EXTREMAL KÄHLER METRICS ON PROJECTIVE BUNDLES OVER A CURVE

VESTISLAV APOSTOLOV, DAVID M. J. CALDERBANK, PAUL GAUDUCHON, AND CHRISTINA W. TØNNESEN-FRIEDMAN

Abstract. Let M = P (E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E → Σ over a compact complex curve Σ of genus ≥ 2. Building on ideas of Fujiki [28], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.

Introduction

Extremal Kähler metrics were first introduced and studied by E. Calabi in [13, 14]. Let (M, J) denote a connected compact complex manifold of complex dimension m. A Kähler metric g on (M, J), with Kähler form ω = g(J·, ·), is extremal if it is a critical point of the functional g ↦ ∫_{M} s_{g}^{2} \frac{ω_{g}^{m}}{m!} , where g runs over the set of all Kähler metrics on (M, J) within a fixed Kähler class Ω = [ω], and s_{g} denotes the scalar curvature of g. As shown in [13], g is extremal if and only if the symplectic gradient K := grad_{ω}s_{g} = J grad_{g}s_{g} of s_{g} is a Killing vector field (i.e., L_{K}g = 0) or, equivalently, a (real) holomorphic vector field (i.e., L_{K}J = 0). Extremal Kähler metrics include Kähler metrics of constant scalar curvature (CSC Kähler metrics) in particular Kähler–Einstein metrics. Clearly, if the identity component Aut_{0}(M, J) of the automorphism group of (M, J) is reduced to {Id}, i.e., if (M, J) has no non-trivial holomorphic vector fields, any extremal Kähler metric is CSC, whereas a CSC Kähler metric is Kähler–Einstein if and only if Ω is a multiple of the (real) first Chern class c_{1}(M, J). In this paper, except for Theorem 1 below, we will be mainly concerned with extremal Kähler metrics of non-constant scalar curvature.

The Lichnerowicz–Matsushima theorem provides an obstruction to the existence of CSC Kähler metrics on (M, J) in terms of the structure of Aut_{0}(M, J), which must be reductive whenever (M, J) admits a CSC Kähler metric; in particular, for any CSC Kähler metric g, the identity component Isom_{0}(M, g) of the group of isometries of (M, g) is a maximal compact subgroup of Aut_{0}(M, J) [55, 49]. The latter fact remains true for any extremal Kähler metric (although Aut_{0}(M, J) is then no longer reductive in general) and is again an obstruction to the existence of extremal Kähler metrics [14, 48]. Another well-known obstruction to the existence of CSC Kähler metrics within a given class Ω involves the Futaki character [30, 14], of which a symplectic version, as developed in [47], will be used in this paper (cf. Lemma 2). Furthermore, it is now known that extremal Kähler metrics within a fixed Kähler class Ω are unique up to the action of the reduced automorphism group^{1} Ãut_{0}(M, J) [10, 15, 20, 52].

Key words and phrases. Extremal and constant scalar curvature Kähler metrics; stable vector bundles; projective bundles; toric fibrations.

It was suggested by S.-T. Yau [75] that a complete obstruction to the existence of extremal Kähler metrics in the Kähler class Ω = 2πc_{1}(L) on a projective manifold (M, J) polarized by an ample holomorphic line bundle L should be expressed in terms of stability of the pair (M, L). The currently accepted notion of stability is the K- (poly)stability introduced by G. Tian [69, 70] and refined by S. K. Donaldson [21]. The Yau–Tian–Donaldson conjecture can then be stated as follows. A polarized projective manifold (M, L) admits a CSC Kähler metric if and only if it is K-polystable. This conjecture is still open, but the implication ‘CSC ⇒ K-polystable’ in the conjecture is now well-established, thanks to work by S. K. Donaldson [20], X. Chen–G. Tian [15], J. Stoppa [64], and T. Mabuchi [53, 54]. In order to account for extremal Kähler metrics of non-constant scalar curvature, G. Székelyhidi introduced [67, 66] the notion of relative K-(poly)stability with respect to a maximal torus of the automorphism group of the pair (M, L) — which is the same as the reduced automorphism group Ãut_{0}(M, J) — and the similar implication ‘extremal ⇒ relatively K-polystable’ was recently established by G. Székelyhidi–J. Stoppa [65].

The Yau–Tian–Donaldson conjecture was inspired by and can be regarded as a nonlinear counterpart of the well-known equivalence for a holomorphic vector bundle over a compact Kähler manifold (M, J, ω) to be polystable with respect to ω on the one hand and to admit a hermitian–Einstein metric [40, 74, 18] on the other. In the case when (M, J) is a Riemann surface this is the celebrated theorem of Narasimhan and Seshadri [56], which, in the geometric formulation given in [9, 17, 28] can be stated as follows. Let E be a holomorphic vector bundle over a compact Riemann surface Σ. Then, E is polystable if and only if it admits a hermitian metric whose Chern connection is projectively-flat.

This paper is mainly concerned with the existence of extremal Kähler metrics on ruled manifolds (M, J), which are total spaces of projective fibre bundles P (E) with E being a holomorphic vector bundle over a compact Kähler manifold (S, J_{S}, ω_{S}) of constant scalar curvature. Notice that this class of complex manifolds includes most explicitly known examples so far of extremal Kähler manifolds of non-constant scalar curvature, starting with the first examples given by E. Calabi in [13]. For complex manifolds of this type one expects stability properties of (M, J) to be reflected in the stability of the vector bundle E. In fact, such a link was established by J. Ross–R. Thomas, with sharper results when the base is a compact Riemann surface of genus at least 1, [61, Thm. 5.12 and Thm. 5.13] (cf. also Remark 2 below.) This suggests that the existence of extremal Kähler metrics on P (E) could be directly linked to the stability of the underlying bundle E. First evidence of such a direct link goes back to the work of Burns– deBartolomeis [12], and many partial results in this direction are now known, see e.g. the works of N. Koiso–Y. Sakane [42, 43], A. Fujiki [28], D. Guan [33], C. R. LeBrun [44], C. Tønnesen-Friedman [71], Y.-J. Hong [36, 37], A. Hwang–M. Singer [39], Y. Rollin– M. Singer [60], J. Ross–R. Thomas [61, 62], G. Székelyhidi [67, 66], and our previous work [5]. However, to the best of our knowledge, a complete understanding of the precise relation is yet to come.

While we principally focus on projective bundles, for which sharper results can be obtained (Theorems 1, 2, 4 below), the techniques and a number of results presented in this paper actually address a much wider class of manifolds, called rigid toric bundles over a semisimple base. These were introduced in our previous paper [4]; we recall their main features in Sect. 2 and prove a general existence theorem (Theorem 3) in Sect. 3.

1Ãut_{0}(M, J) is the unique linear algebraic subgroup of Aut_{0}(M, J) such that the quotient Aut_{0}(M, J)/Ãut_{0}(M, J) is a torus, namely the Albanese torus of (M, J) [27]; its Lie algebra is the space of (real) holomorphic vector fields whose zero-set is non-empty [27, 40, 45, 32].

The simplest situation considered in this paper is the case of a projective bundle over a curve (i.e., a compact Riemann surface). In this case, the existence problem for CSC Kähler metrics can be resolved.

Theorem 1. Let (M, J) = P (E) be a holomorphic projective bundle over a compact complex curve of genus ≥ 2. Then (M, J) admits a CSC Kähler metric in some (and hence any) Kähler class if and only if the underlying holomorphic vector bundle E is polystable.

Remark 1. The ‘if’ part follows from the theorem of Narasimhan and Seshadri: if E is a polystable bundle of rank m over a compact curve (of any genus), then E admits a hermitian–Einstein metric which in turn defines a flat P U(m)-structure on P (E) and, therefore, a family of locally-symmetric CSC Kähler exhausting the Kähler cone of P (E), see e.g. [40], [28]. Note also that in the case when P (E) fibres over ℂP^{1}, E splits as a direct sum of line bundles, and the conclusion of Theorem 1 still holds by the Lichnerowicz–Matsushima theorem, see e.g. [5, Prop. 3].

Remark 2. On all manifolds considered in Theorem 1, rational Kähler classes form a dense subset in the Kähler cone. By the LeBrun–Simanca stability theorem [46, Thm. A] and Lemma 3 below it is then sufficient to consider the existence problem only for an integral Kähler class (or polarization). In this setting, it was shown by Ross–Thomas that any projective bundle M = P (E) over a compact complex curve of genus ≥ 1 is K-polystable (with respect to some polarization) if and only if E is polystable [61, Thm. 5.13]. In view of this theorem, the “only if” part of Theorem 1 can therefore be alternatively recovered — for any genus ≥ 1 — as a consequence of recent papers by T. Mabuchi [53, 54].

By the de Rham decomposition theorem, an equivalent differential geometric formulation of Theorem 1 is that any CSC Kähler metric on (M, J) must be locally symmetric (see [28, Lemma 8] and [44]). It is in this form that we are going to achieve our proof of Theorem 1, building on the work of A. Fujiki [28]. In fact, [28] already proves Theorem 1 in the case when the underlying bundle E is simple, modulo the uniqueness of CSC Kähler metrics, which is now known [15, 20, 52].

In view of this, the main technical difficulty in proving Theorem 1 is related to the existence of automorphisms on (M, J) = P (E) → Σ. The way we proceed is by fixing a maximal torus 𝕋 (of dimension ℓ) in the identity component Aut_{0}(M, J) of the automorphism group, and showing that it induces a decomposition of E = ⨁_{i=0}^{ℓ} E_{i} as a direct sum of ℓ+1 indecomposable subbundles E_{i}, such that 𝕋 acts by scalar multiplication on each E_{i} (see Lemma 1 below). By computing the Futaki invariant of the S^{1} generators of 𝕋, we show that the slopes of E_{i} must be all equal, should a CSC Kähler metric exist on P (E) (see Lemma 3 below).^{2} Then, following the proof of [28, Thm. 3], we consider small analytic deformations E_{i}(t) of E_{i} = E_{i}(0) with E_{i}(t) being stable bundles for t ≠ 0. This induces a 𝕋-invariant Kuranishi family (M, J_{t}) ≅ P (E(t)), where E(t) = ⨁_{i=0}^{ℓ} E_{i}(t), with (M, J) being the central fibre (M, J_{0}). We then generalize in Lemma 4 the stability-under-deformations results of [45, 46, 29], by using the crucial fact that our family is invariant under a fixed maximal torus. This allows us to show that any CSC (or more generally extremal) Kähler metric ω_{0} on (M, J_{0}) can be included into a smooth family ω_{t} of extremal Kähler metrics on (M, J_{t}). As E(t) is polystable for t ≠ 0, the corresponding extremal Kähler metric ω_{t} must be locally symmetric, by the uniqueness results [15, 20, 51]. This implies that ω_{0} is locally symmetric too, and we conclude as in [28, Lemma 8].

2For rational Kähler classes, this conclusion can be alternatively reached by combining [61, Thm. 5.3] and [23].

We next consider the more general problem of existence of extremal Kähler metrics on the manifold (M, J) = P (E) → Σ. Notice that the deformation argument explained above is not specific to the CSC case, but also yields that any extremal Kähler metric ω_{0} on (M, J) = P (E) → Σ can be realized as a smooth limit (as t → 0) of extremal Kähler metrics ω_{t} on (M, J_{t}) = P (E(t)), where E(t) = ⨁_{i=0}^{ℓ} E_{i}(t) with E_{i}(t) being stable (and thus projectively-flat and indecomposable) bundles over Σ for t ≠ 0, and where ℓ is the dimension of a maximal torus 𝕋 in the identity component of the group of isometries of ω_{0}. Unlike the CSC case (where E_{i}(t) must all have the same slope and therefore E(t) is polystable), the existence problem for extremal Kähler metrics on the manifolds (M, J_{t}) is not solved in general. Our main working conjecture is that such a metric ω_{t} must always be compatible with the bundle structure (in a sense made precise in Sect. 2 below). As we observe in §3.4, if this conjecture were true it would imply that the initial bundle E must also split as a direct sum of stable subbundles (and that ω_{0} must be compatible too). We are thus led to believe in the following general statement.

Conjecture 1. A projective bundle (M, J) = P (E) over a compact curve of genus ≥ 2 admits an extremal Kähler metric in some Kähler class if and only if E decomposes as a direct sum of stable subbundles.

