# Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics

Authors: Daniel Guan
Journal: Geometry
 (2014)

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## Abstract

There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.


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## Body

## 1. Introduction

In the study of existence of Kähler-Einstein metrics Mabuchi introduced the geodesic equations in [1]. It turns out that the special homogeneous complex Monge-Ampére equation Semmes considered in [2] is exactly the same equation considered by Mabuchi. Let (M,ω) be a compact Kähler manifold, and let φ(t,m) be a real smooth function with variables of time t and point m∈M. Regarding t as the real part of a complex number z0, φ is a function of z0 and m. If φ satisfies the complex homogeneous Monge-Ampére equation det(ω+∂∂-φ)=0 and ωt=ω+∂M∂-Mφ is a Kähler metric for t∈[0,T], then ωt is a geodesic in the Mabuchi moduli space of the Kähler metrics.Locally ifF(0,m) is the Kähler potential of ω and F(t,m)=F(0,m)+φ, by letting F(z0,m)=F(t,m) we see that the equation is the same as det(∂∂-F)=0.It was observed that the kernel of∂∂-F induced a vector field of a certain Moser type transformation. And the Moser type vector field W can be applied to any general curve in the space of the Kähler metrics. In this paper we will apply this Moser type transformation to give an explicit construction for the parallel transformation along any curve in the Mabuchi moduli spaces of Kähler metrics. The existence of the parallel transformation was proven in [1, page 234] by integration of a vector field and then in [3], without knowing Mabuchi’s first proof, obtained by solving one of the two quasilinear equations which led to the existence of the geodesics therein. That eventually gave a proof of the opposite direction of [4, 5] in 2003 for the cohomogeneity one case. We will give a different proof of the existence of the parallel transformation in this paper. The corresponding quasilinear equation is actually LWh=0. Our argument here is independent of  [3]. While we dealt mostly with the case with a flat Mabuchi moduli space of Kähler metrics in a given Kähler class in [3], we deal with the general case here.One can also easily observe that the geodesic equation is actuallyLWF˙=0.From the properties of the Moser vector fields we also expect that there should be some applications to metrics flows. We will investigate this in the last three sections and in the near future. In particular, we prove that the curvature is parallel on the Mabuchi moduli space of the Kähler metrics of a given Kähler class.After the first version of this paper, many things happened. It is evident that, to define a generalization of Kähler metrics with constant scalar curvature, instead of the scalar curvature being a potential function of a holomorphic vector field as Calabi did, we could consider with a Kähler metrics with a scalar curvature being a function as the restriction of a parallel function on the equivariant Mabuchi moduli space of Kähler metrics. This is motivated by [6]. For example, in the case of cohomogeneity one with a semisimple automorphism group in [7, 8], the Futaki invariants are zero. Therefore, there are no Calabi extremal metrics in general. However, there are nontrivial parallel functions on the equivariant Mabuchi moduli space and therefore, there are many “generalized” Calabi extremal metrics; see also [9]. In particular, any equivariant Kähler metric is a generalized Calabi extremal metric in this sense. Of course, we need more restriction to get a more meaningful generalized Calabi extremal metric. In [10] in 2001, it was proven that Calabi flow was better than the generalized Ricci flow in general. But it seemly was only true for the low dimensional case. When we tried to publish [10], we proposed that the Calabi flow always has the long time existence property just as what Cao and Koiso observed in the Kähler-Ricci flow. However, it seems possible that it is only true for the low dimensional case. For example, in the case of CPn with the standard cohomogeneity one metrics as in [10], as one end has a high codimension n, the Calabi flow might not always have long time existence. Therefore, it is easier to consider a generalized Calabi metric called the m-extremal metric and the related m-Calabi flow [11]. To make the short time existence easier, one might actually consider an exponential Calabi extremal metric or e-extremal metric and e-Calabi flow.


## 2. Preliminaries

Definition 1.
Letωt be a family of symplectic forms. If a vector field X(t,m) satisfies X(ωt)=0, then we call X a Moser vector field.
See also Moser’s original paper [12].
Regarding∂∂-F=(Fij-) as a matrix, let (Aij-) be the adjoint matrix; then Fij-Ajk-=δikdet(∂∂-F) for any i,k∈(0,1,…,n). Applying this formula we obtain the following.Lemma 2.
Y
=
A
0
0
-
(
∂
/
∂
z
0
)
+
A
0
s
-
(
∂
/
∂
z
s
) is in the kernel if det(∂∂-F)=0.LetW=Re(Y/A00-). In general, we let ωt=(ωt,ij-)=ω0+∂∂-φt be a curve in the space of Kähler metrics even if we do not have det(∂∂-F)=0, and we can define Y and W in the same way with Ft=F0+φt. Then by direct calculation we obtain the following.Lemma 3.
We haveLW(∂M∂-MF)|M=0.Proof.
LetX=Y/A00- and denote ∂i=∂/∂zi,∂i-=∂/∂z-i. Then
(1)LX=L∂0+L(A0s-/A00-)∂s.
By
(2)Lai∂i(dzk)(∂l)=Lai∂i(δkl)-dzk([ai∂i,∂l])=dzk(∂l(ai)∂i)=∂l(ak)=ak,l,
similarly we have
(3)Lai∂i(dzk)(∂l-)=ak,l-;
that is, Lai∂i(dzk)=ak,ldzl+ak,l-dz-l. Similarly, we have Lai∂idz-k=0.
Combining this together with the identity(4)F0m-A00-+Frm-A0r-=0
and its differentiations with respect to zs and z-s, we finally obtain
(5)LX(Fsm-dzs∧dz-m)|M=(F˙sm-+A0r-A00-Fsm-r+Frm-(A0r-,sA00--A0r-A00-,sA00-2))×dzs∧dz-m+Frm-(A0r-A00-)s-dz-s∧dz-m=-A00-,sA00-2(F˙m-A00-+Frm-A0r-)dzs∧dz-m-(Frm-s-A0r-A00-+F˙m-s-)dz-m∧dz-s=0
as required.Proposition 4.
W induces a family of Moser differential transformations T(t) such that T(∂M∂-MF|M)=∂M∂-MF|M.By the parallel transformation formula in [1, 3, 6], we have the following.Lemma 5.
A functionh is a parallel transformation, regarded as a vector field in the moduli space of Kähler metrics, along a path ωt=ω0+∂M∂-Mφ if and only if h˙-(1/2)(dh,dF˙)=0.


## 3. Existence of the Parallel Transformations

In general, letωt=(ωt,ij-)=ω0+∂∂-φt be a curve in the space of Kähler metrics (might not be a geodesic curve) we see that;
(6)A00-(h˙-12(dh,dF˙))=ReA0i-hi=A00-LWh.
Therefore, we have the following.Theorem 6.
h is parallel if and only if LWh=0. In particular, ωt is a geodesic if and only if LWF˙=0.Once we have a curve in the moduli space of Kähler metrics we have the orbits of the Moser vector fieldW. Let h0 be any function on M; regarding the initial vector field at ω0, we can let h be constant along the orbits. Then h is parallel along this given curve in the moduli space. In this way, we have an explicit construction of the vector field h from the initial value h0. Therefore, we have the following.Corollary 7.
Given any curveC:ωt with t∈[0,1] in the moduli space of the Kähler metrics in a Kähler class and an initial vector h0 at ω0, there is a unique smooth parallel vector field along C.In [3], without knowing the first proof of Mabuchi in [1, page 234], we had used Corollary 7 to find global parallel vector fields on the compact complex almost homogeneous Kähler manifolds with algebraic reductive groups and to obtain parallel coordinates. Then by integrating the global parallel vector fields we will obtain many smooth geodesics.


## 4. Poisson Bracket and the Curvature Symmetric Property of the Mabuchi Moduli Space of the Kähler Metrics in a Given Kähler Class

For a given Kähler structure we have a symplectic structure. Therefore, there is a Poisson structure{f,g}=ω*(df,dg) or df∧dg∧ωn-1={f,g}ωn. By identifying {f∈C∞(M)|∫M‍fωn=0} with the tangent space at ω we have infinite dimensional Lie algebra structure on each tangent space. It is actually an infinite dimensional Lie algebra with an invariant metric (more like compact Lie algebra of the finite dimensional case; see [13]). Now we would like to see some more global pictures.Letϕ=ϕ(t) be a curve in the Mabuchi moduli space, and let φi be two parallel vector fields along ϕ for i=1,2.Then in view of Lemma3 and Proposition 4, identifying by the Moser transformation we can consider the family of Kähler structures as the same symplectic structure, and φi being regarded as functions of this symplectic manifold are independent of t by Theorem 6. The Poisson bracket {φ1,φ2} also does not depend on t and therefore is parallel along the curve of the Kähler structures. This parallelism can be expressed as follows.Theorem 8.
We have(7)ddt({φ1,φ2})=12(dϕ˙,d{φ1,φ2}).Therefore, by the formula of the curvature in [1] (see also [2, 14], [15, page 195 Theorem A]) we have that
(8)R(f,g)h=-14{{f,g},h}
we also have that the curvature is constant along the curve by the curvature formula.Theorem 9.
On the moduli space∇R=0.We call an infinite dimensional Riemannian manifoldcurvature symmetric if ∇R=0. Theorem 14 just says that the Mabuchi moduli space of the Kähler metrics in a given Kähler class is curvature symmetric.Therefore, the moduli space has a quite good structure. However, we see in [7] that the moduli space, in general, is incomplete. This gives a negative answer to an expectation of Semmes [2] and Donaldson [14] that the moduli space is an infinite dimensional kind of (complete) symmetric space.On the other hand, our result in this section does give some good properties similar to a convex region (see [15]) in a special symmetric space of nonpositive curvature (cf. [16]).I personally do believe that the Mabuchi moduli space of the Kähler metrics is locally symmetric; that is, at each point there is a local isometric map which fixes the given point and induces a map of-1 on the tangent space.


## 5. The Fourth Geodesic Principle and the Geodesic Stability Conjectures

When the curve is an infinite geodesic ray, we can obtain a good picture for the general geodesic stability provided that there is enough smoothness. Instead of using parallel geodesics, which works for the case with the moduli space being flat as in [7], we could use infinite geodesic rays with the same point at infinity. This was also pointed out in [10, Remark 4]. It is just as in the case of finite dimensional symmetric spaces. For example, two maximal geodesic rays γ1 and γ2 with d(γ1(t),γ2(t))<C for some positive constant C go to the same point at infinity. One can use a finite dimensional nonpositively curved symmetric space as a model, for example, the Poincare metric on the upper half plane Imz>0.In the case with the Poincare metric, the geodesics are straight lines perpendicular to thex-axis and the half circles with the centers on the x-axis. Two geodesic rays go to the same point at infinity if their end points on the x-axis are the same.Now let us go back to the infinite dimensional picture on the Mabuchi moduli space. Along each maximal geodesic ray, since∇R=0, we can have a frame of vector fields such that the sectional curvatures are constant and nonpositive. The corresponding Jacobi field of family of geodesics going to the same point at infinity has the form
(9)J=∑i=0+∞‍aie-bitφi,
where φi are parallel along the geodesic and b0=0<b1<⋯<bi<bi+1<⋯ assuming that there exists an orthogonal basis with respect to both the metric and the curvature along the given geodesic (notice that this is not true in general). See [17, page 15] and notice a sign difference for the curvature. The above formula will be J=∑i=1N‍aie-bitφi for the finite dimensional case and we just replace N by +∞ here. We had dealt with some flat cases in [3] in which ai=0 for i>0.Let us have another way to understand this. LetℳK be the Mabuchi moduli space of the invariant Kähler metrics with respect to a maximal compact subgroup K of the complex automorphism group. Rescale it by a sequence of positive numbers Ai→0. Recall that on the tangent space at ω of the Mabuchi moduli space, the Riemannian metric is
(10)g(ϕ1,ϕ2)=∫M‍ϕ1ϕ2ωn.
We let gAi=Aig and fix a Kähler metric ω. We denote the corresponding marked Riemannian manifold by Ai(ℳK,ω). Assuming that we can have an exponential map, we identify Ai(ℳK,ω) as a subset of (ℳK,ω) by the differentiable map
(11)jAi=exp(ℳK,ω)·iAi·expAi(ℳK,ω)-1,
where exp are the exponential maps at ω and iAi is the natural identity map from the tangent space of Ai(ℳK,ω) to that of (ℳK,ω). Then, for any point ω, the marked infinite dimensional Riemannian manifolds Ai(ℳK,ω) converge as subsets to a cone 𝒞 (rather than a sequence of Riemannian manifolds) with ω as the vertex point. The structure of 𝒞 depends on the choice of ω. Any other point ω1 also converges to the vertex point. The condition that an infinite geodesic rayγ1fromω1goes to the same point at infinity as that of a given infinite geodesic rayγfromωis defined aslimi→+∞γ1=limi→+∞γ.A general Jacobi field on(ℳK,ω) has the form
(12)J=(a01φ01+a02tφ02)+∑n=1+∞‍(an1e-bntφn1+an2ebntφn2),
where ank are constants and φnk are parallel functions for k=1,2 (or an2 are zeros except finite of them).ForAi(ℳK,ω), we have ti=Ait and Aiφn,ik=φnk. Therefore,
(13)Ji=(a01Aiφ0,i1+a02tiφ0,i2)+∑n=1+∞‍(Aian1e-Ai-1bntiφn,i1+Aian2eAi-1bntiφn,i2).Therefore, formally we can let(14)J∞=a02t∞φ0,∞+∑n=1+∞‍an2ebnt∞φn,∞.
Other terms tend to zero. Let M(ω1,ω2) be the Mabuchi functional. For any maximal geodesic ray γ, we let
(15)F(γ)=limt→+∞dMdt(γ(0),γ(t)).
We then have the following.Conjecture 10.
Two infinite geodesic raysγ1 and γ2 have the same generalized Futaki invariant if limi→+∞γ1=limi→+∞γ2.The structure of the Jacobi field strongly supports this conjecture. We already have threegeodesic stability principles in [7]; we might call Conjecture 10 the geodesic stability principle 4.This is exactly what we have in Remark 4 of [10]. But we have more detailed picture here.Let𝒞1 be the subcone of 𝒞 of all incomplete maximal geodesic rays, that is, those only parametrized on half lines; as in [7] we have the following.Conjecture 11.
If we identify𝒞1 as part of the tangent space at a given Kähler metric ω0, then there is an extremal metric if and only if the generalized Futaki invariants of maximal geodesic rays are positive and bounded from blow by a given seminorm.As in [6, 7], by the convex property of the modified Mabuchi functional along the geodesics we see that the generalized Futaki invariant cannot be -∞. The generalized Futaki invariant of a maximal geodesic ray might be +∞. We have the following.Conjecture 12.
The subset𝒞′ of maximal geodesic ray c∈𝒞1 with finite generalized Futaki invariant is a subcone and F(c), c∈𝒞′, is linear on 𝒞′ with respect to J∞.Then we have the following.Conjecture 11′.A Kähler class has an extremal metric if and only ifF(c)≥|c|>0,c∈𝒞′for a given seminorm  |·|.We remark here that we need a seminorm here; that is,F(c)>0,  c∈𝒞1, is not enough as we see in [7] and the Ding-Tian type generalized Futaki invariant therein (cf. [18]) might not come from an F(c) as in [7]. We still need to understand what the given seminorm is. However, our picture of the existence of the extremal metrics is quite clear now.Remark 13.
To convince the readers that the geodesic stability is the right stability we should give a philosophical “proof” for the Kähler-Einstein case. If we consider the Kähler-Ricci flow, Perelman’s estimate, also earlier by Cao and Koiso, shows that the change of the metric is bounded. If a subsequence of the metrics converges to a metric of a finite distance, even a singular metric, then, by the first and the second geodesic stability principles in [7] (see also [8] for updated version), it has a finite Mabuchi functional and the sequence of the Kähler metrics converges to a Kähler-Einstein metric. If there is not any finite limit, by the first stability principle there is a unique geodesic connecting g0 and gt. We call it γt. Extend the γt to be a maximal geodesic ray. Let ti be a series of t such that γti converges to a possibly singular maximal geodesic ray γ. Then the stability condition in the third geodesic stability principle (or our Conjecture 11 here) would imply that the Mabuchi functional would increase along γ eventually, a contradiction.


## 6. Global Parallel Vector Fields

Moreover, by [6] the potential functions of the vector fields of the Lie algebra 𝒦 of K are parallel on the moduli space of the invariant metrics. In particular, the vector potential functions of elements of the center 𝒯 of the Lie algebra 𝒦 of K also induce families of invariant Kähler metrics. Notice that no all the potential functions of Lie algebra in the tangent space of the moduli space of the invariant Kähler metrics—they are in the tangent space the bigger (nonnecessary invariant) moduli space. And it is natural for us to restrict our attention to the smaller moduli space. Therefore, these functions from the center are in the tangent space of the Mabuchi moduli space of the invariant Kähler metrics and are parallel. Thus, there might be some global parallel vector fields on ℳK. We denote the set of all the global parallel vector fields by 𝒫. Then by Theorems 6 and 8, we have the following.Theorem 14.
Letf1,…,fk∈𝒫; then, for any analytic function F(x1,…,xk), the composition function F(f1,…,fk) is parallel. Moreover, the set of all these parallel vector fields is closed under the Poisson bracket.For these parallel vector fields we can obtain the geodesics by the method of Section3. However, as what happened for the toric varieties in [7], unlike those in 𝒯, the maximal geodesic might be incomplete and might actually be finite.Moreover, for anyf1,f2∈𝒫, starting from any invariant metric, one should get a 2-dimensional net of geodesics P2. If we can prove this, then P2 is a geodesic submanifold of ℳK. Since both f1,f2 are parallel on P2, we expect that the curvature is zero. We have the following.Theorem 15.
For anyg, {g,f}=0 if f∈𝒫. In particular, the Poisson structure on 𝒫 is trivial; that is, 𝒫 is abelian.Proof.
Letω be any equivariant Kähler metric, and let g be any function in the equivariant tangent space, f∈𝒫, and let ω(s,t) be a 2-parameter family of Kähler metrics such that ω(0,0)=ω and ω(s,0) has tangent g and ω(0,t) has tangent f. Let f∈𝒫; then R(g,f)f=(∇g∇f-∇f∇g)f=0. Since R(g,f)f=-(1/4){{g,f},f} (see [15, page 195]) we get {{g,f},f}=0 and hence (R(g,f)f,g)=0; that is, ∫‍{g,f}2ωn=0. Therefore, {g,f}=0 for any tangent vector g.
In particular, ifg∈𝒫 also, we have that 𝒫 is abelian.
Now we see that𝒫 is in a way similar to 𝒯. But sometimes 𝒫 is bigger than the algebra of functions generated by 𝒯. For the examples in [3, 7], 𝒯=0 but 𝒫 is generated by the function U there. Let k=supm∈Mdimd𝒫|m; then, at a generic point m∈M, we can have k functions fi∈𝒫 which are independent. We call f1,…,fk a parallel coordinate at m with regarding M as a symplectic manifold.Theorem 16.
TheL2 closure of the infinite-dimensional Lie algebra of equivariant functions TK=TℳK splits as an orthogonal direct sum of the closure of 𝒫 and the closure of another infinite-dimensional Lie subalgebra {TK,TK}.Proof.
We want to prove thatf∈TK is perpendicular to {TK,TK} if and only if f∈𝒫. We have
(16)∫M‍f{g,h}ωn=0
for any g,h∈TK if and only if ∫M‍{h,f}gωn=0 for any g∈TK with any given h∈TK; that is, {h,f}=0 for any h∈TK. In the proof of our Theorem 16 we already see that if f∈𝒫, then {h,f}=0. We can also prove the other direction. If {h,f}=0 for any h∈TK, then R(g,h)f=0. We want to prove that f can be globally defined as a parallel vector field on ℳK. We notice that the Lie algebra {TK,TK} is invariant under the parallel transformation and hence so is the function orthogonal to {TK,TK}. Let ω(s,t) be a family of curves in ℳK and ω(s,0)=ω0, ω(s,1)=ω1. Let f(s,t) be family of functions on M such that f(s,0)=f and fs(t) is parallel along the curves Cs(t)=ω(s,t). Then f(s,t) are orthogonal to {TK,TK}. Therefore, R(g,h)f(s,t)=0 for any g,h∈TK. Now we want to prove that f(s,1) does not depend on s. Since f is parallel along each ωs we have ∇tf=0; then
(17)∇t∇sf=(∇t∇s-∇s∇t)f=R(ft,fs)f=0;
that is, ∇sf is also parallel along each ωt. Since ∇sf=0 at t=0, we always have ∇sf=0. In particular, ∇sf=0 at t=1; that is, f(s,1) is independent of s. Therefore, f can be extended as a global parallel vector field; that is, f∈𝒫.
Now, for anyf∈TK, (f,g)g∈𝒫 defines a functional on the closure of 𝒫. But for the property of a Hilbert space we have a f1 in the closure of 𝒫 such that (f1,g)=(f,g) for any g in the closure of 𝒫. Let f2=f-f1; then f2 is in the closure of {TK,TK} and f=f1+f2.For those maximal geodesic rays generated by the functions in𝒫 we already applied our generalized Futaki invariants in [7]. If M is a compact spherical almost homogeneous manifold under an algebraic reductive group, that is, the K invariant functions are generated by an abelian ideal, then TK=𝒫, for example, when M is cohomogeneity one. In that case, any tangent vector at a point can be extended to a global parallel vector field. In these cases, we can apply Mabuchi’s definition to define the Futaki invariants for the maximal geodesic rays. The existence of the Kähler-Einstein metrics should imply the positivity of the Futaki invariant for those geodesics not coming from any holomorphic vector field. Even for the toric manifolds, this gives new obstructions since the function might be a function of the potential functions of holomorphic vector fields but itself is not a potential function of any holomorphic vector field. That is exactly what we did in [7]. Also, for these manifolds, the invariant Mabuchi moduli space is flat. In general, even though not all the tangent vectors in the tangent space of the invariant Mabuchi moduli space come from parallel vector fields, we can use the parallel ones to get new obstructions just like the original Futaki obstruction. We call the related stability parallel stability; then we expect that for the cohomogeneity one and the spherical cases including the toric manifolds the existence is the same as the parallel stability.We can actually expect that the direct sum can be realized asC∞ functions.Conjecture 17.
We haveTK=𝒫+{TK,TK}.While Conjectures10, 11, 12, and (11′) might be very difficult to be proved, Conjecture 17 would be easier. We should try to prove Conjecture 17 in the near future.Remark 18.
When the groupG is Hamiltonian (this is true whenever the manifold is simply connected, e.g., when M is Fano), that is, every element in the Lie algebra 𝒢 of G corresponds to a given function, from (3) and (4) of [13, page 3362], one could easily see that 𝒫 is generated by the functions related to the Lie algebra of the torus which is the complement of H in J. We notice here that G is compact and J is locally a direct product of a torus and H. And the functions on the symplectic reduction produce the major part of {TK,TK}. To make the picture clearer for the reader, let Φ:M→𝒢* be the moment map with Φ(m)(g)=g(m). On each generic orbit G/H, we have the moment map Ψ=Φ|G/H:G/H→G/J. The codimension of G/J=Ψ(G/H) is the same as the dimension of the torus J/H. Therefore, Ψ(M)//G has a dimension of that of J/H. 𝒫 is the pullback of the functions on Ψ(M)//G and the symplectic reduction MG gives the major part of {TK,TK}. That is, locally TK is the functions on Ψ(M)//G×MG. This basically gives a proof for Conjecture 17.Remark 19.
In the case of  [7], one could easily calculate that θ is the square of the norm of Ψ(m) under the standard product metric of CPn×CPn. Therefore, the name phase angle (or square phase angle) is justified. Similarly, U is the square of the norm of Ψ(m) for the corresponding metric.



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