1 Introduction↩︎

A foundational idea in complex geometry is that geometric PDEs arise in a canonical manner in the study of given geometric structures. The most prominent structure where this takes place is that of complex manifolds themselves, where the natural geometric PDE asks for the scalar curvature of a Kähler metric to be constant (producing a cscK metric). The search for cscK metrics is then connected with stability conditions in algebraic geometry through the Yau–Tian–Donaldson conjecture, and is further related to the moduli theory of complex manifolds. In this work, we develop an analogous framework in three related complex-geometric settings: maps between complex manifolds, submersions, and foliations. In each of these three settings the twisted scalar curvature plays a crucial role. This augments the usual scalar curvature \(S(\omega)\) of a Kähler metric \(\omega\) by subtracting the contraction \(\Lambda_{\omega}\alpha\) of an auxiliary closed \((1,1)\)-form \(\alpha\); the form \(\alpha\) is typically assumed to possess positivity properties related to the geometry of the setup.

Beginning in the simplest case, namely that of maps between complex manifolds, we demonstrate that the twisted scalar curvature arises as the natural geometric PDE associated to a Kähler metric \(\omega\) on the source manifold, where the twist \(\alpha\) is induced by pulling back the Kähler metric from the target (so in particular \(\alpha\) is semipositive). The connection between the setting of maps and the settings of submersions is that given a holomorphic submersion, one obtains a map from the base to the moduli space parametrising the fibres, and this moduli space admits a Weil–Petersson form. In this second setting, however, we demonstrate that the natural metric geometry in fact involves a novel coupled system involving both the fibrewise scalar curvature and the twisted scalar curvature, where the twist is given by the Weil–Petersson form. The third setting, namely foliations, is a generalisation of the second, as a holomorphic submersion induces in a natural way a holomorphic foliation which further has compact leaves. Here again a natural coupled system intertwining leafwise and transverse geometry is derived, governing the metric geometry of foliations.

The way in which the scalar curvature is derived as the natural geometric PDE associated to a Kähler metric on a compact complex manifold is through the classical work of Donaldson [1] and Fujiki [2], where the scalar curvature is viewed as a moment map on the space of almost complex structures on a fixed compact symplectic manifold. The present paper is thus devoted to the construction of moment maps in our three settings, namely maps, submersions and foliations. We take two approaches to this; first in an infinite-dimensional framework modeled on the Donaldson–Fujiki picture, and secondly in a finite-dimensional framework modeled on the more recent work of the first author and Hallam [3].

Most of our work is in the setting of submersions, and so we next explain that setting in some detail. Consider a symplectic submersion \[\pi: (M,\omega_M) \to (B,\omega_B)\] between compact manifolds of dimensions \(2m+2n\) and \(2n\) respectively, so that \(\omega_M\) is a closed \(2\)-form on \(M\) which is symplectic on each fibre of \(\pi\), and \(\omega_B\) is a symplectic form on \(B\). Let \(\mathscr J^{\mathop{\mathrm{Int}}}_{\pi}\) be the space of pairs of integrable almost complex structures \((J_M,J_B)\) on \(M\) and \(B\) respectively, compatible with the various structures and making \(\pi\) holomorphic. The space \(\mathscr J^{\mathop{\mathrm{Int}}}_{\pi}\) admits a semipositive closed \((1,1)\)-form \(\Omega\) on \(\mathscr J^{\mathop{\mathrm{Int}}}_{\pi}\) defined through an \(L^2\)-pairing in the spirit of Donaldson–Fujiki.

The space \(\mathscr J^{\mathop{\mathrm{Int}}}_{\pi}\) further admits a natural action of a symplectic gauge group \(\mathcal{G}\), given by the diffeomorphisms of \(M\) and \(B\) preserving the respective forms, in a Hamiltonian fashion compatible with the map \(\pi\). Since our setup assumes the action of \(\mathcal{G}\) is Hamiltonian, we obtain tautological moment maps \(\mu_M: M \to \mathop{\mathrm{Lie}}\mathcal{G}^*\) and \(\mu_B: B \to \mathop{\mathrm{Lie}}\mathcal{G}^*\). We note here that we do not assume that \(\omega_M\) is globally nondegenerate, but merely fibrewise nondegenerate; throughout this work nondegeneracy is not assumed in the definition of a moment map. The semipositive form \(\Omega\) on \(\mathscr J^{\mathop{\mathrm{Int}}}_{\pi}\) is then also \(\mathcal{G}\)-invariant, and so it makes sense to ask for a moment map.

To unwind the notation used above, \(J_M\) induces an almost complex structure on each fibre \((M_b,\omega_b) = \pi^{-1}(b),\) \(S_{V}(\omega_M,J_M)\) is the vertical (or fibrewise) scalar curvature of the resulting relatively Kähler metric, and \(\hat{S}_{V}\) its average. The family of Kähler manifolds \(\pi: (M,\omega_M,J_M) \to (B,J_B)\) induces a Weil–Petersson form on \(B\) denoted by \(\alpha_{\pi}\), with corresponding average twisted scalar curvature \(\hat{S}_{\pi}\).

As mentioned above, we also prove a version of this result for a finite-dimensional family of holomorphic submersions (see Theorem 11). Our proofs in the finite-dimensional and infinite-dimensional settings differ considerably: the former involves more direct tools from equivariant differential geometry, while the latter involves a more indirect adiabatic limit (or asymptotic expansion) argument, which is tightly tied to the geometry of submersions.

In both approaches we obtain a coupled system of equations, intertwining in a natural way the fibrwise and basic geometries of the submersion. It is important to observe that this is not in a natural way a sum of moment maps, and both terms are crucial in obtaining the moment map property. Zeroes of the moment map naturally correspond to fibrewise cscK metrics, such that the resulting twisted scalar curvature on \(B\) is orthogonal to a naturally induced function space (c.f.Remark 9). The form \(\Omega\) is merely semi-positive; it becomes strictly positive on a natural quotient of \(\mathscr J^{\mathop{\mathrm{Int}}}_{\pi}\) where essentially ignore the mixed term in \(J_M\) (as we explain in Section 6.2).

We mention that there is a strong parallel between the theory we develop and the coupled Kähler Yang–Mills equations of [4]: their geometric setting is a holomorphic vector bundle on a compact Kähler manifold, and their moment map setup induces a coupled system of equations intertwining the geometry of the bundle and the geometry of the base. Our system shares many of the features established in their setting, intertwining instead the relative geometry of a submersion with the geometry of the base.

Our primary motivation for the perspective we take comes from the geometric role the twisted scalar curvature plays in certain aspects of complex geometry. This arises most notably through work of Fine [5] (where it relates to cscK metrics on the total space of submersions and which is where the twisted scalar curvature first arose), Song–Tian [6] (in the analytic approach to the minimal model programme through the Kähler–Ricci flow), and Gross–Wilson [7] (where it arises on the base of Calabi–Yau fibrations; see e.g.Tosatti [8] for further developments). In each of these settings, the twisted scalar curvature appears on the base of submersions where the fibres admit cscK metrics (though singularities are typically present in the latter two).

A moment map interpretation of the twisted scalar curvature was presented, motivated in part by these results, by Stoppa [9], essentially by summing the moment map interpretations for the scalar curvature and the “twisting” term \(\Lambda_{\omega}\alpha\). This work motivates connections between twisted scalar curvature and algebro-geometric stability conditions, and as part of this work, Stoppa produced the first algebro-geometric obstruction to the existence of such metrics. From the geometric perspective taken here, the interpretation given by Stoppa appears less related to the role the twisted scalar curvature has played in the theory of submersions, which is our primary motivation. We note in addition that in the work of Stoppa one does not see a coupled system as the natural moment map. In particular, our approach geometrises the appearance of the twisted scalar curvature within the theory of submersions, and in particular its role in connection to the geometry of the base of families. This appearance has traditionally been viewed as arising from the resulting map to the moduli space of cscK manifolds from the base of a submersion, and our work explains this from a moment map perspective.

A related motivation for our work is the understanding of the global geometry of the Kähler manifold \((M,J_M,\omega_M+k\omega_B)\) for \(k\gg 0\), which is the “adiabatic limit” [5], [10]. Here, from an analytic perspective, the geometry splits into three pieces: the fibrewise geometry, the base geometry of the map from the base to the moduli space parametrising the fibres, and the global “twisting” geometry of the total space. This third geometry corresponds to the theory of optimal symplectic connections [11], [12], generalising the Hermite–Einstein condition when \(M=\mathbb{P}(E)\) is the projectivisation of a vector bundle, and a satisfactory moment map interpretation for this equation has been provided by Ortu [13] extending the classical Atiyah–Bott picture.