Remark 3. This conjecture turns out to be true in the case when E is of rank 2 and Σ is a curve of any genus, cf. [7] for an overview.

A partial answer to Conjecture 1 is given by the following result which deals with Kähler classes far enough from the boundary of the Kähler cone (noting that H^{2}(M, ℝ) is 2-dimensional so the Kähler cone up to positive scale is just a ray).

Theorem 2. Let p: P (E) → Σ be a holomorphic projective bundle over a compact complex curve Σ of genus ≥ 2 and [ω_{Σ}] be a primitive Kähler class on Σ. Then the following are equivalent:

• there exists k_{0} ≥ 0 such that for any k > k_{0} the Kähler class Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{Σ}] on (M, J) = P (E) admits an extremal Kähler metric;

• E splits as a direct sum of stable subbundles. In the case when E decomposes as the sum of at most two indecomposable subbundles,^{3} Conjecture 1 is true.

The proof of Theorem 2, given in §3.4, will be deduced from a general existence theorem established in the much broader framework of rigid and semisimple toric bundles introduced in [4], whose main features are recalled in Sect. 2 below. As explained in Remark 7, this class of manifolds is closely related to the class of multiplicity-free manifolds recently discussed in Donaldson’s paper [25]. Our most general existence result can be stated as follows.

Theorem 3. Let (g, ω) be a compatible Kähler metric on M, where M is a rigid semisimple toric bundle over a CSC local product Kähler manifold (S, g_{S}, ω_{S}) with fibres isomorphic to a toric Kähler manifold (W, ω_{W} , g_{W}), as defined in Sect. 2. Suppose, moreover, that the fibre W admits a compatible extremal Kähler metric. Then, for any k ≫ 0, the Kähler class Ω_{k} = [ω] + kp^{∗}[ω_{S}] admits a compatible extremal Kähler metric.

3This is equivalent to requiring that the automorphism group of P (E) has a maximal torus of dimension ≤ 1.

The terms of this statement, in particular the concept of a compatible metric, are introduced in Sect. 2. The proof, given in Sect. 3, uses in a crucial way the stability under small perturbations of existence of compatible extremal metric (Proposition 2) which constitutes the delicate technical part of the paper. Another important consequence of Proposition 2 is the general openness theorem given by Corollary 1.

A non-trivial assumption in the hypotheses of Theorem 3 above is the existence of compatible extremal Kähler metric on the (toric) fibre W. This is solved when W ≅ ℂP^{r} and M = P (E) with E being holomorphic vector bundle of rank r + 1, which is the sum of ℓ + 1 projectively-flat hermitian bundles, as a consequence of the fact that the Fubini–Study metric on ℂP^{r} admits a non-trivial hamiltonian 2-form of order ℓ ≤ r (cf.[3]). We thus derive in §3.3 the following existence result.

Theorem 4. Let p: P (E) → S be a holomorphic projective bundle over a compact Kähler manifold (S, J_{S}, ω_{S}). Suppose that (S, J_{S}, ω_{S}) is covered by the product of constant scalar curvature Kähler manifolds (S_{j}, ω_{j}), j ∈ I (a finite index set), and E = ⨁_{i=0}^{ℓ} E_{i} is the direct sum of projectively-flat hermitian bundles. Suppose further that for each i c_{1}(E_{i})/rk(E_{i}) − c_{1}(E_{0})/rk(E_{0}) pulls back to ∑_{j∈I} p_{ji}[ω_{j}] on ∏_{j∈I} S_{j} (for some constants p_{ji}). Then there exists k_{0} ≥ 0 such that for any k > k_{0} the Kähler class Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{S}] admits a compatible extremal Kähler metric.

Remark 4. Theorem 4 is closely related to the results of Y.-J. Hong in [36, 37] who proves, under a technical assumption on the automorphism group of S, that for any hermitian–Einstein (i.e., polystable) bundle E over a CSC Kähler manifold S, the Kähler class Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{S}] for k ≫ 0 admits a compatible CSC Kähler metric if and only if the corresponding Futaki invariant 𝔉_{Ω_{k}} vanishes. However, in the case when E is not simple (i.e., has automorphisms other than multiples of identity) the condition 𝔉_{Ω_{k}} ≡ 0 is not in general satisfied for these classes, see [5, §3.4 & 4.2] for specific examples. Thus, studying the existence of extremal rather than CSC Kähler metrics in Ω_{k} is essential. Another useful remark is that although the hypothesis in Theorem 4 that E is the sum of projectively-flat hermitian bundles over S is rather restrictive when S is not a curve, the above result suggests a general setting for the existence of extremal Kähler metrics in Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{S}] for k ≫ 0, where E is a direct sum of stable bundles (with not necessarily equal slopes) over a CSC Kähler base S.

In the concluding Sect. 4, we develop further our approach by extending the leading conjectures [21, 67] about existence of extremal Kähler metrics on toric varieties to the more general context of compatible Kähler metrics that we consider in this paper. Thus motivated, we explore in a greater detail examples when M is a projective plane bundle over a compact complex curve Σ. We show that when the genus of Σ is greater than 1, Kähler classes close to the boundary of the Kähler cone of M do not admit any extremal Kähler metric. In Appendix A, we introduce the notion of a compatible extremal almost Kähler metric (the existence of which is conjecturally equivalent to the existence of a genuine extremal Kähler metric) and show that if the genus of Σ is 0 or 1, then any Kähler class on M admits an explicit compatible extremal almost Kähler metric.

The first author was supported in part by an NSERC discovery grant, the second author by an EPSRC Advanced Research Fellowship and the fourth author by the Union College Faculty Research Fund. The authors are grateful to the referees for remarks which improved the presentation of the paper.

1. Proof of Theorem 1

As we have already noted in Remark 1, the ‘if’ part of the theorem is well-known. So we deal with the ‘only if’ part.

Let (M, J) = P (E), where E → Σ is a holomorphic vector bundle of rank m over a compact curve Σ of genus ≥ 2. We want to prove that E is polystable if (M, J) admits a CSC Kähler metric ω. Also, by Remark 1, we will be primarily concerned with the case when the connected component of the identity Aut_{0}(M, J) of the automorphisms group of (M, J) is not trivial. Note that, as the normal bundle to the fibres of P (E) → Σ is trivial and the base is of genus ≥ 2, the group Aut_{0}(M, J) reduces to H^{0}(Σ, P GL(E)), the group of fibre-preserving automorphisms of E, with Lie algebra 𝔥(M, J) ≅ H^{0}(Σ, 𝔰𝔩(E)). As any holomorphic vector field in 𝔥(M, J) has zeros, the Lichnerowicz–Matsushima theorem [49, 55] implies 𝔥(M, J) = 𝔦(M, g) ⊕ J𝔦(M, g), where 𝔦(M, g) is the Lie algebra of Killing vector fields of (M, J, ω). Thus, Aut_{0}(M, J) ≠ {Id} iff 𝔥(M, J) ≠ {0} iff 𝔦(M, g) ≠ {0}. We will fix from now on a maximal torus 𝕋 (of dimension ℓ) in the connected component of the group of isometries of (M, J, ω). Note that 𝕋 is a maximal torus in Aut_{0}(M, J) too, by the Lichnerowicz–Matsushima theorem cited above.

We will complete the proof in three steps, using several lemmas.

We start with the following elementary but useful observation which allows us to relate a maximal torus 𝕋 ⊂ Aut_{0}(M, J) with the structure of E.

Lemma 1. Let (M, J) = P (E) → S be a projective bundle over a compact complex manifold S, and suppose that the group H^{0}(S, P GL(E)) of fibre-preserving automorphisms of (M, J) contains a circle S^{1}. Then E decomposes as a direct sum E = ⨁_{i=0}^{ℓ} E_{i} of subbundles E_{i} with ℓ ≥ 1, such that S^{1} acts on each factor E_{i} by a scalar multiplication.

In particular, any maximal torus 𝕋 ⊂ H^{0}(S, P GL(E)) arises from a splitting as above, with E_{i} indecomposable and ℓ = dim(𝕋).

Proof. Any S^{1} in H^{0}(S, P GL(E)) defines a ℂ^{×} holomorphic action on (M, J), generated by an element Θ ∈ 𝔥(M, J) ≅ H^{0}(S, 𝔰𝔩(E)). For any x ∈ S, exp (tΘ(x)), t ∈ ℂ generates a ℂ^{×} subgroup of SL(E_{x}) and so Θ(x) must be diagonalizable. The coefficients of the characteristic polynomial of Θ(x) are holomorphic functions of x ∈ S, and therefore are constants. It then follows that Θ gives rise to a direct sum decomposition E = ⨁_{i=0}^{ℓ} E_{i} where E_{i} correspond to the eigenspaces of Θ at each fibre.

The second part of the lemma follows easily. □

Because of this result and the discussion preceding it, we consider the decomposition E = ⨁_{i=0}^{ℓ} E_{i} as a direct sum of indecomposable subbundles over Σ, corresponding to a fixed, maximal ℓ-dimensional torus 𝕋 in the connected component of the isometry group of (g, J, ω). We note that the isometric action of 𝕋 is hamiltonian as 𝕋 has fixed points (on any fibre).

Our second step is to understand the condition that the Futaki invariant [30], with respect to the Kähler class Ω = [ω] on (M, J), restricted to the generators of 𝕋 is zero. Hodge theory implies that any (real) holomorphic vector field with zeros on a compact Kähler 2m-manifold (M, J, ω, g) can be written as X = grad_{ω}f − Jgrad_{ω}h, where f +ih is a complex-valued smooth function on M of zero integral (with respect to the volume form ω^{m}), called the holomorphy potential of X, and where grad_{ω}f stands for the hamiltonian vector field associated to a smooth function f via ω. Then the (real) Futaki invariant associates to X the real number

𝔉_{ω}(X) = ∫_{M} fScal_{g} ω^{m},

where Scal_{g} is the scalar curvature of g. Futaki shows [30] that 𝔉_{ω}(X) is independent of the choice of ω within a fixed Kähler class Ω, and that (trivially) 𝔉_{ω}(X) = 0 if Ω contains a CSC Kähler metric. A related observation will be useful to us: with a fixed symplectic form ω, the Futaki invariant is independent of the choice of a compatible integrable almost complex structure within a path component.

Lemma 2. Let J_{t} be a smooth family of integrable almost-complex structures compatible with a fixed symplectic form ω, which are invariant under a compact group G of symplectomorphisms acting in a hamiltonian way on the compact symplectic manifold (M, ω). Denote by 𝔤_{ω} ⊂ C^{∞}(M) the finite dimensional vector space of smooth functions f such that X = grad_{ω}f ∈ 𝔤, where 𝔤 denotes the Lie algebra of G.^{4} Then the L^{2}-orthogonal projection of the scalar curvature Scal_{g_{t}} of (J_{t}, ω, g_{t}) to 𝔤_{ω} is independent of t.

Proof. By definition, any f ∈ 𝔤_{ω} defines a vector field X = grad_{ω}f which is in 𝔤, and is therefore Killing with respect to any of the Kähler metrics g_{t} = ω(·, J_{t}·). To prove our claim, we have to show that ∫_{M} fScal_{g_{t}}ω^{m} is independent of t. Using the standard variational formula for scalar curvature (see e.g. [11, Thm. 1.174]), we compute

(1)  \frac{d}{dt}Scal_{g_{t}} = Δ(tr_{g_{t}}h) + δδh − g_{t}(r, h) = δδh,

where h denotes \frac{d}{dt}g_{t}, while Δ, δ and r are the riemannian laplacian, the codifferential and the Ricci tensor of g_{t}, respectively. Note that to get the last equality, we have used the fact that h is J_{t}-anti-invariant (as all the J_{t}’s are compatible with ω) while the metric and the Ricci tensor are J_{t}-invariant (on any Kähler manifold). Integrating against f, we obtain

\frac{d}{dt}(∫_{M} fScal_{g_{t}}ω^{m}) = ∫_{M}(δδh)fω^{m} = ∫_{M} g_{t}(h, Ddf)ω^{m},

where D is the Levi–Civita connection of g_{t}; however, as f is a Killing potential with respect to the Kähler metric (g_{t}, J_{t}), it follows that Ddf is J_{t}-invariant, and therefore ∫_{M} fScal_{g_{t}}ω^{m} is independent of t. □

Remark 5. One can extend Lemma 2 to any smooth family of (not necessarily integrable) G-invariant almost complex structures J_{t} compatible with ω. Then, as shown in [47], the L^{2}-projection to 𝔤_{ω} of the hermitian scalar curvature of the almost Kähler metric (ω, J_{t}) (see Appendix A for a precise definition) is independent of t. This gives rise to a symplectic Futaki invariant associated to a compact subgroup G of the group of hamiltonian symplectomorphisms of (M, ω).

Lemma 2 will be used in conjunction with the Narasimhan–Ramanan approximation theorem (see [57, Prop. 2.6] and [58, Prop. 4.1]), which implies that any holomorphic vector bundle E over a compact curve Σ of genus ≥ 2 can be included in an analytic family of vector bundles E_{t}, t ∈ D_{ε} (where D_{ε} = {t ∈ ℂ, |t| < ε}) over Σ, such that E_{0} := E and E_{t} is stable for t ≠ 0. Such a family will be referred to in the sequel as a small stable deformation of E.

Lemma 3. Suppose that the vector bundle E = U ⊕V → Σ splits as a direct sum of two subbundles, U and V. Consider the holomorphic S^{1}-action on (M, J) = P (E), induced by fibrewise scalar multiplication by exp (iθ) on V, and let X ∈ 𝔥(M, J) be the (real) holomorphic vector field generating this action. Then the Futaki invariant of X with respect to some (and therefore any) Kähler class Ω on (M, J) vanishes if and only if U and V have the same slope.

Proof. We take some Kähler form ω on (M, J) = P (E) and, by averaging it over S^{1}, we assume that ω is S^{1}-invariant. As the S^{1}-action has fixed points, the corresponding real vector field X is J-holomorphic and ω-hamiltonian, i.e., X = grad_{ω}f for some smooth function f with ∫_{M} fω^{m} = 0.

4We will tacitly identify throughout the Lie algebra 𝔤 of a group G acting effectively on M with the Lie algebra of vector fields generated by the elements of 𝔤.

We now consider small stable deformations U_{t}, V_{t}, t ∈ D_{ε} of U and V, and put E_{t} = U_{t} ⊕ V_{t}. Considering the projective bundle P (E_{t}), we obtain a non-singular Kuranishi family (M, J_{t}) with J_{0} = J. By the Kodaira stability theorem (see e.g. [41]) one can find a smooth family of Kähler metrics (ω_{t}, J_{t}) with ω_{0} = ω. Using the vanishing of the Dolbeault groups H^{2,0}(M, J_{t}) = H^{0,2}(M, J_{t}) = 0, Hodge theory implies that by decreasing the initial ε if necessary, we can assume [ω_{t}] = [ω] in H_{dR}^{2}(M). Note that any J_{t} is S^{1}-invariant so, by averaging over S^{1}, we can also assume that ω_{t} is S^{1}-invariant. Applying the equivariant Moser lemma, one can find S^{1}-equivariant diffeomorphisms, Φ_{t}, such that Φ_{t}^{∗}ω_{t} = ω. Considering the pullback of J_{t} by Φ_{t}, the upshot of this construction is that we have found a smooth family of integrable complex structures J_{t} such that: (1) each J_{t} is compatible with the fixed symplectic form ω and is S^{1}-invariant;

(2) J_{0} = J; (3) for t ≠ 0, the complex manifold (M, J_{t}) is equivariantly biholomorphic to P (U_{t} ⊕ V_{t}) → Σ with U_{t} and V_{t} stable (and therefore projectively-flat) hermitian bundles.

If U and V have equal slopes, then E_{t} = U_{t} ⊕ V_{t} becomes polystable for t ≠ 0, and (M, J_{t}) has a CSC Kähler metric in each Kähler class. It follows that the Futaki invariant of X on (M, J_{t}, ω) is zero for t ≠ 0.

Conversely, if U and V have different slopes, it is shown in [5, §3.2] that the Futaki invariant of X is different from zero for any Kähler class on (M, J_{t}), t ≠ 0.

We conclude using Lemma 2. □

This lemma shows that all the factors in the decomposition E = ⨁_{i=0}^{ℓ} E_{i} must have equal slope, should a CSC Kähler metric exists. As in the proof of Lemma 3, we consider small stable deformations E_{i}(t) of E_{i} and our assumption for the slopes insures that E(t) = ⨁_{i=0}^{ℓ} E_{i}(t) is polystable for t ≠ 0; furthermore, by acting with 𝕋-equivariant diffeomorphisms, we obtain a smooth family of 𝕋-invariant complex structures J_{t} compatible with ω, such that for t ≠ 0, the complex manifold (M, J_{t}) has a locally-symmetric CSC Kähler metric in each Kähler class; by the uniqueness of the extremal Kähler metrics modulo automorphisms [15, 52], any extremal Kähler metric on (M, J_{t}) is locallysymmetric when t ≠ 0. The third step in the proof of Theorem 1 is then to show that the initial CSC Kähler metric (J_{0}, ω) must be locally symmetric too. This follows from the next technical result which generalizes arguments of [46, 29] and fits with the general approach to the deformation theory of CSC Kähler metrics given in [68, Thm. 2].

Lemma 4. Let J_{t} be a smooth family of integrable almost-complex structures compatible with a symplectic form ω on a compact manifold M, which are invariant under a torus 𝕋 of hamiltonian symplectomorphisms of (M, ω). Suppose, moreover, that (J_{0}, ω) define an extremal Kähler metric and that 𝕋 is a maximal torus in the reduced automorphism group of (M, J_{0}). Then there exists a smooth family of extremal Kähler metrics (J_{t}, ω_{t}, g_{t}), defined for sufficiently small t, such that ω_{0} = ω and [ω_{t}] = [ω] in H_{dR}^{2}(M).

Proof. Recall that [32, 45] on any compact Kähler manifold (M, J), the reduced automorphism group, Ãut_{0}(M, J), is the identity component of the kernel of the natural group homomorphism from Aut_{0}(M, J) to the Albanese torus of (M, J); it is also the connected closed subgroup of Aut_{0}(M, J), whose Lie algebra 𝔥_{0}(M, J) ⊂ 𝔥(M, J) is the ideal of holomorphic vector fields with zeros.

We denote by 𝔱 the Lie algebra of 𝕋 and by 𝔥 (resp. 𝔥_{0}) the Lie algebra of the complex automorphism group (resp. reduced automorphism group) of the central fibre (M, J_{0}). As 𝕋 acts in a hamiltonian way, we have 𝔱 ⊂ 𝔥_{0}. By assumption, 𝔱 is a maximal abelian subalgebra of 𝔦_{0}(M, g_{0}) = 𝔦(M, g_{0})∩𝔥_{0}, where 𝔦(M, g_{0}) is the Lie algebra of Killing vector fields of (M, J_{0}, ω, g_{0}).

As in the Lemma 2 above, we let 𝔱_{ω} ⊂ C^{∞}(M) be the finite dimensional space of smooth functions which are hamiltonians of elements of 𝔱. As the Kähler metric (J_{0}, ω, g_{0}) is extremal (by assumption), its scalar curvature Scal_{g_{0}} is hamiltonian of a Killing vector field X = grad_{ω}(Scal_{g_{0}}) ∈ 𝔦_{0}(M, g_{0}). Clearly, such a vector field is central, so X ∈ 𝔱 (by the maximality of 𝔱) and therefore Scal_{g_{0}} ∈ 𝔱_{ω}.

For any 𝕋-invariant Kähler metric (J̃, ω̃, g̃) on M, we denote by 𝔱ω̃ the corresponding space of Killing potentials of elements of 𝔱 (noting that any X ∈ 𝔱 has zeros, so that 𝕋 belongs to the reduced automorphism group of (M, J̃)), and by Πω̃ the L^{2}-orthogonal projection of smooth function to 𝔱ω̃, with respect to the volume form ω̃^{m}. Obviously, if the scalar curvature Scal g̃ of g̃ belongs to 𝔱ω̃, then g̃ is extremal.

Following [46], let C_{⊥}^{∞}(M)^{𝕋} denote the Fréchet space of 𝕋-invariant smooth functions on M, which are L^{2}-orthogonal (with respect to the volume form ω^{m}) to 𝔱_{ω}, and let U be an open set in ℝ × C_{⊥}^{∞}(M)^{𝕋} of elements (t, f) such that ω + dd_{t}^{c}f is Kähler with respect to J_{t} (here d_{t}^{c} denotes the d^{c}-differential corresponding to J_{t}). We then consider the map Ψ: U → ℝ × C_{⊥}^{∞}(M)^{𝕋}, defined by

Ψ(t, f) = (t, (Id − Π_{ω}) ∘ (Id − Πω̃)(Scal g̃)),

where ω̃ := ω + dd_{t}^{c}f and Scal g̃ is the scalar curvature of the Kähler metric g̃ defined by (J_{t}, ω̃). One can check that this map is C^{1} and compute (as in [45], by also using (1)) that its differential at (0, 0) ∈ U is

(T_{(0,0)}Ψ)(t, f) = (t, tδδh − 2δδ(Ddf)^{−}),

where D and δ are respectively the Levi–Civita connection and the codifferential of g_{0}, h = (\frac{dg_{t}}{dt} )_{t=0} and (Ddf)^{−} denotes the J_{0}-anti-invariant part of Ddf. Note that L(f) := δδ((Ddf)^{−}) is a fourth order (formally) self-adjoint 𝕋-invariant elliptic linear operator (known also as the Lichnerowicz operator, see e.g. [32]). When acting on smooth functions, L annihilates 𝔱_{ω} (because any Killing potential f satisfies (Ddf)^{−} = 0). It then follows that L leaves C_{⊥}^{∞}(M)^{𝕋} invariant and, by standard elliptic theory, we obtain an L^{2}-orthogonal splitting C_{⊥}^{∞}(M)^{𝕋} = ker(L) ⊕ im(L). However, any smooth 𝕋invariant function f in ker(L) gives rise to a Killing field X = grad_{ω}f in the centralizer of 𝔱 ⊂ 𝔦_{0}(M, g_{0}). As 𝔱 is a maximal abelian subalgebra of 𝔦_{0}(M, g_{0}) we must have X ∈ 𝔱, i.e., f ∈ 𝔱_{ω}. It follows that the kernel of L restricted to C_{⊥}^{∞}(M)^{𝕋} is trivial, and therefore L is an isomorphism of the Fréchet space C_{⊥}^{∞}(M)^{𝕋}.

This understood, we are in position to apply standard arguments, using the implicit function theorem for the extension of Ψ to the Sobolev completion L_{⊥}^{2,k}(M)^{𝕋} (with k ≫ 1) of C_{⊥}^{∞}(M)^{𝕋}, together with the regularity result for extremal Kähler metrics, precisely as in [45, 46, 29]. We thus obtain a family (t, ω_{t}) of smooth, 𝕋-invariant extremal Kähler metrics (J_{t}, ω_{t}) (defined for t in a small interval about 0) which converge to the initial extremal Kähler metric (J_{0}, ω) (in any Sobolev space L^{2,k}(M), k ≫ 1, and hence, by the Sobolev embedding, in C^{∞}(M)). □

The uniqueness argument thus also applies at t = 0, and the initial metric is locally symmetric. We can now conclude the proof of Theorem 1 by a standard argument using the de Rham decomposition theorem (see [28, Lemma 8] and [44]). This realizes the fundamental group of Σ as a discrete subgroup group of isometries of the hermitian symmetric space ℂP^{m−1}×ℍ and thus defines a projectively flat structure on P (E) → Σ.

2. Rigid toric bundles and the generalized Calabi construction

In this section, we recall the notions of rigid and semisimple isometric hamiltonian torus actions on compact Kähler manifolds, as well the generalized Calabi construction of compatible Kähler metrics on such manifolds [4]. Compatible Kähler metrics on rigid toric bundles over a semisimple base provide a framework to search for extremal Kähler metrics which parallels (and extends) the theory of toric extremal Kähler metrics developed in [21, 22, 24]. We shall apply the construction to projective bundles of the form P (E_{0} ⊕ ⋯ ⊕ E_{ℓ}) → S, where E_{i} is a projectively-flat hermitian bundle over a Kähler manifold (S, ω_{S}), and prove the existence of compatible extremal Kähler metrics in classes sufficiently far from the boundary of the Kähler cone, cf. Theorems 3 and 4.

2.1. Rigid and semisimple torus actions. Most of the material in this section is taken from [4, Sect. 2] to which we refer the Reader for further details.

Definition 1. Let (M, g, J, ω) be a connected Kähler 2m-manifold with an effective isometric hamiltonian action of an ℓ-torus 𝕋 with momentum map z: M → 𝔱^{∗}. We say the action is rigid if for all x ∈ M, R_{x}^{∗}g depends only on z(x), where R_{x}: 𝕋 → 𝕋 · x ⊂ M is the orbit map.

Equivalently, the action is rigid if, for any two generators X_{ξ}, X_{η} (with ξ, η ∈ 𝔱) the smooth function g(X_{ξ}, X_{η}) is constant on the levels of the momentum map z.

The standard m-torus action on a toric Kähler 2m-manifold is rigid. If M is compact, as we shall assume henceforth, many features of toric Kähler manifolds are shared by rigid torus actions. In particular:

• the image of M under the momentum map z is a Delzant polytope Δ ⊂ 𝔱^{∗} [4, Prop. 4];

• the points in the interior Δ^{0} are the regular values of z;

• the complex (stable) quotient Ŝ of M by the complexified torus 𝕋^{c} (which is a priori a 2(m − ℓ)-dimensional complex orbifold) is smooth [4, Prop. 5];

• M^{0} := z^{−1}(Δ^{0}) is a principal 𝕋^{c} bundle over Ŝ. The Delzant construction [16], associates to Δ (and hence M) a smooth compact symplectic 2ℓ-manifold (V, ω_{V}), with a toric 𝕋-action, and there are complex structures J_{V} making V into a toric variety under the action of the complexified torus 𝕋^{c}.

The inverse image under z of the union of the codimension one faces of Δ is a (reducible) subvariety Ξ of M such that the blow-up of M along Ξ is 𝕋^{c}-equivariantly biholomorphic to the toric bundle M̂ := M^{0} ×_{𝕋^{c}} V → Ŝ. In simple examples, Ξ is a divisor and M̂ is biholomorphic to M. In general, M, like M̂, is a 𝕋^{c}-equivariant compactification of the principal 𝕋^{c}-bundle M^{0} over Ŝ; we shall therefore abuse language and refer to M as a rigid toric bundle.

The general form of M will be used in applications to projective bundles, but only in the special case that the (rigid) torus action is also “semisimple” in the following sense.

Definition 2. An isometric hamiltonian torus action on a Kähler manifold (M, g, J, ω) with momentum map z and complex (stable) quotient^{5} Ŝ is semisimple if at any regular value of z, the z-derivative of the family ωŜ(z) of Kähler forms (induced by the symplectic quotient at momentum level z) is parallel and diagonalizable with respect to ωŜ(z).

This definition essentially implies that the universal cover of Ŝ is a Kähler product ∏_{j∈I}(S_{j}, ω_{j}) (I finite) such that the pullback of ωŜ(z) is homothetic to ω_{j} on each factor by functions f_{j}(z). The distributions on Ŝ induced by this product structure are simultaneous eigenspaces of ωŜ(z), and in the rigid case the functions f_{j} are affine in z.

5This quotient is well-defined within a connected component of the regular values of z.

In this case each component of Ξ with codimension d + 1 > 1 in M corresponds to a ℂP^{d} factor (with constant holomorphic sectional curvature) in the Kähler product.

Conversely, these features lead to a general construction of rigid toric bundles with semisimple torus actions. Let 𝕋 = 𝔱/2πΛ be a real (compact) ℓ-torus, with Lie algebra 𝔱, let Δ be a Delzant polytope in the dual space 𝔱^{∗}, with codimension one faces F_{b} : b ∈ B and primitive inward normals u_{b} ∈ 𝔱, let (V, J_{V} , ω_{V} , 𝕋) be a toric Kähler 2ℓ-manifold, with momentum polytope Δ, and let S be compact Kähler manifold whose universal cover is a Kähler product ∏_{a∈A}(S_{a}, ω_{a}) (with finite index set A disjoint from B).

For each b ∈ B, fix an integer d_{b} ≥ 0, and let Ŝ be the total space of a fibre product (for b ∈ B) of flat unitary ℂP^{d_{b}}-bundles over S. The universal cover of Ŝ is then a Kähler product ∏_{j∈I}(S_{j}, ω_{j}), where I = A ∪ {b ∈ B : d_{b} > 0} and for each b ∈ B, S_{b} = ℂP^{d_{b}} with Fubini–Study metric ω_{b} normalized so that [ω_{b}] = 2πc_{1}(O_{ℂP^{d_{b}}}(1)) (i.e., ω_{b} has scalar curvature 2d_{b}(d_{b} + 1), or equivalently, holomorphic sectional curvature 2).

Let P̂ be a principal 𝕋-bundle over Ŝ, such that −2πc_{1}(P̂) ∈ H^{2}(Ŝ, 𝔱), is diagonalizable with respect to the local product structure of Ŝ, i.e., is of the form ∑_{j∈I}[ω_{j}] ⊗ p_{j} = ∑_{a∈A}[ω_{a}] ⊗ p_{a} + ∑_{b∈B}[ω_{b}] ⊗ u_{b}, where the p_{j} are (constant) elements of 𝔱 with p_{b} = u_{b} for b ∈ B. Let M̂ = P̂ ×_{𝕋} V be the associated toric bundle over Ŝ.

It remains to describe the blow-down map M̂ → M, which is characterized by its behaviour over each fibre of Ŝ → S. Each such fibre is biholomorphic to S′ = ∏_{b∈B} ℂP^{d_{b}}, on which P̂ restricts to a principal 𝕋-bundle P′ with c_{1}(P′) = ∑_{b∈B} c_{1}(O_{ℂP^{d_{b}}}(−1))⊗u_{b}. The blow-down W of Ŵ := P′ ×_{𝕋} V over S′ may be described as a restricted toric quotient [4]. For this, recall that the Delzant construction realizes V as a symplectic quotient of ℂ^{B} := ∏_{b∈B} ℂ by a subgroup G of the diagonal torus action; now replace ℂ^{B} with ∏_{b∈B} ℂ^{d_{b}+1}, which has a diagonal action of the same torus (by scalar multiplication on each factor), hence also an action of G. The symplectic quotient by G is the blowdown W. We refer to [4, §1.6] for details.^{6}

Definition 3. A blow-down of M̂ = P̂ ×_{𝕋} V → Ŝ is a (locally trivial) fibre bundle M over S which fits into a 𝕋-equivariant commutative diagram

[figure: M̂ = P̂ ×_{𝕋} V; Ŝ; M; S,]

and is 𝕋-equivariantly biholomorphic to Ŵ → W over each fibre of Ŝ → S.

A complex manifold (M, J, 𝕋) obtained in this way will be called a rigid toric bundle over a semisimple base.

In [4, §2.5], this blow-down construction was discussed in terms of the universal covers of M̂ → M and Ŝ → S, the latter cover being a trivial S′ bundle. More generally, if P is a principal 𝕋-bundle on S whose first Chern class satisfies −2πc_{1}(P) = ∑_{a∈A}[ω_{a}] ⊗ p_{a}, then M̂ = P̂ ×_{𝕋} V ≅ P ×_{𝕋} Ŵ and so M = P ×_{𝕋} W satisfies Definition 3.

2.2. Projective bundles as rigid toric bundles. The constructions of the previous paragraph will now be specialized to the prominent case of projective bundles, when the Delzant polytope Δ is a simplex in 𝔱^{∗} ≅ ℝ^{ℓ}, with codimension one faces F_{j}, j ∈ B = {0, 1, … ℓ}. The associated complex toric variety V is then ℂP^{ℓ} and M̂ is 𝕋^{c}- equivariantly biholomorphic to a ℂP^{ℓ}-bundle over Ŝ; since M̂ comes from a principal 𝕋^{c}-bundle, it has the form P (L_{0} ⊕ ⋯ ⊕ L_{ℓ}) → Ŝ, where L_{i} are hermitian holomorphic line bundles (the 𝕋^{c} action is induced by scalar multiplication on L_{i}).

6A simple illustration of this construction is W = ℂP^{2} seen as a fibrewise S^{1}-equivariant blow-down of the first Hirzebruch surface Ŵ = P (O ⊕ O(−1)) → ℂP^{1}.

The blow-down of M̂ is encoded by the realization of Ŝ as a fibre product of flat projective unitary ℂP^{d_{b}}-bundles over a Kähler manifold S. We shall only consider flat projective bundles of the form P (E), where E is a projectively-flat hermitian vector bundle over S (the obstruction to the existence of such an E is a torsion element of H^{2}(S, O^{∗}); in particular, it always exists if S is a Riemann surface). We thus write Ŝ = P (E_{0}) ×_{S} ⋯ ×_{S} P (E_{ℓ}) → S, where each E_{i} → S is a projectively-flat hermitian bundle of rank d_{i} + 1, and we assume that c_{1}(E_{i})/(d_{i} + 1) − c_{1}(E_{0})/(d_{0} + 1) pulls back to ∑_{a∈A} p_{ia}[ω_{a}] on the covering space ∏_{a∈A}(S_{a}, ω_{a}) of S.

In this case, M̂ = P (O(−1)_{E_{0}} ⊕⋯⊕O(−1)_{E_{ℓ}}) → Ŝ, where O(−1)_{E_{i}} is the (fibrewise) tautological line bundle over P (E_{i}) → S — trivial over the other factors of Ŝ — whereas M = P (E_{0} ⊕ ⋯ ⊕ E_{ℓ}) → S: in each fibre over S, the blow-down is biholomorphic to the map from the total space of P (⨁_{j=0}^{ℓ} O(−1)_{ℂ^{d_{j} +1}}) → ∏_{j=0}^{ℓ} ℂP^{d_{j}+1} to ℂP^{d}, given by a direct sum decomposition ℂ^{d+1} = ⨁_{j=0}^{ℓ} ℂ^{d_{j}+1}, cf., [4].

2.3. The generalized Calabi construction. A compact Kähler manifold with rigid semisimple isometric hamiltonian action of an ℓ-torus 𝕋 is biholomorphic to a rigid toric bundle (M, J) over a semisimple base. According to [4, Thm. 2], on any such rigid toric bundles, the compatible Kähler metrics (g, ω) (for which the ℓ-torus action is rigid, semisimple, isometric and hamiltonian) may be described using a generalized Calabi construction with three main ingredients — only two if there is no blow-down.

The first ingredient is the choice of a compatible 𝕋-invariant Kähler metric g_{V} on the symplectic toric manifold (V, ω_{V} , 𝕋) with Delzant polytope Δ. The description of such metrics is well known (see e.g. [1, 2, 22, 34]). On the union V^{0} of the generic 𝕋 orbits (the inverse image of the interior Δ^{0} under the momentum map z: V → Δ ⊂ 𝔱^{∗}) there are “angular coordinates” t: V^{0} → 𝔱/2πΛ, defined up to an additive constant, with dt(JX_{ξ}) = 0 and dt(X_{ξ}) = ξ (the orthogonal distribution to the 𝕋 orbits is integrable). The action-angle coordinates (z, t) identify each tangent space with 𝔱 ⊕ 𝔱^{∗} and the symplectic form on V is ω_{V} = ⟨dz ∧dt⟩, where the angle brackets denote contraction of 𝔱 and 𝔱^{∗}. Compatible Kähler metrics may then be written in the form

(2)  g_{V} = ⟨dz, G, dz⟩ + ⟨dt, H, dt⟩,

where G is a positive definite S^{2}𝔱-valued function on Δ^{0}, H is its inverse in S^{2}𝔱^{∗} and ⟨·, ·, ·⟩ denotes the pointwise contraction 𝔱^{∗} × S^{2}𝔱 × 𝔱^{∗} → ℝ or the dual contraction. The corresponding almost complex structure is defined by

(3)  Jdt = −⟨G, dz⟩

from which it follows that J is integrable if and only if G is the hessian of a function U (the symplectic potential) on Δ^{0} [34].

In order for such a metric on V^{0} to compactify on V, U must satisfy boundary conditions [2, 4, 22], which we describe following [4, Prop. 1]. For any face F ⊂ Δ, let 𝔱_{F} ⊂ 𝔱 denote the vector subspace spanned by inward normals u_{i} ∈ 𝔱 to codimension one faces of Δ containing F; the codimension of 𝔱_{F} is dimension of F, and the annihilator 𝔱_{F}^{0} ⊂ 𝔱^{∗} is isomorphic to (𝔱/𝔱_{F})^{∗}. A smooth function U on Δ^{0} corresponds to a 𝕋invariant, ω_{V} -compatible Kähler metric g_{V} via (2) if and only if the S^{2}𝔱^{∗}-valued function H = Hess(U)^{−1} on Δ^{0} satisfies the following conditions:

• [smoothness] H is the restriction to Δ^{0} of a smooth S^{2}𝔱^{∗}-valued function on Δ;

• [boundary values] for z in a codimension one face F_{i} ⊂ Δ with inward normal u_{i},

(4)  H_{z}(u_{i}, ·) = 0  and  (dH)_{z}(u_{i}, u_{i}) = 2u_{i},

where the differential dH is viewed as a smooth S^{2}𝔱^{∗} ⊗ 𝔱-valued function on Δ; • [positivity] for any point z in the interior of a face F ⊆ Δ, H_{z}(·, ·) is positive definite when viewed as a smooth function with values in S^{2}(𝔱/𝔱_{F})^{∗}. We denote by S(Δ) the space of all symplectic potentials U on Δ satisfying these conditions: any U ∈ S(Δ) defines a 𝕋-invariant, ω_{V} -compatible Kähler metric on V.

The second ingredient, nontrivial only if “blow-downs” are present, is to transfer the metric on V to the restricted toric quotient variety W, which is birational to Ŵ = P′ ×_{𝕋} V, where P′ is a 𝕋-principal bundle over ∏_{b∈B} ℂP^{d_{b}}. For this we choose a connection 1-form θ′ on P′, with curvature dθ′ = ∑_{b∈B} ω_{b}⊗u_{b}, where ω_{b} is the (normalized) Fubini– Study metric on ℂP^{d_{b}} of scalar curvature 2d_{b}(d_{b} + 1), and u_{b} ∈ 𝔱 is the inward normal to the codimension one face F_{b} ⊂ Δ (on which ⟨u_{b}, z⟩ + c_{b} = 0). We still denote by θ′ ∈ Ω^{1}(W^{0}, 𝔱) the induced 1-form on the open dense subset W^{0} := P′ ×_{𝕋}V^{0} of Ŵ. The Kähler structure on W^{0} is then defined by

(5)  \begin{aligned}g_{W} & = ∑_{b∈B}(⟨u_{b}, z⟩ + c_{b})g_{b} + ⟨dz, G, dz⟩ + ⟨θ′, H, θ′⟩, \\ ω_{W} & = ∑_{b∈B}(⟨u_{b}, z⟩ + c_{b})ω_{b} + ⟨dz ∧ θ′⟩,  dθ′ = ∑_{b∈B} ω_{b} ⊗ u_{b}, \end{aligned}

with G = Hess(U) = H^{−1}. By [4, Prop. 2], (g_{W} , ω_{W}) extends to a smooth, 𝕋-invariant, Kähler structure on W.

The third and final ingredient builds on the second by a similar construction of a Kähler structure on M^{0} = P̂ ×_{𝕋} V^{0}, using a connection on P̂, with curvature (covered by) ∑_{a∈A} ω_{a} ⊗ p_{a} + ∑_{b∈B} ω_{b} ⊗ u_{b}. The restriction of this connection to each fibre of Ŝ → S is then isomorphic to (P′, θ′) over ∏_{b∈B} ℂP^{d_{b}}. We denote by θ̂ ∈ Ω^{1}(M^{0}, 𝔱) the induced connection 1-form on M^{0} = P̂ ×_{𝕋} V^{0}, on which we define a Kähler structure

(6)  \begin{aligned}g & = ∑_{j∈I}(⟨p_{j}, z⟩ + c_{j})g_{j} + ⟨dz, G, dz⟩ + ⟨θ̂, H, θ̂⟩, \\ ω & = ∑_{j∈I}(⟨p_{j}, z⟩ + c_{j})ω_{j} + ⟨dz ∧ θ̂⟩,  dθ̂ = ∑_{j∈I} ω_{j} ⊗ p_{j}, \end{aligned}

where:

• G = Hess(U) = H^{−1}, where U is the symplectic potential of the chosen toric Kähler structure g_{V} on V;

• for each b ∈ B, p_{b} = u_{b} and the real number c_{b} is such that ⟨p_{b}, z⟩ + c_{b} = 0 on the codimension one face F_{b};

• for each a ∈ A, ⟨p_{a}, z⟩ + c_{a} is positive on Δ.

Clearly, (6) extends to a smooth tensor on M̂ and by [4, Thm. 2], it is the pullback of a smooth metric on the blow-down M. This is obvious when Ŝ → S is the trivial S′-bundle (for instance, when M̂ is simply connected), since the connection θ̂ on P̂ → Ŝ = S′ ×S is then isomorphic to the sum of (pulled back) connections θ on P → S and θ′ on P′ → S′ with curvatures ∑_{a∈A} ω_{a} ⊗ p_{a} and ∑_{b∈B} ω_{b} ⊗ u_{b} respectively. Hence M ≅ P ×_{𝕋} W and M^{0} ≅ P ×_{𝕋} W^{0} where W^{0} = P′ ×_{𝕋} V^{0}. It follows that the metric (g, ω) (6) restricts to (g_{W} , ω_{W}) (5) on each W^{0} fibre; the latter compactifies smoothly on W, and ⟨p_{a}, z⟩ + c_{a} are strictly positive on M, so (6) defines a Kähler structure on M. To handle the general case, one can consider the universal covers of M̂ and M and use the previous argument, noting that the smooth extension of the metric is a local property; a direct argument in the case of the projective bundles described in §2.2 can be given along the lines of [5, §1.3]. This completes the generalized Calabi construction according to [4].

Assuming that the metrics (g_{j}, ω_{j}), the connection 1-form θ̂, the polytope Δ and the constants (p_{j}, c_{j}) are all fixed, (6) defines a family of Kähler metrics parametrized by symplectic potentials U ∈ S(Δ) (or, equivalently, by toric Kähler metrics on (V, ω_{V} , 𝕋)). We note that for this family, the symplectic 2-form ω remains unchanged, so we obtain a family of 𝕋-invariant ω-compatible Kähler metrics corresponding to different complex structures. However, any two such complex structures are biholomorphic, under a 𝕋equivariant diffeomorphism in the identity component: this is well-known [2, 21] in the case of a symplectic toric manifolds (i.e., on (V, ω_{V} , 𝕋)), and the same argument holds (fibrewise) on W and M (see [5, §1.4]). The pullbacks of the symplectic form ω under such diffeomorphisms belong to a common Kähler class Ω on a fixed complex manifold (M, J) (we can take J to be the complex structure on M introduced in Definition 3: it corresponds to the standard symplectic potential U_{0}, see [2, 34]).

Definition 4. A compatible Kähler metric (g, ω) on a rigid toric bundle M over a semisimple base S is one arising via the generalized Calabi construction (and hence given explicitly by (6) on M^{0} for a symplectic potential U ∈ S(Δ)). The corresponding Kähler classes on (M, J) are called compatible Kähler classes.

We shall further assume that the metrics (g_{j}, ω_{j}) are fixed and have constant scalar curvature Scal_{j} (with Scal_{b} = 2d_{b}(d_{b} + 1) for b ∈ B),^{7} and that Δ and p_{j} are fixed. Recall that for b ∈ B, the constants c_{b} are also fixed by requiring ⟨u_{b}, z⟩ + c_{b} = 0 on the codimension one face F_{b} ⊂ Δ. The real constants c_{a}, for a ∈ A, can vary (on a given manifold (M, J)) and they parametrize the compatible Kähler classes.

2.4. The isometry Lie algebra. For a compact Kähler manifold (M, g), we denote by 𝔦_{0}(M, g) the Lie algebra of all Killing vector fields with zeros; this is equivalently the Lie algebra of all hamiltonian Killing vector fields.

Lemma 5. Let (g, ω) be a compatible Kähler metric on M, where the stable complex quotient Ŝ is equipped with the local product Kähler metric (gŜ, ωŜ) covered by ∏_{j∈I}(S_{j}, ω_{j}). Denote by p̂: M^{0} → Ŝ the principal 𝕋^{c}-fibre structure of the regular part M^{0} of 𝕋 action on M. Let 𝔷(𝕋, g) be the centralizer in 𝔦_{0}(M, g) of the ℓ-torus 𝕋.

Then the vector space 𝔷(𝕋, g) is the direct sum of a lift of 𝔦_{0}(Ŝ, gŜ) and the Lie algebra 𝔱 ⊂ 𝔦_{0}(M, g) of 𝕋 in such a way that the natural homomorphism p̂_{∗}: 𝔷(𝕋, g) → 𝔦_{0}(S, g_{S}) is a surjection.

In the case ℓ = 1, this result is proven in [5, Prop. 3] and the proof there generalizes easily; for the convenience of the Reader, we here supply details for arbitrary ℓ.

Proof. Denote by K = grad_{ω}z ∈ C^{∞}(M, T M) ⊗ 𝔱^{∗} the family of hamiltonian Killing vector fields generated by 𝕋: thus, the span of K realizes the Lie algebra 𝔱 of 𝕋 as a subalgebra of 𝔦_{0}(M, g).

Let X be a holomorphic vector field on Ŝ which is hamiltonian with respect to ωŜ; then the projection X_{j} of X onto the distribution H_{j} (induced by T S_{j} on the universal cover ∏_{j∈I} S_{j} of S) is a Killing vector field with zeros, so ι_{X_{j}}ωŜ = −df_{j} for some function f_{j} (with integral zero). Thus ∑_{j∈I} f_{j}p_{j} is a family of hamiltonians for X with respect to the family of symplectic forms covered by ∑_{j∈I} ω_{j} ⊗ p_{j}: since this is the curvature dθ̂ of the connection on M^{0}, X lifts to a holomorphic vector field X̃ = X_{H} + ∑_{j∈I} f_{j}⟨p_{j}, K⟩ on M^{0}, which is hamiltonian with potential ∑_{j∈I}(⟨p_{j}, z⟩ + c_{j})f_{j} and commutes with the components of K. (Here X_{H} is the horizontal lift to M^{0} with respect to θ̂.) As the metric g extends to M and X̃ is Killing with respect to g, it extends to M too (note that M \ M^{0} has codimension ≥ 2). It is not difficult to see that X̃ has zeros on M (in fact, if s_{0} ∈ Ŝ is a zero of X then X̃ − ∑_{j∈I} f_{j}(s_{0})⟨p_{j}, K⟩ vanishes on M^{0}) so that X̃ is an element of 𝔦_{0}(M, g). Of course, this shows that the Killing potential ∑_{j∈I}(⟨p_{j}, z⟩+c_{j})f_{j} extends as a smooth function on M.

7Presumably, the Kähler metrics (g_{j}, ω_{j}) must be CSC in order to obtain an extremal Kähler metric (g, ω) as above. We do not prove this here, but this fact has been established for ℓ = 1 in [6, Prop. 14].

Conversely, any X̃ ∈ 𝔷(𝕋, g) is a 𝕋^{c}-invariant holomorphic vector field, so its restriction to M^{0} is projectable to a holomorphic vector field X ∈ 𝔥_{0}(Ŝ). This allows to reverse the above arguments: for X̃ = X_{H} + f⟨p, K⟩ + hJ⟨q, K⟩ (where p, q ∈ 𝔱 and f, g ∈ C^{∞}(Ŝ)) be Killing with respect to the metric (6), we must have q = 0 and X be Killing with respect to gŜ. Such a vector field maps to zero iff it is a constant multiple of a component of K. This gives a projection to 𝔦_{0}(Ŝ, gŜ) splitting the inclusion just defined. □

This is the main ingredient in the proof of the following result.

Proposition 1. Let (g, ω) be a compatible Kähler metric on M, where the factors (S_{j}, ω_{j}) in the covering ∏_{j∈I} S_{j} of the stable quotient Ŝ have constant scalar curvature. Then g is invariant under a maximal torus G of the reduced automorphism group Ãut_{0}(M, J).

Proof. Let G be a maximal torus in the group of hamiltonian isometries Isom_{0}(M, g), containing the ℓ-torus 𝕋. By Lemma 5, G is the product of a maximal torus in the group of hamiltonian isometries Isom_{0}(Ŝ, gŜ) and the ℓ-torus 𝕋. Denote by 𝔤 ⊂ 𝔦_{0}(M, g) the corresponding Lie algebra. We are going to show that 𝔤^{ℂ} = 𝔤 + J𝔤 is a maximal abelian subalgebra of 𝔥_{0}(M, J).

As in the proof of Lemma 5, we consider natural homomorphism p̂_{∗}: 𝔷(𝕋, J) → 𝔥_{0}(Ŝ) from the centralizer 𝔷(𝕋, J) of 𝕋 in 𝔥_{0}(M, J) to 𝔥_{0}(Ŝ). The proof of Lemma 5 shows that the restriction of p̂_{∗} to 𝔷(𝕋, g) is surjective onto 𝔦_{0}(Ŝ, gŜ).

By assumption, the induced Kähler metric (gŜ, ωŜ) on Ŝ is of constant scalar curvature, so by the Lichnerowicz–Matsushima theorem [49, 55], 𝔥_{0}(Ŝ) is the complexification of 𝔦_{0}(Ŝ, gŜ). It follows that p̂_{∗}: 𝔷(𝕋, J) → 𝔥_{0}(S) is also surjective. As 𝔤 ⊂ 𝔷(𝕋, g) is a maximal abelian subalgebra, its projection to 𝔦_{0}(S, g_{S}) must also be a maximal abelian subalgebra, so is then the image p̂_{∗}(𝔤^{ℂ}) ⊂ 𝔥_{0}(Ŝ) (by using the Lichnerowicz–Matsushima theorem again). It follows that 𝔤^{ℂ} ⊂ 𝔥_{0}(M, J) is maximal abelian iff 𝔤^{ℂ} ∩ 𝔥Ŝ(M̂) is a maximal abelian subalgebra of the complex algebra of fibre-preserving holomorphic vector fields 𝔥Ŝ(M̂). But the fibre V is a toric variety under 𝕋, so 𝔤^{ℂ}∩𝔥Ŝ(M̂) = 𝔱^{ℂ} = 𝔱+J𝔱, which is clearly a maximal abelian subalgebra of 𝔥(V, J_{V}) and hence also of 𝔥Ŝ(M̂). □

2.5. The extremal vector field. For convenience, we will introduce at places a basis of 𝔱 (resp. of 𝔱^{∗}), for example by taking ℓ generators of the lattice Λ (where 𝕋 = 𝔱/2πΛ). This identifies the vector space 𝔱 with ℝ^{ℓ} (and 𝔱^{∗} with (ℝ^{ℓ})^{∗}), and fixes a basis of Poisson commuting hamiltonian Killing fields K_{1}, …, K_{ℓ} in K. Thus, a S^{2}𝔱^{∗}-valued function H on Δ can be seen as an ℓ × ℓ-matrix of functions (H_{rs}) = H on Δ. Similarly, we write z = (z_{1}, …, z_{ℓ}) for the momentum coordinates with respect to K_{1}, …, K_{ℓ}.

An important technical feature of the Kähler metrics given by the generalized Calabi construction (6) is the simple expression of their scalar curvature in terms of the geometry of (V, g_{V}) and (Ŝ, gŜ) (see e.g. [3, p. 380]):

(7)  Scal_{g} = ∑_{j∈I} \frac{Scal_{j}}{⟨p_{j}, z⟩ + c_{j}} − \frac{1}{p(z)} ∑_{r,s=1}^{ℓ} \frac{∂^{2}}{∂z_{r}∂z_{s}} (p(z)H_{rs}),

where p(z) = ∏_{j∈I}(⟨p_{j}, z⟩ + c_{j})^{d_{j}}. This formula generalizes the expression obtained by Abreu [1] in the toric case (when Ŝ is a point).

Another immediate observation is that the volume form Vol_{ω} = ω^{m} is given by

(8)  ω^{m} = p(z)(ωd ; Ŝ ∧ ⟨dz ∧ θ̂⟩^{∧ℓ}) = p(z)(⋀_{j} ω_{j}^{∧d_{j}}) ∧ ⟨dz ∧ θ̂⟩^{∧ℓ},

where ∑_{j∈I} d_{j} = d = m−ℓ. It follows that integrals over M of functions of z (pullbacks from Δ) are given by integrals on Δ with respect to the volume form p(z) dv, where dv is the (constant) euclidean volume form on 𝔱^{∗}, obtained by wedging any generators of the lattice Λ.

We now recall the definition [31] of the extremal vector field of a compact Kähler manifold (M, J, g, ω). Let G be a maximal connected compact subgroup of the reduced automorphism group Ãut_{0}(M, J).^{8} Following [31], the extremal vector field of a Ginvariant Kähler metric (g, J, ω) on M is the Killing vector field whose Killing potential is the L^{2}-projection of the scalar curvature Scal_{g} of g to the space 𝔤_{ω} of all Killing potentials (with respect to g) of elements of the Lie algebra 𝔤. As shown by Futaki and Mabuchi [31], this definition is independent of the choice of a G-invariant Kähler metric within the given Kähler class Ω = [ω] on (M, J). The extremal vector field is necessarily in the centre of 𝔤, so it can also be defined by taking G to be a maximal torus in Ãut_{0}(M, J). This remark is useful for the Kähler metrics in (6), as we have seen in Proposition 1 that they are automatically invariant under such a torus G. In this case, by Lemma 5, 𝔤_{ω} is the direct sum of 𝔱_{ω} (which in turn is identified with the space of affine functions of z) and a subspace of Killing potentials of zero integral of lifts of Killing vector fields on (Ŝ, gŜ). We have shown in the proof of Lemma 5 that the latter potentials are all of the form ∑_{j}(⟨p_{j}, z⟩ + c_{j})f_{j} where f_{j} is a function on Ŝ of zero integral with respect to ωd ; Ŝ. As the scalar curvature of a compatible metric is a function of z only (assuming Scal_{j} are constant—see (7)) it follows from (8) that the L^{2}-projection of Scal_{g} to 𝔤_{ω} lies in 𝔱_{ω}. This shows that the extremal vector field is in 𝔱 and the projection of Scal_{g} orthogonal to the Killing potentials of g takes the form:

Scal_{g}^{⊥} = ⟨A, z⟩ + B + Scal_{g},

where

\begin{cases}∑_{s} α_{s}A_{s} + αB + 2β & = 0, \\ ∑_{s} α_{rs}A_{s} + α_{r}B + 2β_{r} & = 0, \end{cases} (9)

α = ∫_{Δ} p(z)dv,  α_{r} = ∫_{Δ} z_{r}p(z)dv,  α_{rs} = ∫_{Δ} z_{r}z_{s}p(z)dv, with
β = \frac{1}{2} ∫_{Δ} Scal_{g}p(z)dv = ∫_{∂Δ} p(z)dσ + \frac{1}{2} ∫_{Δ}(∑_{j} \frac{Scal_{j}}{⟨p_{j}, z⟩ + c_{j}} )p(z)dv,

β_{r} = \frac{1}{2} ∫_{Δ} Scal_{g}z_{r}p(z)dv = ∫_{∂Δ} z_{r}p(z)dσ + \frac{1}{2} ∫_{Δ}(∑_{j} \frac{Scal_{j}}{⟨p_{j}, z⟩ + c_{j}} )z_{r}p(z)dv.

Here dσ is the (ℓ − 1)-form on ∂Δ with u_{i} ∧ dσ = −dv on the face F_{i} with normal u_{i}. These formulae are immediate once one applies the divergence theorem and the boundary conditions (4) for H, noting that the normals are inward normals, which introduces a sign compared to the usual formulation of the divergence theorem.

The extremal vector field of (M, g, J, ω) is −⟨A, K⟩, where K ∈ C^{∞}(M, T M) ⊗ 𝔱^{∗} is the generator of the 𝕋 action.

8By a well-known result of Calabi [14], any extremal Kähler metric is invariant under such a G.

3. Stability of the extremal equation and Theorems 2–4

3.1. Stability of solutions of the extremal equation under small perturbation. It follows from the considerations in §2.5 that on a given manifold M of the type we consider, finding a compatible extremal Kähler metric (g, ω) of the form (6) reduces to solving the equation (for a unknown symplectic potential U ∈ S(Δ))

(10)  ⟨A, z⟩ + B + ∑_{j∈I} \frac{Scal_{j}}{c_{j} + ⟨p_{j}, z⟩} − \frac{1}{p(z)} ∑_{r,s} \frac{∂^{2}}{∂z_{r}∂z_{s}} (p(z)H_{rs}) = 0,

where

• (H_{rs}) = H = (Hess(U))^{−1};

• (c_{j}, p_{j}, Scal_{j}) are fixed constants;

• p(z) = ∏_{j∈I}(c_{j} + ⟨p_{j}, z⟩)^{d_{j}} is strictly positive on Δ^{0} but vanishes on the blow-down faces F_{b}, b ∈ B;

• A and B are expressed in terms of (c_{j}, p_{j}, Scal_{j}) by (9).

Recall from §2.3 that the real constants c_{a} (a ∈ A) parametrize compatible Kähler classes on a given manifold M. A general result of LeBrun–Simanca [46] affirms that Kähler classes admitting extremal Kähler metric form an open subset of the Kähler cone. We want to obtain a relative version of this result, by showing that compatible Kähler classes which admit a compatible extremal Kähler metric is an open condition on the parameters c_{a}.

We will state and prove our stability result in a slightly more general setting, by considering (10) as a family of differential operators on S(Δ), parametrized by λ ∈ {(c_{a}, p_{a}, Scal_{a}), a ∈ A} (thus λ takes values in a (2 + ℓ)|A|-dimensional euclidean vector space). For any λ such that ⟨p_{a}, z⟩ + c_{a} > 0 on Δ, we consider

(11) P_{λ}(U) = ⟨A_{λ}, z⟩ + B_{λ} + ∑_{j∈I} \frac{Scal_{j}}{c_{j} + ⟨p_{j}, z⟩} − \frac{1}{p_{λ}p_{0}} ∑_{r,s=1}^{ℓ} \frac{∂^{2}}{∂z_{r}∂z_{s}} (p_{λ}p_{0}H_{rs}),

where (H_{rs}) = Hess(U)^{−1}, p_{λ}(z) = ∏_{a∈A}(⟨p_{a}, z⟩ + c_{a})^{d_{a}}, p_{0}(z) = ∏_{b∈B}(⟨u_{b}, z⟩ + c_{b})^{d_{b}}, and A_{λ}, B_{λ} are introduced by (9). The central result of this section is the following one.

Proposition 2. Let (g_{0}, ω_{0}) be a compatible extremal Kähler on M, with symplectic potential U_{0} and parameters λ_{0} = (c_{a}^{0}, p_{a}^{0}, Scal_{a}^{0}), a ∈ A. Then there exists ε > 0 such that for any λ with |λ − λ_{0}| < ε there exists a symplectic potential U_{λ} ∈ S(Δ) such that P_{λ}(U_{λ}) = 0 on Δ^{0}.

The proof of this proposition has several steps and will occupy the rest of this section. It is not immediately clear from (11) that P_{λ} is a well-defined differential operator:

in the presence of blow-downs, the terms \frac{Scal_{b}}{c_{b}+⟨p_{b},z⟩} and \frac{1}{p_{0}(z)} become degenerate on the boundary of Δ.^{9} Of course, for λ = λ_{0} we know from (10) that P_{λ_{0}}(U) = Scal_{g}^{⊥} where g is the compatible metric on M corresponding to U, and Scal_{g}^{⊥} is the L^{2}-projection of the scalar curvature to the space of functions orthogonal to the Killing potentials of g. However, for generic values of λ the data (c_{a}, p_{a}, Scal_{a}) are not longer associated with a compatible Kähler class on a smooth manifold: for this to be true p_{a} and Scal_{a} must satisfy integrality conditions. To overcome this technical difficulty, we are going to rewrite our equation on the smooth compact manifold W. (Note that for b ∈ B, p_{b} = u_{b}, c_{b}, Scal_{b} = 2d_{b}(d_{b} + 1) are fixed in our construction.)

9This does not affect the principal part of P_{λ}, which is concentrated in the scalar curvature Scal_{V} = − ∑_{r,s} \frac{∂^{2}}{∂z_{r}∂z_{s}} H_{rs} of the induced Kähler metric g_{V} on V [1], and is manifestly independent of λ.

Recall from §2.3 that any symplectic potential U ∈ S(Δ) introduces a compatible Kähler metric (g_{W} , ω_{W}) on the manifold W obtained by blowing down Ŵ = P′ ×_{𝕋} V. Thus, (W, g_{W} , ω_{W}) itself is obtained by the generalized Calabi construction with S being a point.

By a well-known result of G. W. Schwarz [63], the space C^{∞}(V)^{𝕋} of 𝕋-invariant smooth functions on the toric symplectic manifold (V, ω_{V} , 𝕋) is identified with the space of pullbacks (via the momentum map z) of smooth functions C^{∞}(Δ) on Δ; similarly, the space of smooth 𝕋-invariant functions on W (resp. on M) which are constant on the inverse images of the momentum map z is identified with the space C^{∞}(Δ). We will use implicitly these identification throughout. Occasionally, when we want to emphasize the dependence of this identification on z, we will denote these isomorphisms by S_{z}. With this convention, we have

Lemma 6. Let U ∈ S(Δ) be a symplectic potential of a compatible Kähler metric g_{V} on (V, ω_{V} , 𝕋) and (g_{W} , ω_{W}) be the corresponding compatible Kähler metric on W. Then, for any λ such that ⟨p_{a}, z⟩ + c_{a} > 0 on Δ,

P_{λ}(U) =⟨A_{λ}, z⟩ + B_{λ} + ∑_{a∈A} \frac{Scal_{a}}{c_{a} + ⟨p_{a}, z⟩} + Scal_{W}

− \frac{1}{p_{λ}(z)} ∑_{r,s=1}^{ℓ}((\frac{∂^{2}p_{λ}}{∂z_{r}∂z_{s}} )(z)g_{W} (K_{r}, K_{s}))

+ \frac{2}{p_{λ}(z)} ∑_{r=1}^{ℓ}((\frac{∂p_{λ}}{∂z_{r}} )(z)Δ_{W} z_{r}),

where Scal_{W} and Δ_{W} respectively denote the scalar curvature and the riemannian laplacian of g_{W}, and dz_{r} = −ω_{W} (K_{r}, ·).

Proof. We work on the open dense subset W^{0} = P′ ×_{𝕋} V^{0} where the compatible metric (g_{W} , ω_{W}) takes the explicit form (5). The formula (7) for the scalar curvature of the compatible metric g_{W} then specifies to

Scal_{W} = ∑_{b∈B} \frac{Scal_{b}}{⟨p_{b}, z⟩ + c_{b}} − \frac{1}{p_{0}(z)} ∑_{r,s=1}^{ℓ} \frac{∂^{2}}{∂z_{r}∂z_{s}} (p_{0}(z)H_{rs}).

Still using the explicit form (5) of the Kähler structure, we calculate that for the pullback to W of a smooth function f(z) on Δ

\begin{aligned}dd_{W}^{c} f & = d( ∑_{r,s=1}^{ℓ} \frac{∂f}{∂z_{s}} H_{rs}θ′_{r}) \\ & = ∑_{k,r,s=1}^{ℓ} \frac{∂}{∂z_{k}} (\frac{∂f}{∂z_{s}} H_{rs})dz_{k} ∧ θ′_{r} + ∑_{b∈B}( ∑_{r,s=1}^{ℓ} \frac{∂f}{∂z_{s}} H_{rs}p_{br})ω_{b}, \end{aligned} ; (12)

where the decompositions θ′ = (θ′_{1}, …, θ′_{ℓ}) and p_{b} = (p_{b1}, …, p_{bℓ}) are with respect to the chosen basis of 𝔱 and 𝔱^{∗}. Wedging with ω_{W}, we obtain the following expression for the laplacian

(13)  Δ_{W} f = −\frac{1}{p_{0}(z)} ∑_{r,s=1}^{ℓ} \frac{∂}{∂z_{r}} (p_{0}(z)\frac{∂f}{∂z_{s}} H_{rs}).

Specifying (13) to f = z_{r} and putting the above formulae back in (11) implies the lemma. □

Note that \frac{1}{p_{λ}(z)} and \frac{Scal_{a}}{c_{a}+⟨p_{a},z⟩} pull back to smooth functions on W for λ such that c_{a} + ⟨p_{a}, z⟩ > 0 on Δ, and A_{λ} and B_{λ} are well-defined and depend smoothly on λ (at least for λ close to λ_{0}). Thus, Lemma 6 implies that P_{λ} is a fully non-linear fourth order differential operator which depends smoothly on λ (for λ sufficiently close to λ_{0}). It follows that P_{λ}(U) ∈ C^{∞}(Δ) for any U ∈ S(Δ).

Our problem is formulated in terms of compatible Kähler metrics on V (or, equivalently, on W and M) with respect to a fixed symplectic form ω_{V} (resp. ω_{W} and ω). This introduces the space of symplectic potentials S(Δ) where we have to work with smooth functions on Δ^{0} which have a prescribed boundary behaviour on ∂Δ. Our lack of understanding of the convergence in this space (with respect to suitable Sobolev norms) leads us to make an additional technical step and reformulate our initial problem as an existence result on a suitable subspace of the space M_{Ω}(M)^{G} ≅ {f ∈ C_{0}^{∞}(M)^{G} : ω_{0} + dd^{c}f > 0} of G-invariant Kähler metrics in the Kähler class of (g_{0}, J_{0}, ω_{0}), where C_{0}^{∞}(M)^{G} denotes the space of G-invariant smooth functions on M of zero integral with respect to ω_{0}^{m} (thus M_{Ω}(M)^{G} is viewed as an open set in C_{0}^{∞}(M)^{G} with respect to || · ||_{C^{2}}). Once this interpretation is achieved, we will apply the implicit function theorem along the lines of the proof of Lemma 4.

First of all, note that the Frechét space C^{∞}(Δ) pulls back via z to a closed subspace in C^{∞}(V)^{𝕋}, C^{∞}(W)^{T} and C^{∞}(M)^{G}, where T (resp. G) is a maximal torus in Ãut_{0}(W) (resp. Ãut_{0}(M)) containing 𝕋, as in Proposition 1: this follows easily from the description of the Lie algebras of T and G given in Lemma 5. Furthermore, by (8), the corresponding normalized subspaces of functions with zero integral for the measures p_{λ}(z)p_{0}(z)Vol_{ω_{V}^{0}} , p_{λ}(z)Vol_{ω_{W}^{0}} and Vol_{ω_{0}}, respectively, are identified with the space C_{0}^{∞}(Δ) of smooth functions of zero integral with respect to the volume form dμ_{0} = p_{λ_{0}}(z)p_{0}(z)dv on Δ^{0}: this normalization will be used throughout.

Secondly, to adopt the classical point of view of Kähler metrics within a given Kähler class on a fixed complex manifold, we consider the Fréchet space M_{Ω}(V)^{𝕋} ≅ {f ∈ C_{0}^{∞}(Δ) : ω_{V}^{0} + dd_{V}^{c} f > 0} of 𝕋-invariant Kähler metrics in the Kähler class Ω = [ω_{V}^{0} ], where the complex structure on V (resp. on W and M) is determined (and will be fixed throughout) by the initial compatible metric (g_{0}, ω_{0}); similarly, we introduce the spaces M_{Ω}(W)^{T} and M_{Ω}(M)^{G} of Kähler metrics in the given Kähler class which are invariant under a maximal torus (see Proposition 1). These three spaces are interrelated by the generalized Calabi construction as follows.

Lemma 7. Let ω̃_{V} = ω_{V}^{0} + dd_{V}^{c} f be a Kähler metric in M_{Ω}(V)^{𝕋}. Then ω̃_{W} = ω_{W}^{0} + dd_{W}^{c} f and ω̃ = ω_{0}+dd_{M}^{c}f define Kähler metrics in M_{Ω}(W)^{T} and M_{Ω}(M)^{G} respectively, such that ω̃_{V} , ω̃_{W} and ω̃ are linked by the generalized Calabi construction on M, with respect to the data (Δ, Ŝ, θ̂, ω_{j}) of the initial metric ω_{0}, but with momentum co-ordinate z̃ = z + d_{V}^{c} f(K).

Proof. A direct calculation based on the expressions of dd_{V}^{c} f, dd_{W}^{c} f and dd_{M}^{c}f, see (12); we leave the details to the Reader. □

Lemma 7 allows us to introduce subspaces of compatible Kähler metrics M_{Ω}^{comp}(W) = M_{Ω}(W)^{T} ∩ C_{0}^{∞}(Δ) and M_{Ω}^{comp}(M) = M_{Ω}(M)^{G} ∩ C_{0}^{∞}(Δ) (within a fixed Kähler class Ω) and identify each of them with the space M_{Ω}(V)^{𝕋}. The correspondence which associates to any ω̃_{W} ∈ M_{Ω}^{comp}(W) (resp. ω̃ ∈ M_{Ω}^{comp}(M)) the corresponding symplectic potential Ũ ∈ S(Δ)^{10} allows us to reformulate our existence problem on the space M_{Ω}^{comp}(W) as follows: for any λ sufficiently close to λ_{0} (so that A_{λ}, B_{λ} are well-defined and ⟨p_{a}, z⟩ + c_{a} > 0 on Δ), we consider the family of differential operators on M_{Ω}(W)^{T}

10For a metric ω̃_{V} = ω_{V}^{0} + dd_{V}^{c} f ∈ M^{𝕋}(V) the corresponding symplectic potential Ũ is linked to f by a Legendre transform [2, 34]; this is true fibrewise for metrics in M_{Ω}^{comp}(W) and M_{Ω}^{comp}(M).

Q_{λ}(ω̃_{W}) = \frac{p_{λ}(z̃)}{p_{λ_{0}}(z̃)}[⟨A_{λ}, z̃⟩ + B_{λ} + ∑_{j∈I} \frac{Scal_{j}}{c_{j} + ⟨p_{j}, z̃⟩}

+ S̃cal_{W} − \frac{1}{p_{λ}(z̃)} ∑_{r,s}((\frac{∂^{2}p_{λ}}{∂z_{r}∂z_{s}} )(z̃)g̃_{W} (K_{r}, K_{s}))

+ \frac{2}{p_{λ}(z̃)} ∑_{r}((\frac{∂p_{λ}}{∂z_{r}} )(z̃)Δ̃_{W} (z̃_{r}))],

(14)

where z̃ = z + d_{W}^{c} f(K) is the momentum map of 𝕋 with respect to the Kähler form ω̃_{W} = ω_{W}^{0} + dd_{W}^{c} f of the Kähler metric g̃_{W}, and S̃cal_{W} (resp. Δ̃_{W}) denote the scalar curvature (resp. laplacian) of g̃_{W}. Thus, by Lemmas 6 and 7, any Kähler metric ω̃_{W} ∈ M_{Ω}^{comp}(W) for which Q_{λ}(ω̃_{W}) = 0 gives rise to a symplectic potential Ũ ∈ S(Δ) solving P_{λ}(Ũ) = 0.

The positive factor \frac{p_{λ}(z̃)}{p_{λ_{0}}(z̃)} in front of Q_{λ} is introduced so that for any compatible

metric ω̃_{W} ∈ M_{Ω}^{comp}(W), the function Sz̃(Q_{λ}(ω̃_{W})) is L^{2}-orthogonal with respect to the measure dμ_{0} = p_{λ_{0}}p_{0}dv on Δ to the space of affine functions on 𝔱^{∗}, where, we recall, Sz̃ denotes the identification of 𝕋-invariant smooth functions on W which are constant on the inverse images of z̃ (equivalently of z) with pullbacks via z̃ of smooth functions on Δ. Indeed, by Lemma 6, p_{λ_{0}}(z̃)p_{0}(z̃)Q_{λ}(ω̃_{W}) = P_{λ}(Ũ)p_{λ}(z̃)p_{0}(z̃), so integrating the r.h.s. of (11) by parts and using (4) we get

∫_{Δ} P_{λ}(U)f(z)p_{λ}(z)p_{0}(z)dv = − ∫_{Δ}⟨H, Hess(f)⟩p_{λ}(z)p_{0}(z)dv

+ ∫_{Δ}(⟨A, z⟩ + B + ∑_{j∈I} \frac{Scal_{j}}{c_{j} + ⟨p_{j}, z⟩})f(z)p_{λ}(z)p_{0}(z)dv

+ 2 ∫_{∂Δ} f(z)p_{λ}(z)p_{0}(z)dσ,

which holds for any smooth function f(z). When f is affine, the first term on the r.h.s is clearly zero, while by the definition (9) of A_{λ} and B_{λ} the sum of the two other terms is zero too; our claim then follows by Lemma 7 and the expression (8) for the volume form of the compatible metric ω̃_{W}.

Let Π_{0} denote the orthogonal L^{2}-projection of C^{∞}(Δ) to the finite dimensional subspace of affine functions of 𝔱^{∗} with respect to the measure dμ_{0} = p_{λ_{0}}p_{0}dv on Δ, and C_{⊥}^{∞}(Δ) be the kernel of Π_{0}. We then consider the map Ψ: U → ℝ^{(2+ℓ)|A|} × C_{⊥}^{∞}(Δ), defined in a small neighbourhood U of (λ_{0}, 0) ∈ ℝ^{(2+ℓ)|A|} × C_{⊥}^{∞}(Δ) by

Ψ(λ, f) = (λ, (Id − Π_{0})(S_{z}(Q_{λ}(ω̃_{W}))),

where ω̃_{W} = ω_{W}^{0} + dd_{W}^{c} f is a compatible metric on M_{Ω}^{comp}(W). Note that if f has sufficiently small C^{1}-norm, the equation (Id − Π_{0}) ∘ (S_{z}(Q_{λ}(ω̃_{W})) = 0 is satisfied if and only if Q_{λ}(ω̃_{W}) = 0: this follows from the fact that Π_{0} ∘ Sz̃ ∘ S_{z}^{−1} defines a continuous family of linear endomorphisms of the finite dimensional space of affine functions on 𝔱^{∗}, with the identity corresponding to ω̃_{W} = ω_{W}^{0} ; thus Π_{0} ∘ Sz̃ ∘ S_{z}^{−1} ∘ Π_{0} is invertible for ω̃_{W} close to ω_{W}^{0} , and hence (by using that Π_{0}(Sz̃(Q(ω̃_{W})) = 0) we get

Π_{0} ∘ Sz̃ ∘ S_{z}^{−1} ∘ (Id − Π_{0})(S_{z}(Q_{λ}(ω̃_{W})) = −Π_{0} ∘ Sz̃ ∘ S_{z}^{−1} ∘ Π_{0}(S_{z}(Q_{λ}(ω̃_{W})))

which is zero iff S_{z}(Q_{λ}(ω̃_{W}) = 0 i.e., Q_{λ}(ω̃_{W}) = 0.

By the discussion above, we are in position to complete the proof of Proposition 2 by applying the inverse function theorem to the extension of Ψ to suitable Sobolev spaces, together with elliptic regularity (as in [46], see also the proof of Lemma 4) in order to find a family ω̃_{W}^{λ} = ω_{W}^{0} + dd_{W}^{c} f_{λ} of smooth compatible metrics satisfying Ψ(λ, f_{λ}) = (λ, 0) for |λ − λ_{0}| < ε.

Let us first introduce the functional spaces we will work on. Recall that C^{∞}(Δ) is seen as a (closed) Fréchet subspace of the space of T -invariant smooth functions on W (resp. G-invariant smooth functions on M) which are constant on the inverse images of the momentum map z for the sub-torus 𝕋. It follows from the description of the Lie algebra of T (resp. G) given in Lemma 5 that C_{⊥}^{∞}(Δ) is precisely the intersection of C^{∞}(Δ) with the space C_{⊥}^{∞}(W)^{T} of T -invariant smooth functions on W which are L^{2}-orthogonal with respect to p_{λ_{0}}Vol_{ω_{W}^{0}} to Killing potentials of g_{W}^{0} (resp. the space C_{⊥}^{∞}(M)^{G} of G-invariant smooth functions on M which are L^{2}-orthogonal with respect to Vol_{ω_{0}} to Killing potentials of g_{0}). We let L_{⊥}^{2,k}(W, Δ) (resp. L_{⊥}^{2,k}(M, Δ)) be the closure of C_{⊥}^{∞}(Δ) with respect to the Sobolev norm || · ||_{2}^{k} on W for the measure p_{λ_{0}}(z)Vol_{ω_{W}^{0}} and riemannian metric g_{W}^{0} (resp. the Sobolev norm || · ||_{2}^{k} on M with respect to Vol_{ω_{0}} and g_{0}). For k ≫ 1, the Sobolev embedding L_{⊥}^{2,k+4}(W, Δ) ⊂ C_{⊥}^{3}(Δ) allows us to extend the differential operator Ψ to a C^{1}-map from a neighbourhood of (λ_{0}, 0) ∈ ℝ^{(2+ℓ)|A|} × L_{⊥}^{2,k+4}(W, Δ) into L_{⊥}^{2,k}(W, Δ), such that Ψ(λ_{0}, 0) = 0; furthermore, as the principal part of Q_{λ} is concentrated in the term S̃cal_{W}, one can see that Ψ is a fourth order quasi-elliptic operator [46].

The following lemma now suffices to apply the inverse function theorem.

Lemma 8. Let T_{0} : C_{⊥}^{∞}(Δ) → C_{⊥}^{∞}(Δ) be the linearization at ω_{W}^{0} ∈ M_{Ω}^{comp}(W) of Q_{λ_{0}}. Then T_{0} is an isomorphism of Fréchet spaces.

Proof. Let (g_{0}, J_{0}, ω_{0}) be the compatible extremal Kähler metric on M corresponding to the initial value λ = λ_{0}. For any function f ∈ C_{⊥}^{∞}(Δ) we consider the compatible Kähler metric g̃ on M, with Kähler form ω̃ = ω_{0} + dd_{M}^{c}f and the compatible Kähler metric g̃_{W} on W with Kähler form ω̃_{W} = ω_{W}^{0} + dd_{W}^{c} f. We saw already in §2.5 that for λ = λ_{0}, Q_{λ_{0}}(ω̃_{W}) = P_{λ_{0}}(Ũ) = Scal ⊥ ; g̃, where Ũ and Scal ⊥ ; g̃ are the symplectic potential and normalized scalar curvature of g̃. It then follows from [32, 45] that the linearization T_{0} of Q_{λ_{0}} (at ω_{W}^{0} ) is equal to −2 times the Lichnerowicz operator L of (g_{0}, ω_{0}) acting on the space of pullbacks (via z) of functions in C_{⊥}^{∞}(Δ). We have already observed in the proof of Lemma 4 that L is an isomorphism when restricted to the space C_{⊥}^{∞}(M)^{G} of G-invariant smooth functions L^{2}-orthogonal to Killing potentials of g_{0}. The main point here is to refine this by showing that L is an isomorphism when restricted to subspace C_{⊥}^{∞}(Δ), the only missing piece being the surjectivity.

Suppose for a contradiction that L: C_{⊥}^{∞}(Δ) → C_{⊥}^{∞}(Δ) is not surjective. Considering the extension of L to an operator between the Sobolev spaces L_{⊥}^{2,4}(M, Δ) → L_{⊥}^{2}(M, Δ) (by elliptic theory L is a closed operator), our assumption is then equivalent to the existence of a non-zero function u ∈ L_{⊥}^{2}(M, Δ) such that, for any ϕ ∈ C_{⊥}^{∞}(Δ), L(ϕ) is L^{2} orthogonal to u. As any sequence of functions converging in L^{2}(M) has a point-wise converging subsequence, u = u(z) is (the pullback to M of) a L^{2}-function on Δ, and using (8) we have

(15)  ∫_{M} L(ϕ)uω_{0}^{m} = ∫_{Δ^{0}} L(ϕ)u(z)p(z)dv = 0.

We claim that (15) implies

(16)  ∫_{M} L(f)u ω_{0}^{m} = 0

for any f ∈ C_{⊥}^{∞}(M)^{G}. This would be a contradiction because L extends to an isomorphism between the closures L_{⊥}^{2,4}(M)^{G} and L_{⊥}^{2}(M)^{G} of C_{⊥}^{∞}(M)^{G} in the corresponding Sobolev spaces on M.

It is enough to establish (16) by integrating on M^{0} = z^{−1}(Δ^{0}) (which is the complement of the union of submanifolds of real codimension at least 2).

The Lichnerowicz operator L has the following general equivalent expression [32, 45]

(17)  L(f) = \frac{1}{2}Δ_{g_{0}}^{2}f + g_{0}(dd^{c}f, ρ_{g_{0}}) + \frac{1}{2}g_{0}(df, dScal_{g_{0}}),

where ρ_{g_{0}} is the Ricci form of (g_{0}, J_{0}) and Δ_{g_{0}} is its laplacian. We will use the specific form (6) of g_{0} to express the r.h.s of the above equality in terms of the geometry of (V, g_{V}^{0} ) and (Ŝ, gŜ).

Any G-invariant (and hence 𝕋-invariant) smooth function f on M can be written as a function f_{s}(z) of the momentum map z depending smoothly on s ∈ Ŝ (as the pullback of f to M̂ is smooth, f_{s} is a smooth function on Δ, not only on Δ^{0}). Similarly, for any z ∈ Δ^{0}, f_{z}(s) = f(z, s) will denote the corresponding smooth function on Ŝ.

Using [4, Prop. 7] and the specific form (6) of g_{0}, it is straightforward to check that on M^{0} we have

dd^{c}f = ∑_{k,r,t=1}^{ℓ} \frac{∂}{∂z_{k}} (\frac{∂f}{∂z_{t}} H_{rt})dz_{k} ∧ θ̂_{r} + ∑_{j∈I}( ∑_{r,t=1}^{ℓ} \frac{∂f}{∂z_{t}} H_{rt}p_{jr})ω_{j}

+ ∑_{r=1}^{ℓ}(dŜ(∑_{s=1}^{ℓ} \frac{∂f}{∂z_{t}} H_{rt}) ∧ θ̂_{r} + dc ; Ŝ(∑_{t=1}^{ℓ} \frac{∂f}{∂z_{t}} H_{rt}) ∧ Jθ̂_{r})

+ dŜdc ; Ŝf_{z}

Δ_{g_{0}}f =Δ_{Ŝ,z}f_{z} + Δ_{g_{0}}f_{s};
ρ_{g_{0}} = ∑_{j∈I} ρ_{j} − ∑_{k,r,t=1}^{ℓ} \frac{∂}{∂z_{k}} (\frac{1}{2p(z)} \frac{∂(p(z)H_{tr})}{∂z_{t}})dz_{k} ∧ θ̂_{r}

− \frac{1}{2p(z)} ∑_{j∈I}( ∑_{r,t=1}^{ℓ} \frac{∂(p(z)H_{rt})}{∂z_{t}}p_{jr})ω_{j},

Scal_{g_{0}} =Scal_{Ŝ,z} − \frac{1}{p(z)} ∑_{r,t=1}^{ℓ} \frac{∂^{2}}{∂z_{r}∂z_{t}} (p(z)H_{rt}),

where

• p(z) = ∏_{j∈I}(⟨p_{j}, z⟩ + c_{j})^{d_{j}};

• θ̂ = (θ̂_{1}, …, θ̂_{ℓ}) and p_{j} = (p_{j1}, …, p_{jℓ}) with respect to the chosen basis of 𝔱;

• dŜ and dc ; Ŝ are the differential and the d^{c}-operator acting on functions and forms on Ŝ;

• (g_{j}, ω_{j}) are the product CSC Kähler factors of the Kähler metric (gŜ, ωŜ), with respective Ricci forms ρ_{j} and laplacians Δ_{g_{j}};

• g_{Ŝ,z} = ