fundamental vector field and (4.6).Finally, since Aut (M, [ω]) K is connected by smooth arcs, in order to check that it preserves the symplectic form Ω, it is sufficient to prove that L X * Ω = 0 for any X ∈ aut (M, [ω]) K .Therefore, fix X ∈ aut (M, [ω]) K , J ∈ J alm (ω) K and take two basic vector fields â, b on J alm (ω) K .A straightforward computation shows that [X * , â] = L X a and then, by the infinite dimensional Cartan formula, we compute Therefore, by (4.5), we get and so the thesis follows from (3.2).
Notice that the existence of a complex structure J ∈ J (ω) K such that µ(J) = µ forces F to vanish and thus it is an obstruction to the existence of these special locally conformally Kähler metrics.where π : M → M denotes the projection and f : M → R is the smooth function that verifies π * ω = e f ω o .Therefore, the moment map µ defined in (4.8) does not correspond to the scalar curvature map on the minimal symplectic covering.
Any vector field of the form (3.3) will be called basic.As in [16, Equation 9.2.7], we prove the following Lemma 3.3.The Lie bracket of two basic vector fields â, b on J alm (ω) K is basic and given by [â, b] = [a, b] , with [a, b] = ab − ba .

. 4 )
By the very definitions, it is immediate to check that both aut (M, [ω]) K , ham(M, [ω]) K verify the property ( ) and that Aut (M, [ω]) K , Ham(M, [ω]) K are the generalized Lie transformation groups generated by them.The first ingredient to construct a moment map in our setting is the following Proposition 4.1.The following properties hold true.a) The Lie algebra aut (M, [ω]) K preserves ω, i.e.L X ω = 0 for any X ∈ aut (M, [ω]) K .(4.5)