What our work suggests is that one cannot decouple the remaining two geometric aspects, and that to obtain a satisfactory moment map interpretation one must actually consider a coupled system intertwining the fibrewise and basic geometry. This necessity is closely related to algebro-geometric counterparts of these various conditions, where the basic analogue of the Yau–Tian–Donaldson conjecture involves a notion of algebro-geometric stability for the map from the base to the moduli space [14]. It is also closely related to an observation of the first and third authors [14] in which it is more natural to consider this moduli space as a moduli stack, which can loosely be viewed as a corresponding algebro-geometric coupling.

Moment map interpretations have many applications, as they suggest deep links with algebraic geometry and further to moduli theory. In a more practical direction, moment map results are closely related to (and imply) the existence of Lie algebra characters on a fixed structure (so for us, on a fixed holomorphic submersion), in the same manner as the classical Futaki invariant associated to holomorphic vector fields on a compact Kähler manifold [15]. We construct an analogous coupled invariant, associated to a holomorphic vector field on \(X\), which we call the transverse Futaki invariant. This invariant is defined relative to a choice of Kähler metric on the base; we prove:

As well as being a formal consequence of our moment map results, we also give a direct argument, analogously to the original work of Futaki. We do so for two reasons: firstly, this result will motivate some of the basic ideas of our moment map results, particularly in our adiabatic arguments; secondly, the concrete calculations demonstrate some of the basic analyic features of our coupled system, such as its linearisation, that we expect to play a foundational role in the surrounding theory.

In the final part of this work, we turn to the problem of extending our results to a more general setting. Rather than a submersion \(\pi: (M,\omega_M) \to (B,\omega_B)\), we consider a foliation \(\mathcal{F}\subset TM\) along with a leafwise symplectic form \(\omega_M\) and a transverse symplectic form \(\omega_B\). After defining a natural analogues of the space of integrable almost complex structure, we prove an analogous claim to Theorem 1, namely that a coupled system involving both the leafwise and transverse scalar curvature appears as the natural moment map; Theorem 2 is similarly proven in the generality of foliations.

Organisation of the paper↩︎

We begin in Section 2 with background material concerning equivariant differential geometry, and review how to view the scalar curvature as a moment map. We consider two frameworks, the first through ideas from equivariant differential geometry (where what we explain is primarily finite dimensional), and the second from the classical Donaldson–Fujiki approach (which is infinite dimensional). In Section 3, we prove variants of these moment map results for maps between complex manifolds, both in the finite-dimensional and infinite-dimensional frameworks; these are relatively straightforward applications of ideas from equivariant differential geometry. Section 4 relates to the geometry of submersions, and proves the finite-dimensional version of Theorem 1. The proofs here use the equivariant differential geometry approach, and in Section 5 we turn to the adiabatic limit approach to derive the analogue of the Futaki invariant. This latter approach is used in Section 6 to prove Theorem 1 in its general, infinite-dimensional form. Lastly, in Section 7, we extend our results to the setting of foliations.

Acknowledgements:↩︎

This work was conducted through support of the American Institute of Mathematics (AIM) under the SQuaREs program. We are particularly grateful to AIM and their staff for this support and excellent working environment. RD thanks Calum Spicer for several helpful discussions around the birational geometry of foliations. RD was funded by a Royal Society University Research Fellowship (URF\R1\201041), and JR is partially supported by the National Science Foundation (DMS 1749447). LMS is funded by a Starting Grant from the Swedish Research Council (grant 2022-04574). XW research was partially supported by Simons Foundation Grant 631318 and MPS-TSM-00006636.

2 Preliminary Material↩︎

2.1 Equivariant differential geometry↩︎

We recall the basic ideas of equivariant differential geometry, and refer to [16] for further details. Throughout, we let \(K\) be a compact Lie group acting on a manifold \(M\), and denote by \(\mathfrak k\) the Lie algebra of \(K\).

As with differential forms, there is a natural exterior derivative on equivariant differential forms. In the following, as is standard, we identify an element \(v\in\mathfrak{k}\) with its induced vector field on \(M\).

Suppose \(\omega\) is a closed two-form on \(M\) and that \(K\) acts on \(M\) preserving \(\omega\).

The reader will easily check that this agrees with the usual definition of a moment map. Usually one considers moment maps in the case that \(\omega\) is a symplectic form, but throughout we allow the definition of a moment map to include \(\omega\) that are not necessarily positive.

We will use the following two properties of equivariant differential forms extensively, the proofs of which are straightforward calculations.

We will use a small number of examples of equivariant differential forms in the present work.

2.2 The scalar curvature as a moment map↩︎

We recall from [3] how to employ equivariant differential geometry to cast the scalar curvature as a moment map. In the present section we explain this in the finite-dimensional framework of a proper holomorphic submersion between complex manifolds, and in the subsequent section we consider the infinite-dimensional framework of the space of all almost complex structures compatible with a given symplectic form.

We thus return to the setting of Example 1, so \(\pi\colon (X,\omega)\to B\) is a proper holomorphic submersion of relative dimension \(m\), equivariant with respect to the holomorphic action of a compact (finite-dimensional) Lie group \(K\), with \(\omega\) relatively Kähler and \(K\)-invariant.

On a given fibre \((X_b,\omega_b)\), with \(\omega_b := \omega_X|_{X_b}\) the restriction (which is assumed Kähler), we will be interested in the scalar curvature \[S(\omega_b) = m\frac{\mathop{\mathrm{Ric}}\omega_b\wedge \omega_b^{m-1}}{\omega_b^m},\] whose (topological) average is given by \[\hat{S}_b = m\frac{c_1(X_b)\cdot[\omega_b]^{m-1}}{[\omega_b]^{m}},\] which is independent of \(b\in B\).

Thus we obtain a closed, \(K\)-invariant \((1,1)\)-form on the base \(B\) of the holomorphic submersion. The Weil–Petersson form \(\alpha_{\pi}\) is not necessarily Kähler in general, but it is positive in various natural settings. See [17] for foundational results on its positivity, and [18] for examples in which it cannot be even semipositive.

Assuming there is a moment map \(\mu: X \to \mathfrak{k}^*\) for the \(K\)-action on \((X,\omega)\), we have the following, whose proof we recall from [3].

2.3 The space of almost complex structures↩︎

We turn to the analogous infinite-dimensional theory. Consider a compact \(2m\)-dimensional symplectic manifold \((M,\omega)\), and denote by \(\mathscr J(M,\omega)\) the space of almost complex structures on \(M\) that are compatible with \(\omega\). As explained in [21] (to which we refer for details on the present section), the space \(\mathscr J(M,\omega)\) can be identified with the space of sections of a certain fibre bundle, and as such it admits a natural structure of a complex Fréchet manifold, whose tangent space at a point \(J\in \mathscr J(M,\omega)\) is \[T_J \mathscr J(M,\omega) = \{A \in \mathop{\mathrm{End}}(TM) : JA + AJ =0 \text{ and } \omega(AX,JY) = \omega(JX,AY)\}.\] The natural almost complex structure \(\mathbb{J}\) on \(\mathscr J(M,\omega)\) is then given by \[\mathbb{J}(A) = JA \text{ for } A\in T_J \mathscr J(M,\omega)\label{eq:holomorphicstructureonmathscrJ};\tag{1}\] this almost complex structure is integrable, and corresponds to the complex Fréchet manifold structure on \(\mathscr J(M,\omega)\).

Inside \(\mathscr J(M,\omega)\) sits the subspace of integrable almost complex structures, described by the vanishing of the Nijenhuis tensor \[N_J(X,Y) = [X,Y] + J([JX,Y]+[X,JY]-[JX,JY])\] which we denote by \[\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega) :=\{J\in \mathscr J(M,\omega) : N_J =0 \}.\] This is space is possibly singular and since we do not wish, or need, to go into the details of Fréchet structures which may not be smooth, we instead work formally declaring that the “tangent space" to a point \(J\) in \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\) consists of those \(A\in T_J \mathscr J(M,\omega)\) such that \(N_{J+tA} = O(t^2)\). Explicitly if we set \[P_A(X,Y): = J([AX,Y] + [X,AY]) + A[JX,Y] + A[X,JY] - [AX,JY] - [JX,AY]\] then \[T_J\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega) := \{ A\in T_J \mathscr J(M,\omega) : P_A=0\}.\] A calculation, using \(N_J(X,Y)=0\), yields \[P_{JA}(X,Y) = -JP_A(JX,JY) \text{ for } A\in T_J\mathscr J_{\mathop{\mathrm{Int}}}(M,\omega)\] which implies that if \(A\) is in \(T_J\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\) then so is \(JA\). That is, \(\mathbb{J}\) formally induces an almost complex structure on \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\).

Now let \(\mathcal{G}\) be the group of exact symplectomorphisms of \((M,\omega)\), so the Lie algebra of \(\mathcal{G}\) is the space \(C^{\infty}(M)/\mathbb{R}\) of functions modulo constants which is endowed with the Poisson bracket. Note that the group \(\mathcal{G}\) group acts on \(\mathscr J(M,\omega)\) by conjugation. Define a Kähler form \(\Omega\) on \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\) whose associated Kähler metric is given by \[\label{mabuchi} \langle A_1,A_2 \rangle := \int_M \langle A_1, A_2\rangle_{g_J} \omega^m,\tag{2}\] at a point \(J \in \mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\), where the pairing \(\langle A_1, A_2\rangle_{g_J}\) is defined using the Riemannian metric \(g\) on \(M\) induced from \(\omega\) and \(J\). We note that this Kähler structure is actually the restriction of such a structure from the ambient Fréchet manifold \(\mathscr J(M,\omega)\).

To tie this into the framework of equivariant differential forms, we note that there is a universal family \(\mathcal{U}{\to} \mathscr J(M,\omega)\) whose fibre \(\mathcal{M}_J\) over a point \(J\in \mathscr J(M,\omega)\) is the almost complex manifold \((M,\omega,J)\) [3]. As a Fréchet manifold \[\mathcal{U} = M\times \mathscr J(M,\omega)\] which is given the almost complex structure \[\mathbb{J}(v,A) = (Jv, JA) \text{ for } v\in T_x M \text{ and } A\in T_J\mathscr J(M,\omega).\] In turn this induces an almost complex structure on \(M\times \mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\), and with this structure it is easy to check that \(\pi:\mathcal{U}\to \mathscr J(M,\omega)\) is holomorphic and further \(\mathcal{G}\)-equivariant.

The universal family \(\mathcal{U}\) admits a natural relatively Kähler metric \(\omega_{\mathcal{U}}\), which is simply the pullback \(\pi_M^*\omega\) of \(\omega\) through the natural projection \(\pi_M: \mathcal{U}\to M\). The action of \(\mathcal{G}\) on \(\mathcal{U}\) is then Hamiltonian, with tautological moment map \(\mu_{\mathcal{U}}: \mathcal{U} \to \mathop{\mathrm{Lie}}\mathcal{G}^*\) defined by \(\langle \mu_{\mathcal{U}}, v\rangle = \pi_M^*h_v\). One may then also prove Theorem 7 using the geometry of the universal family and the approach of Section 2.2 [3].

3 Maps↩︎

3.1 Maps in finite dimensions↩︎

Consider a proper holomorphic submersion of relative dimension \(m\) \[\pi: X\to B\] between complex manifolds \(X\) and \(B\), not necessarily compact. Let \(K\) be a compact Lie group acting holomorphically on \(X\) and \(B\), making \(\pi\) a \(K\)-equivariant map, and fix a \(K\)-invariant relatively Kähler metric \(\omega\) on \(X\). Fix an additional Kähler manifold \((Y,\alpha)\) and a map \[p: X\to Y\] that is \(K\)-invariant, i.e.for all \(g\in K\), \(p\circ g = p.\) We view this data as a \(K\)-equivariant family of maps \(p: (X_b,\omega_b) \to (Y,\alpha)\), parametrised by \(B\). We are motivated by the case that \(Y\) is a moduli space endowed with a Weil–Petersson form (for example parametrising the fibres of a holomorphic submersion), so we retain the notation \(\alpha\) for the Kähler metric on the target \(Y\).

We will be interested in the twisted scalar curvature on each fibre \(X_b\), given by \[S_{p}(x) := S(\omega_{b}) - \Lambda_{\omega_{b}}p^*\alpha,\] where \(b=\pi(x)\) and we omit that \(p^*\alpha\) has been restricted to the fibre \(X_b\). We also let \(\hat{S}_{p}\) denote its fibrewise average (which we note is in general different from \(\hat{S}_b\)).

We next turn to the base geometry, and view \(B\) as being the base of the family of maps \(p_b: X_b\to B.\) Define \[\Omega_{\alpha} := \int_{X/B} \omega^m\wedge p^*\alpha,\] which is a \(K\)-invariant closed \((1,1)\)-form that is semipositive.

To involve the scalar curvature, set \[\Omega_p := -\int_{X/B}(\rho+p^*\alpha)\wedge\omega^m + \frac{\hat{S}_{p}}{m+1} \int_{X/B}\omega^{m+1}.\]

3.2 Returning to the space of almost complex structures↩︎

We now discuss an infinite-dimensional analogue of the preceding section. Consider the universal family \[\mathcal{U} \to \mathscr J^{\mathop{\mathrm{Int}}}(M,\omega).\] Our discussion will apply verbatim to any complex subspace of \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\) invariant under an appropriate subgroup of \(\mathcal{G}\) in the same manner, but for simplicity of notation we do not include this generalisation.

Assume in addition that we have a holomorphic map \[p: \mathcal{U}\to Y\] with \((Y,\omega_Y)\) a finite-dimensional Kähler manifold (where we note that such maps are more abundant on complex subspaces of \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\)). We denote the stabiliser of \(p\) under \(\mathcal{G}\) by \[\mathcal{G}_p : = \{ g\in\mathcal{G} : g\circ p = g\},\] and set \[\Omega_{\alpha}: = \int_{\mathcal{U}/\mathscr J(M,\omega)} p^*\alpha \wedge \omega^n\] which is a semipositive \((1,1)\)-form on \(\mathscr J(M,\omega)\). Here, as in Section 2.3 we may view the form \(\omega\) on \(M\) as a relatively Kähler metric on \(\mathcal{U} \to {\mathscr J}^{\mathop{\mathrm{Int}}}(M,\omega)\) (so that the underlying smooth structure of \(\mathcal{U}\) is \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\times M\) and we simply pull back \(\omega\) from \(M\)). Setting \[\Omega_p: = \Omega - \Omega_{\alpha},\] where \(\Omega\) is the usual Fujiki–Donaldson Kähler metric on \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega)\) with averaging constant \(\hat{S}_{p}\) instead of the average scalar curvature of the fibres, we obtain the following.

4 Submersions: a finite-dimensional framework↩︎

Broadly speaking, the previous section constructed moment maps where we incorporated the additional data of a map to some fixed space \(Y\). In geometric applications, one is frequently interested in the situation that \(Y\) is a moduli space, in which case it is usually more appropriate to consider families parameterised by some space \(C\), instead of the the induced map to the moduli space. In the language of moduli theory, maps from a \(C\) to moduli spaces are naturally in bijection with families over \(C\) only when the moduli space is fine (namely when the objects involved have no nontrivial automorphisms). Since this frequently fails to hold, it is more actually more natural to consider families directly.

We thus turn to families of holomorphic submersions, where we shall see that the geometry is richer than merely involving the twisted scalar curvature. Our setting shall be a finite-dimensional family of proper holomorphic submersions, parametrised by \(C\) , which we write as the commutative diagram

We emphasise here that we assume \(\gamma\) and \(\pi\) are both proper, but we do not assume \(C\) is compact. Thus for each \(c\in C\) we obtain a holomorphic submersion, which we write \(\pi_c: X_c\to B_c\). We let \(\pi\) have relative dimension \(m\), and let \(\gamma\) have relative dimension \(n\), so that \(X_c\) has dimension \(m+n\). In a subsequent section, we will return to the analogous infinite-dimensional framework.

Let \(K\) be a compact Lie group acting on \(X, B\) and \(C\), making each map equivariant. Endow \(X\) with a \(\pi\)-relatively Kähler metric \(\omega_X\) and \(B\) with a \(\gamma\)-relatively Kähler metric \(\omega_B\), both \(K\)-invariant. We also assume that there exist moment maps \(\mu_{B}: B\to\mathfrak{k}^*\) and \(\mu_{X}: X\to\mathfrak{k}^*\). The relatively Kähler metric \(\omega_{X}\) induces a Weil-Petersson form \(\alpha_{\pi}\) on \(B\), the base of the holomorphic submersion \(\pi: X\to B\), defined as before by \[\alpha_{\pi} := -\int_{X/B}\rho_{X}\wedge\omega_{X}^m + \frac{\hat{S}_b}{m+1}\int_{X/B}\omega_{X}^{m+1},\] and where \(\rho_X\in c_1(-K_{X/B})\) is defined as before through \(\omega_X\). A moment map \(\nu_{\pi}: B\to\mathfrak{k}^*\) for the \(K\)-action on \((B,\alpha_{\pi})\) is given from Theorem 4 by \[\langle\nu_{\pi},v\rangle(b) = \int_{X_b}\langle \mu_{X},v\rangle (S(\omega_b)-\hat{S}_b)\omega_b^m.\] Here \(\omega_b\) is the restriction of \(\omega_X\) to the fibre \(X_b\) of \(\pi: X\to B\), and \(\hat{S}_b\) is the fibrewise average scalar curvature. Said equivalently, the form \(\alpha_{\pi}+\nu_{\pi}\) is equivariantly closed on \(B\).

We induce a moment map for the base \(C\) of the family of submersions. We obtain two Weil–Petersson forms: \(\alpha_{\pi}\) on \(B\) and \(\alpha_{\gamma}\) on \(C\). To define the moment map, denote by \[\hat{S}_{\pi,c} = \frac{\int_{B_c}(S(\omega_c) - \Lambda_{\omega_c} \alpha_{\pi,c})\omega_c^n}{\int_{B_c}\omega_c^n}\] the average twisted scalar curvature (using the obvious notation for restriction, so that \(\alpha_{\pi,c}\) is the restriction of \(\alpha_{\pi}\) to \(B_c\) and \(\omega_c\) is the restriction of \(\omega_C\) to \(B_c\)) and denote further \[\Omega_C = \alpha_{\gamma} + \int_{B/C} \alpha_{\pi}\wedge \omega_B^n - \left(\frac{\int_{B_c}\Lambda_{\omega_c}\alpha_{\pi,c}\omega_c^n}{(n+1)\int_{B_c}\omega_c^n}\right)\int_{B/C}\omega_B^{n+1},\] which is a \(K\)-invariant closed \((1,1)\)-form on \(C\).

We have thus produced a moment map which is a coupled system, involving both the fibrewise average scalar curvature and the twisted scalar curvature on the base.

This in particular proves a finite-dimensional version of Theorem 1, establishing a moment map property for this coupled system, involving the twisted scalar curvature of the base and the fibrewise scalar curvature. It is crucial in the argument that the base \(C\) is smooth, in order to employ various tools from equivariant differential geometry. In the actual setting of Theorem 1, the corresponding base—which is the space of integrable almost complex structures on the submersion—is singular. In fact, it is not even naturally a subspace of a smooth space on which the moment map and group action extend. This means that the tools from equivariant differential geometry do not apply, leading us to take a more robust approach.

5 The transverse Futaki invariant↩︎

Our interest is in the construction of moment maps in various settings. In order to prove such results in the generality we desire we work with an adiabatic limit. For a fixed submersion, this means comparing the geometry of a submersion \(\pi: (X,\omega_X)\to (B,\omega_B)\) to the geometry of \((X, k\omega_X+\pi^*\omega_B)\) for \(k\gg 0\).

The goal of the present section is to develop tools to tackle this class of problem, in a simplified setting. Rather than constructing moment maps, we produce a main outcome of the existence of a moment map by constructing a numerical invariant—the Futaki invariant—associated to a holomorphic vector field preserving the submersion structure. The tools and ideas we develop will then be later reapplied to moment map results.

In fact, the results of the present section also apply in more generality than submersions. Endow a fixed compact Kähler manifold \(X\) with a foliation, where a foliation is defined as a holomorphic subbundle \(\mathcal{F}\subset TX\) of the holomorphic tangent bundle. The key example is a holomorphic submersion \(\pi: X \to B\), where the vertical subbundle \(V \subset TX\) given by the kernel of the differential \(d\pi\) defines a foliation. While a foliation has leaves which are complex submanifolds, and by Frobenius’ theorem locally admits the structure of a submersion, in general foliations are not induced by submersions and do not necessarily have compact leaves. By analogy with our interest in the geometry of the base \(B\) of a holomorphic submersion, we are interested in the transverse geometry of a foliation: the geometry orthogonal to the leafwise geometry.

Postponing the precise definitions for the moment, here we are interested in assigning numerical invariants to holomorphic vector fields on \(X\). Fixing such a vector field \(v\), suppose we are given a leafwise Kähler metric \(\omega\) on \(X\) (which, by definition, is a closed \((1,1)\)-form which is positive on each leaf) and a transverse Kähler metric \(\beta\), assumed invariant under the part of \(v\) lying in a compact torus. To this data we associate what we call the transverse Futaki invariant, which is defined as a sum of \(L^2\)-pairings of the leafwise average scalar curvature with a fibrewise Hamiltonian, and the twisted transverse average scalar curvature with a transverse Hamiltonian. In particular, this Futaki invariant vanishes when our system has leafwise constant scalar curvature and constant twisted transverse scalar curvature. Our goal in the present section is to prove, in Theorem 15, that the transverse Futaki invariant depends only on the cohomology class of \(\beta\). Thus, much as with the classical Futaki invariant, we obtain an equivariant-cohomological, algebro-geometric obstruction to the existence of solutions of our coupled system. We also provide two motivations for our definition of the transverse Futaki invariant, one arising from moment map theory, and the other arising from an adiabatic limit result.

5.1 Transverse Kähler geometry↩︎

We turn to the foundational setup of the Kähler geometry of foliations. Let \(\mathcal{F}\subset TX\) be a foliation of rank \(m\) and co-rank \(n\) (where our notation means that \(\mathcal{F}\) is a holomorphic subbundle of the holomorphic tangent bundle), so that \(X\) has dimension \(m+n\). A closed form \(\omega \in \Omega^{1,1}(X)\) is defined to be leafwise Kähler on \((X,\mathcal{F})\) if the restriction of \(\omega\) to \(\mathcal{F}{\oplus} \mathcal{F}\) is positive. Given such a leafwise Kähler metric \(\omega\) on \((X,\mathcal{F})\), the orthogonal complement \(H \subset TX\) is well-defined, where the fibre \(H_x\) over \(x \in X\) is given by \[H_x = \{ w \in T_x X : \omega(w,v) =0 \textrm{ for all } v \in \mathcal{F}_x \}.\] The splitting \(TX = \mathcal{F}\oplus H\) naturally induces the splitting \(\Omega^2(X) = \Omega_{\mathcal{F}} \oplus \Omega_{H} \oplus \Omega_{\textrm{mix}}\) of \(2\)-forms into purely leafwise, purely horizontal, and mixed type. For a two-form \(\eta\) we denote \[\eta = \eta_{\mathcal{F}} + \eta_H + \eta_{\textrm{mix}}\] the decomposition of \(\eta\) into these types. Observe that \(\omega = \omega_{\mathcal{F}} + \omega_{H}\); the mixed term vanishes as \(\omega\) itself is used to define the splitting. Moreover, \(\Lambda^{m+n} TX \cong \Lambda^m \mathcal{F}\otimes \Lambda^n H\).

Note that the mixed and horizontal components of \(\rho\) may be nontrivial, while its purely vertical component is characterised by its restriction to each leaf, where it agrees with the Ricci curvature of the induced Kähler metric on the given leaf.

The definition of a foliation as a subbundle of the tangent bundle implies that there is a canonical notion of a transverse differential form:

The condition that \(\beta\) be transverse is equivalent to asking that \(\iota_v\beta\) and \(\mathcal{L}_v\beta\) both vanish. Another general characterisation is that \(\beta\) is transverse if and only if \(\beta = \beta_H\), where we note it follows from the preceding characterisation that \(\beta\) being transverse is independent of choice of the horizontal subspace \(H\) defined by \(\omega\). Transverse forms are sometimes also called basic forms, notably in the theory of fibre bundles and principal bundles. The set of transverse functions (i.e. functions \(f\) with \(v(f) = 0\) for all \(v \in \Gamma(X, \mathcal{F})\), which simply means \(f\) is constant along leaves) will be denoted \(C^{\infty}_H (X)\). We denote by \(\Omega_H^k(X)\subset \Omega^k(X)\) the vector space of transverse differential forms.

The associated curvature quantity is as follows, where we view \(H\) as a holomorphic vector bundle over \(X\) by defining it as the quotient \(TX/\mathcal{F}\).

Explicitly, the horizontal contraction is characterised for \(\eta\in \Omega^2(X)\) by \[(\Lambda_{\beta}\eta)\beta^n = n\eta_H\wedge\beta^{n-1},\] so that the transverse scalar curvature is given by \(S(\beta) = \Lambda_{\beta} \mathop{\mathrm{Ric}}\beta\). In addition, the \(\partial \bar\partial\) lemma admits the following refinement in the foliated setting.

We note that usual \(\partial \bar\partial\)-lemma implies that such a \(\phi\) exists, and as the difference of two transverse differential forms, it follows that \(i\partial \bar\partial\phi\in \Omega^2_H(X)\).

Such transverse \(\phi\) will be called transverse Kähler potentials. Given a transverse Kähler form \(\beta\), the transverse Laplacian defined on functions by \(\Delta_{\beta} f = \Lambda_{\beta} ( (i\partial \bar\partial f)_H)\) will play a central role.

We also give an interpretation of \(\Delta_{\beta}\) in terms of adjoints. Denote by \(d^* : H^* \to C^{\infty}_H\) be the formal adjoint of \(d\) with respect to \(\beta\).

5.2 Transverse holomorphy potentials↩︎

We next incorporate holomorphic vector fields. Fix a leafwise Kähler metric \(\omega\), a holomorphic foliation \(\mathcal{F}\subset TX\), and a transverse Kähler metric \(\beta\). Now set \(k>0\) such that \(\omega_k = \omega + k \beta\) is Kähler. We consider a holomorphic vector field \(v\) whose Lie derivative preserves both \(\omega\) and \(\beta\) (thus the closure of the flow of the vector field induces a compact torus of biholomorphisms of \(X\)). A classical result in Kähler geometry then states that if \(v\) is a holomorphic vector field on \(X\) which vanishes somewhere, then it admits a holomorphy potential with respect to \(\omega_k\), namely a function \(h_k\) such that \(\iota_{v} \omega_k = \bar\partial h_k\), or equivalently \(\nabla^{1,0}_k h_k = v\) where \(\nabla_k\) is the usual gradient with respect to \(\omega_k\) and we take the \((1,0)\)-component.

Transversality of \(\beta\) implies that the function \(h\) is transverse, and hence that \(h\) is unique up to the addition of a constant. It also follows that the condition for \(h\) to be a transverse holomorphy potential can be written \(\iota_{v} \beta = \bar\partial_H h\). Linearity of the usual holomorphy potential construction also implies that \(h\) is a transverse holomorphy potential precisely if there exists a \(k\) such that \(h = h_{k+1} - h_k\), up to the addition of a constant.

We also note that if \(v\) and \(v'\) are holomorphic vector fields on \(X\) with zeros with the same transversal component, then their transverse holomorphy potentials agree up to addition of a constant, as follows from the fact that \(\iota_{v} \beta = \iota_{v_H} \beta\) as \(\beta\) is transverse. Next, we explain the dependence of the transverse holomorphy potential on the transverse Kähler metric.

5.3 The transverse Futaki invariant↩︎

We next define and motivate the definition of the transverse Futaki invariant. The motivation will be twofold: firstly, we will demonstrate that the transverse Futaki invariant appears as the leading order term in an asymptotic expansion of the usual Futaki invariant. Secondly, in the special case of submersions, we will relate the natural quantity arising from moment map theory to our transverse Futaki invariant.

Before defining the quantities of interest, we require additional notation. We let \(S_{\mathcal{F}}(\omega)\) denote the leafwise scalar curvature of \(\omega_{\mathcal{F}}\), which is the leafwise contraction of the leafwise Ricci curvature \(\rho\) of \(\omega\), and let \[\hat{S}_{\mathcal{F}} =\frac{m \int_X \rho \wedge \omega^{m-1} \wedge \beta^n}{\int_X \omega^{m} \wedge \beta^n}.\] In principle, this quantity depends on the cohomology class of \(\beta\), though we note it is independent of this choice in the setting of submersions as it is then the fibrewise average of the scalar curvature. We additionally define a constant \(\lambda\) by \[\label{gammaeqn} \lambda = \frac{n\int_X ( \mathop{\mathrm{Ric}}\beta + \rho)\wedge \beta^{n-1}\wedge\omega^m}{\int_X \omega^m\wedge\beta^n}.\tag{3}\]

Now write the holomorphy potential of \(v \in \mathfrak{h}\) with respect to \(\omega+k\beta\) as \[\label{h95k} h_k = kh + h_\mathcal{F}.\tag{4}\]

5.3.1 The transverse Futaki invariant through an adiabatic expansion↩︎

We next motivate our definition through the usual Futaki invariant of the Kähler metric \(\omega_k=\omega + k \beta\) for \(k \gg 0\), which is defined as \[\mathop{\mathrm{Fut}}_k(v) = \int_X h_k(S(\omega_k) - \hat{S}_k)\omega_k^{m+n},\] where \(\hat{S}_k\) denotes the average of the scalar curvature \(S(\omega_k)\) of \(\omega_k\). Futaki’s classical result is that \(\mathop{\mathrm{Fut}}_k(v)\) depends on the class of \([\omega_k]\) (and in particular on \(k\) itself), but not on the specific representative \(\omega_k\).

In order to understand the asymptotic behaviour of this quantity as \(k \to \infty\), we first consider the scalar curvature.

Similarly, by the definition given in Equation 3 of \(\lambda,\) the average scalar curvature of \(\omega_k\) expands as \[\hat{S}_k = \hat{S}_{\mathcal{F}} + k^{-1} \lambda + O(k^{-2}).\] Using that \(h_k = kh + h_\mathcal{F}\) and that \[\omega_k^{m+n} = \binom{m+n}{n} k^n \omega^m \wedge \beta^n + k^{n-1} \binom{m+n}{n-1} \omega^{m+1} \wedge \beta^{n-1} + O(k^{n-2}),\] we see that \[\begin{align} \label{Fut-expan} \frac{\mathop{\mathrm{Fut}}_k(v)}{\binom{m+n}{n} k^n} &= \binom{m+n}{n}^{-1} k^{-n} \int_X h_k (S(\omega_k) - \hat{S}_k) \omega_k^{m+n}, \\ &= k\int_Xh (S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}) \omega^m \wedge \beta^n+ \int_X h_{\mathcal{F}} (S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}) \omega^m \wedge \beta^n\nonumber\\ &\;\;+ \int_X h \left(S(\beta) + \Lambda_{\beta} (\rho_H) - \lambda + \Delta_{V} (\Lambda_{\beta}(\omega)) \right) \omega^m \wedge \beta^n \nonumber\\ &\;\;+ \frac{n}{m+1}\int_X h \left(S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}\right) \omega^{m+1} \wedge \beta^{n-1} + O(k^{-1}), \nonumber\\ &= k\int_Xh (S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}) \omega^m \wedge \beta^n+ \int_X h_{\mathcal{F}} (S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}) \omega^m \wedge \beta^n\nonumber \\ &\;\;+ \int_X h \left(S(\beta) + \Lambda_{\beta} (\rho) - \lambda \right) \omega^m \wedge \beta^n \nonumber \\ &\;\;\;+ \frac{n}{m+1}\int_X h \left(S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}\right) \omega^{m+1} \wedge \beta^{n-1} + O(k^{-1})\nonumber, \end{align}\tag{5}\] where we have used that, since \(h\) is transverse, its vertical Laplacian vanishes, along with a self-adjointness property of the vertical Laplacian.

Thus, the transverse Futaki invariant appears as a subleading order term in an asymptotic expansion. We next show that the leading order term \(\int_X h (S_{\mathcal{F}}(\omega) - \hat{S}_{\mathcal{F}}) \omega^m \wedge \beta^n\) in the expansion actually vanishes automatically, meaning that the transverse Futaki invariant governs the Futaki invariant for \(k\gg 0\). This is essentially a consequence of the fact that \(h\) is constant along leaves, which would trivially imply the result in the submersion setting, by combining the theorems of Ehresmann and Fubini. In the absence of a global submersion structure, we argue instead using the transverse Laplacian.

We have now shown the following:

5.3.2 The transverse Futaki invariant through moment maps↩︎

We give a second motivation for our definition of the tranverse Futaki invariant, by comparing it with the moment map geometry of submersions, namely through Theorem 11. This result implies the existence of an analogue of the Futaki invariant by general moment map theory: one simply pairs the moment map with a holomorphic vector field on \(\pi: X\to B\) itself. We call this quantity the submersion Futaki invariant.

5.4 Cohomological invariance of the tranverse Futaki invariant↩︎

We next turn to one of the main results of our work, which is as follows.

We begin with some preliminary results concerning the linearisation of the transverse scalar curvature. Key to this will be the transverse version of the classical Lichnerowicz operator. Before defining this operator, we begin with a basic definition.

In what follows, for a section \(v\) of the smooth tangent bundle of \(X\), we denote by \(v^{1,0}\) its \((1,0)\)-component, which is a section of the holomorphic tangent bundle \(TX\) of \(X\).

A transverse holomorphy potential satisfies \(v_H=\nabla^{1,0}_{\beta} h\), and since \(v\) is holomorphic, \(\bar \partial (\nabla^{1,0}_{\beta} h) = 0\). Thus, by self-adjointness, the kernel of the transverse Lichnerowicz operator consists precisely of transverse holomorphy potentials.

We aim to understand how the various terms appearing in the definition of the transverse Futaki invariant, namely Definition 10, depend on the transverse Kähler structure. We thus modify the transverse Kähler metric \(\beta\) by adding \(ti\partial \bar\partial\phi\) and we will be interested in the derivative of various quantities when \(t=0\). Being a pointwise result, the derivation of the following is a consequence of the analogous result in the submersion setting, see e.g. [23]–[25].

As in the submersion case, this follows from the following: \[\begin{align} \frac{d}{dt}\bigg|_{t=0}\left(\omega_{\mathcal{F}}^m \wedge (\beta+i\partial \bar\partial\phi)^n\right) &= (\Delta_{\beta}\phi ) \omega_{\mathcal{F}}^m \wedge \beta^n, \label{eq:volderivative} \\ \frac{d}{dt}\bigg|_{t=0}\left( S(\beta_t)\right) &= - \Delta_{\beta}^2(\phi) + \langle \mathop{\mathrm{Ric}}(\beta), i\partial \bar\partial(\phi) \rangle_{\beta}. \end{align}\tag{10}\]

Next we compute the derivatives of the global quantities appearing in the transverse Futaki invariant.

Note that \(h_t = h + tv(\phi)\) is the potential with respect to \(\beta_t\), by Lemma 4.

We next relate the linearised operator to transverse holomorphy potentials.

We are now ready to show that the transverse Futaki invariant is independent of the representative in the class.

6 Submersions: an infinite-dimensional framework↩︎

We return to the moment map geometry of submersions, where we consider the infinite-dimensional geometry to the space of almost complex structures on a submersion. In the subsequent section, we generalise our results to the setting of foliations. We thus define a suitable analogue version of the space of almost complex structures compatible with a smooth symplectic submersion, introduce a gauge group action on this space, and prove that a version of our coupled system arises as a moment map. A key tool in our work will be to employ the robust adiabatic arguments of Section 5.3.1 to deduce the moment map property.

6.1 The space of almost complex structures compatible with a submersion↩︎

We fix a smooth symplectic submersion, by which we mean the data of \(\pi:(M,\omega_M)\to (B,\omega_B)\), with \(\pi: M \to B\) being a smooth proper submersion between compact manifolds, with \(\omega_B\) being symplectic and with \(\omega_M\) being relatively symplectic (that is, a closed two-form whose restriction to each fibre is symplectic).

We begin with the relevant space of almost complex structures.

Thus an element of \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) is a pair of integrable almost complex structures on \(M\) and \(B\), which make the smooth map \(\pi: M \to B\) holomorphic. Our notation \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega_M)\) is a minor extension of the usual notation, referring to integrable almost complex structures on \(M\) which preserve \(\omega_M\), which may not be globally symplectic. Recall that the tangent space to \(\mathscr J^{\mathop{\mathrm{Int}}}(M,\omega_M)\) at some complex structure \(J_M\) consists of the \(A_M \in \mathcal{A}^0(\mathop{\mathrm{End}}TM)\) such that \(A_M\) is compatible with the complex structure, i.e. \(A_M \circ J_M = - J_M \circ A_M\), is compatible with the symplectic form, i.e. \(\omega_M(J_M(\cdot), A_M(\cdot))=-\omega_M(A_M(\cdot), J_M(\cdot))\) and further \(\bar \partial_{J_M}(A_M^{1,0})=0\), where \(A_M^{1,0}\) is the \((1,0)\)-component of \(A_M\) with respect to \(J_M\).

We view \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) as a version of a complex subspace of \(\mathscr J(M,\omega_M)\times \mathscr J(B,\omega_B)\), but lacking a robust infinite-dimensional theory of singular complex spaces we give a more concrete treatment and view \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) as a set with a well-defined (formal) tangent space.

Note that the condition \(A_B \circ d \pi = d \pi \circ A_M\) is preserved by \((J_M \circ A_M, J_B \circ A_B)\), so the above almost complex structure sends \(T_{(J_M,J_B)}\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) to itself.

The space \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) admits a natural gauge group action.

The Lie algebra \(\mathfrak h_{\pi}\) of \(\mathop{\mathrm{Ham}}(\pi)\) thus consists of pairs of Hamiltonian vector fields \((v_M, v_B)\), such that the pushforward of \(v_M\) under \(d\pi\) equals \(v_B\).

We next endow \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) with a natural closed semipositive \((1,1)\)-form, by defining for \(A^j = (A^j_m,A^j_B) \in T_{(J_M, J_B)}\mathscr J^{\mathop{\mathrm{Int}}}_\pi\), \[\Omega_{\mathscr J_\pi}(A^1, A^2):= \int_M\left(\omega_B(A_B^1, A_B^2)+ \omega_M\big|_{V}(A_M^1, A_M^2)\right)\omega_M^{m}\wedge \omega_B^n,\] where \(\omega_M\big|_{V}\) is \(\omega_M\) restricted to the purely vertical component \(\mathop{\mathrm{End}}(\ker \pi)\).

We will show that this is semipositive, but note that it is not necessarily strictly positive. In order to see this, and to construct a moment map, we embed \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) into a sequence of infinite-dimensional Kähler manifolds.

The closedness and semi-positivity of \(\Omega_{\mathscr J_{\pi}}\) now follows from an adiabatic argument.

We are now in a position to ask for a moment map for the \(\mathop{\mathrm{Ham}}(\pi)\)-action on \((\mathscr J^{\mathop{\mathrm{Int}}}_\pi,\Omega_{\mathscr J_\pi})\).

The spaces \((\mathscr J(M, \omega_M+k\pi^*\omega_B),\Omega_k)\) each admit a natural moment map with respect to their respective Hamiltonian symplectomorphism group actions, given by the scalar curvature. This induces a moment map for the subgroup \(\mathrm{Ham}(\pi)\) by composing with the map \(\mathfrak{h}_k^* \to \mathfrak{h}_{\pi}^*\), dual to the inclusion \(\mathfrak{h}_{\pi} \subset \mathfrak{h}_k\), where \(\mathfrak{h}\) is the Lie algebra of the exact symplectomorphisms of \((M, \omega_k)\). This simply amounts to restricting the moment map with respect to \(\Omega_k\) to the vector fields in \(\mathfrak{h}_{\pi}\). Since the moment map on an invariant subspace is simply the restriction of the moment map, we have a \(k\)-dependent sequence of Kähler metrics \(\Omega_k\) and a sequence of moment maps on \(\mathscr J^{\mathop{\mathrm{Int}}}_\pi\) itself. It is then straightforward to see that, provided the sequence of Kähler metrics and moment maps admit an asymptotic expansion in \(k\), the moment map property is preserved for each term in the asymptotic expansion (see for example [26]). This is the main idea in the following proof.

6.2 A formal space with positivity↩︎

The moment map given by Theorem 18 for the \(\mathrm{Ham}(\pi)\)-action on \(\left(\mathscr J_\pi,\Omega_{\mathscr J_\pi}\right)\) is a moment map with respect to a symplectic form which is degenerate, as it vanishes on the mixed terms corresponding to the change of almost complex structure on \(X\). If one changes perspective and varies the Kähler form instead of the complex structure, this degeneracy can be seen by the fact that the equation does not change when adding a form pulled back from the base of the submersion \(X \to B\) to \(\omega\).

We next construct a quotient of \(\left(\mathscr J^{\mathop{\mathrm{Int}}}_\pi,\Omega_{\mathscr J_\pi}\right)\) on which the symplectic form becomes positive. Again we argue formally, and consider spaces which are given by a set together with a specified vector space at each point acting as the formal tangent space.

Define an equivalence relation \(\sim\) on \(\mathscr J_\pi\) by setting \[J\sim J' \iff J_V = J'_V \;\text{and} \;\; J_B = J'_B.\] Here \(J_V\) is the purely vertical part of the almost complex structure \(J_M\) on \(M\). Thus, \(J\) and \(J'\) are equivalent precisely when the complex structures on \(M\) agree on each fibre of the map \(\pi\), and \(J_B=J'_B\), which implies that for \(J\sim J'\), the almost complex structures \(J_M\) and \(J'_M\) only differ in their mixed components. The underling set of our space is then \[\widetilde{\mathscr J}_\pi = \mathscr J_\pi / \mathord{\sim}\] and we denote by \(\phi : \mathscr J_\pi \to \widetilde{\mathscr J}_\pi\) the quotient map. Now define the tangent space at a point \([J]\) to be the subspace of \(T_J \mathscr J_\pi \subset \mathop{\mathrm{End}}(TM)\) generated by purely vertical and purely horizontal endomorphisms. Note that this is well-defined, as \(T_J \mathscr J_\pi\) does not depend on \(J\). Thus each tangent space of \(\widetilde{\mathscr J}_\pi\) can be identified with a fixed vector space (that we call \(W\)) that is independent of \([J]\), so the “tangent bundle" to \(\widetilde{\mathscr J}_\pi\) is just the trivial bundle modeled on \(W\).

Having defined the tangent spaces at each point of \(\widetilde{\mathscr J}_\pi\), we can also define their usual tensor products. We will abuse notation and write \(W\) for the tangent bundle of \(\widetilde{\mathscr J}_\pi\) as well, and similarly for the tensor bundles.

One may define the exterior derivative \(d : \Gamma(\Lambda^j W) \to \Gamma(\Lambda^{j+1} W)\) in the following manner. We say that a \(j\)-form \(\alpha\) on \(\widetilde{\mathscr J}_\pi\) is smooth if its pullback \(\hat{\alpha}\) to \(\mathscr J_\pi\) is smooth. For such a \(j\)-form, we then have the exterior derivative \(d \hat{\alpha}\). Over a point \([J] \in \widetilde{\mathscr J}_\pi\), the form \(d \hat{\alpha}\) may depend on the representative. However, since \(\hat{\alpha}\) itself has no mixed terms (as it is pulled back from \(\widetilde{\mathscr J}_\pi\)), this dependence on \(J\) is only in the mixed terms of \(d \hat{\alpha}\). Thus, the \(\Gamma(\Lambda^{j+1} W)\)-component of \(d \hat{\alpha}\) is independent of the representative of \(J \in \phi^{-1}([J])\) and this allows us to define \(d \alpha\) at a point \([J]\) as the \(\Gamma(\Lambda^{j+1} W)\)-component of \(d \hat{\alpha}\) at a representative \(J\) in \(\mathscr J_{\pi}\).

We obtain a two form \(\Omega_{\widetilde{\mathscr J}_\pi} \in \Gamma(\Lambda^2T\widetilde{\mathscr J}_\pi)\) on \(\widetilde{\mathscr J}_\pi\) by restricting \(\Omega_{\mathscr J_\pi}\) to \(W\). Note that this is well-defined as if \(J \sim J'\), then the purely vertical and purely horizontal components of \(J\) and \(J'\) agree. Hence, the trace terms with respect to \(\omega_B\) and \(\omega_M\) in the expression for \(\Omega_{\mathscr J_\pi}\) will agree for \(J\) and \(J'\) on elements of \(W\). Note also that \(\Omega_{\widetilde{\mathscr J}_\pi}\) is closed, since \(\phi^* \Omega_{\widetilde{\mathscr J}_\pi} = \Omega_{\mathscr J_\pi}\) and \(\Omega_{\mathscr J_\pi}\) is closed. Moreover, \(\Omega_{\widetilde{\mathscr J}_\pi}\) is now a positive form, in contrast with \(\Omega_{\mathscr J_\pi}\).

Now, \(\mathrm{Ham}(\pi)\) acts on \(\widetilde{\mathscr J}_\pi\) making the quotient map \(\phi :\mathscr J_\pi \to \widetilde{\mathscr J}_\pi\) equivariant. Indeed, if \((\phi_M, \phi_B) \in \mathrm{Ham}(\pi)\) and \(J \sim J'\), then \(J_B = J_B'\), so \(\phi_B\circ J_B = \phi_B \circ J_B'\), and moreover the vertical components of \(\phi_M \circ J_M\) and \(\phi_M \circ J_M'\) will agree. Thus \((\phi_M \circ J_M, \phi_B \circ J_B)\) and \((\phi_M \circ J'_M, \phi_B \circ J'_B)\) map to the same point in \(\widetilde{\mathscr J}_\pi\). As we have the notions of a tangent space, the various tensor bundles and a differential on \(\widetilde{\mathscr J}_\pi\), it makes sense to ask for the \(\mathrm{Ham}(\pi)\)-action on \(\widetilde{\mathscr J}_\pi\) to be Hamiltonian. All the terms in the expression in the statement of Theorem 18 for \(\mu_\pi(J)\) are independent of \(J \in \phi^{-1}([J])\). Thus we can define \(\mu_{\pi} : \widetilde{\mathscr J}_\pi \to \mathfrak{ham}(\pi)^*\) by its value on a representative in \(\mathscr J_\pi\) of a given point in \(\widetilde{\mathscr J}_\pi\). Unraveling the definitions, we see that the moment map property holds on \(\widetilde{\mathscr J}_\pi\), since it does so on \(\mathscr J_{\pi}\). Thus, one may view the equation of finding a zero of \(\mu_{\pi}\) as a moment map equation with respect to a positive form, provided we pass to the space \(\widetilde{\mathscr J}_\pi\).

With the definitions given above, we have shown the following:

6.3 Zeroes of the moment map↩︎

We next wish to characterise zeroes of the moment map \(\mu_{\pi}\). The group \(\mathrm{Ham}(\pi)\) fits into an exact sequence \[\begin{align} \label{eq:ES} 0 \to \mathrm{Ham}_0 (M, \omega) \to \mathrm{Ham}(\pi) \to \mathrm{Ham}(B, \omega_B), \end{align}\tag{13}\] where the Lie algebra \(\mathfrak{ham}_0(M,\omega)\) of \(\mathrm{Ham}_0 (M, \omega)\) consists of the fibrewise Hamiltonian vector fields that are purely vertical, while the Lie algebra \(\mathfrak{ham}(B,\omega_{B})\) of \(\mathrm{Ham}(B, \omega_B)\) consists of the Hamiltonian vector fields on \(B\). Note that a Hamiltonian vector field on \(B\) may not lift to \(X\) as a Hamiltonian vector field. This would require the vector field to lift and to admit a Hamiltonian with respect to the form \(\omega\). So, the sequence may not be a short exact sequence.

Now, we can identify the Lie algebra \(\mathfrak{ham}_0(M,\omega)\) with the space \(C^{\infty}_0(M)\) of functions of average \(0\) on each fibre of \(M \to B\). We do this by associating to such a vector field \(v\) the function obtained by gluing together the fibrewise Hamiltonians \(h_b\) of average \(0\) for \(v_b\) on \(M_b\). Since \(\mathrm{Ham}_0 (M, \omega)\) is a subgroup, this means that a zero of the moment map in particular satisfies \[\begin{align} \int_M h_{M} \left(S_{V}(\omega)-\hat{S}_{V} \right) \omega^m \wedge \beta^n = 0 \end{align}\] for every \(h_{M} \in C^{\infty}_0(M)\). This implies that \(S_{V}(\omega)-\hat{S}_{V} =0\), i.e. at a zero of the moment map, \(\omega\) is fibrewise cscK. This further implies that at a zero of the moment map, the moment map has expression \[\int_M h_B (\Lambda_{\beta}( \mathop{\mathrm{Ric}}(\beta) + \rho_H) - \hat{S}_{\pi}) \omega^m \wedge \beta^n.\] Thus, at a zero of the moment map the above is zero for all \((h_{M},h_B) \in \mathfrak{ham}(\pi)\), which means that the twisted scalar curvature \(\Lambda_{\beta}( \mathop{\mathrm{Ric}}(\beta) + \rho_H)\) lies in the orthogonal complement \(\mathfrak{k}\) to the image of the map \[\mathfrak{ham}(\pi) \to \mathfrak{ham}(B,\omega_B).\] Thus a zero of the moment map is the pair of a fibrewise cscK metric on \(M \to B\) together with a base metric whose twisted scalar curvature lies in \(\mathfrak{k}\).

7 Foliations↩︎

We now generalise the above to the case of foliations. Let \(\mathcal{F}\subset TX\) be a foliation, \(\omega\) a leafwise symplectic form and \(\beta\) a transversal symplectic form. We will define a space \(\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\) of integrable almost complex structures compatible with the foliation \(\mathcal{F}\) and endow it with a Kähler structure so that the coupled quantity involving the leafwise scalar curvature of \(\omega\) and the twisted scalar curvature of \(\beta\) becomes a moment map with respect to a certain group action.

The approach mirrors that of the submersion case closely and is in particular based on an adiabatic argument. There are several reasons why the case of foliations is treated separately from the case of submersions. First, the spaces, groups and so on that are considered in this section have more concrete interpretations in the submersion case. Second, the twisted scalar curvature equation becomes an equation on \(X\) in the foliation case, whereas it is an equation on the base \(B\) when we have a foliation, see Corollary 4. Finally, in the submersion case, we have an interpretation of having a zero of the moment map in terms of the twisted cscK equation involving the Weil–Petersson form as twist. This comes from the fact when there is a fibrewise cscK metric, the submersion induces a map from the base \(B\) to the moduli space of cscK manifolds, and the twisted equation on the base then has the Weil–Petersson form pulled back via this map as its twist.

We begin with the definitions of the relevant space of almost complex structures. Fix a compact symplectic manifold \((M,\omega)\), a foliation \(\mathcal{F}\subset TM\) of the (smooth) tangent bundle of \(M\), and a transverse closed two-form \(\beta\).

In particular, \(J\) sends \(\mathcal{F}\) to itself, which means its \(\textrm{Hom}(\mathcal{F}, H)\)-component vanishes. Moreover, as a consequence, one has \[\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}} \subset \mathscr J(M,\omega+k\beta),\] just as in Lemma 8. Note that \(\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\) depends on both \(\omega\) and \(\beta\), even though this is not reflected in the notation.

A foliation is locally a holomorphic submersion. We employ a metric enhancement of this classical fact, for almost complex structures \(J \in \mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\), so that \(J\) preserves \(\beta\in \Omega^{2}_H(M)\), which is hence a \((1,1)\)-form on the complex manifold \((M,J)\). Choose a holomorphic neighbourhood \(U\) around any point of \(X\) with coordinates \(x= (y,z) \in \mathbb{C}^{m+n}\), such that leaves of \(\mathcal{F}\) are precisely the fibres of the local submersion \(x \mapsto z\). Then, since \(d\beta = 0\), the form \(\beta\) is locally the pullback of a Kähler metric from the base of the local submersion. Thus, for each \(J \in J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\), the induced metrised foliation is locally a metrised holomorphic submersion. This being a local claim, we illustrate it with the following example, with the general case being identical.

Next, we define the relevant gauge group that acts on \(\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\). We first define a subgroup of \(\mathrm{Ham}(X,\omega)\), which does not depend on \(\beta\), as follows. First set \[\begin{align} \mathrm{Ham}(X,\omega,\mathcal{F}):=\left\{ \phi\in\mathrm{Ham}(X,\omega) \left | \phi^\ast \mathcal{F}= \mathcal{F}\right.\right\}, \end{align}\] where \(\phi^\ast \mathcal{F}\subset TX\) is regarded as the pullback subbundle of \(\mathcal{F}\subset TX\), meaning we have a diagram \[\xymatrix{ \phi^\ast\mathcal{F}\ar@{>}[rr]^{} \ar@{>}[d]^{}& & \mathcal{F}\ar@{>}[d]^{} \\ X\ar@{>}[rr]^{\phi} & & X. }\] The group of Hamiltonian symplectomorphisms of \(\mathcal{F}\) will be a subgroup of this group.

Note that this depends on both \(\omega\) and \(\beta\), but as with \(\mathrm{Ham}(\pi)\) in the case of submersions we omit this from the notation. The Lie algebra of \(\mathrm{Ham}(\mathcal{F})\) is given by \[\mathfrak{h}_{\mathcal{F}}=\left\{ v \in\mathfrak{ham}(X,\omega,\mathcal{F}) \left | \iota_{v}\beta=df \text{ for some }f\in C^\infty_H(X) \right.\right\}.\]

We can view \(\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\) as a symplectic manifold by expanding the restriction of the symplectic form of \(\mathscr J^{\mathop{\mathrm{Int}}} (X,\omega_k)\) to \(\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\) asymptotically in \(k\). The leading order term in this expansion is of order \(k^n\) and is given by

\[\Omega_{\mathscr J_\mathcal{F}}(A^1, A^2) =\int_X\left(\beta(A^1, A^2)+ \omega_{\mathcal{F}}(A^1, A^2)\right)\omega^{m}\wedge \beta^n,\] for \(A^j \in T_{(J_M, J_B)}\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\). As \(\Omega_{\mathscr J_\mathcal{F}}\) arises from an asymptotic expansion of a sequence Kähler forms, this form is then closed and semi-positive on \(\mathscr J^{\mathop{\mathrm{Int}}}_{\mathcal{F}}\) just as was the case in Lemma 9, but again it is not in general positive.

As a consequence of the calculation following Definition 10, we obtain the following, with the analogous notation:

This follows in exactly the same way as Theorem 18, using the sequence of Kähler metrics \(\omega + k \beta\) on \(X\) and the corresponding expansion of the associated moment map on the space of almost complex structures on \(X\). Note that we may further also define a formal quotient space \(\widetilde{\mathscr J}_{\mathcal{F}}\) on which we obtain a positive symplectic form for which \(\mu\) is a moment map, as in Section 6.2. We note in addition that it is straightforward to see that this result generalises the corresponding result for submersions.

Finally, a zero of the moment map \(\mu_{\mathcal{F}}\) has a similar interpretation in terms of the twisted scalar curvature. We have a subgroup \(\mathrm{Ham}_0 (M, \omega_{\mathcal{F}}) \subset \mathrm{Ham}(\mathcal{F})\) generated by the vector fields that are sections of \(\mathcal{F}\subset TX\). We can then define a space \(\mathfrak{k}\) which is the cokernel of the map from \(\mathfrak{h}_{\mathcal{F}}\) to the horizontal Hamiltonian functions with respect to \(\beta\). The vanishing of the Futaki invariant then in particular implies the vanishing of \[\int_X h_{\mathcal{F}} \left(S_{\mathcal{F}}(\omega)-\hat{S}_{\mathcal{F}} \right) \omega^m \wedge \beta^n\] for all \(h_{\mathcal{F}} \in \mathfrak h_{\mathcal{F}}\). As this equals the integral of the inner product of the leafwise gradients of \(h_{\mathcal{F}}\) and \(S_{\mathcal{F}}(\omega)\) over \(M\) (with volume form \(\omega^m \wedge \beta^n\)), this forces the leafwise gradient \(\nabla_{\mathcal{F}} S_{\mathcal{F}}(\omega)\) of the leafwise scalar curvature to vanish if there is a zero of the moment map, and so \(S_{\mathcal{F}}(\omega)\) must be constant as it has no transversal component. This in turn implies that at a zero of the moment map, the twisted transversal scalar curvature must land in \(\mathfrak{k}\).

Note that the resulting condition on \((J,\beta)\) is an equation on the total space of the foliation, in contrast to the setting of submersions, where the condition is one formulated on the base \(B\). By analogy, the condition that the twisted scalar curvature of \((J, \beta)\) lies in \(\mathfrak{k}\) can be viewed as a basic condition.

References↩︎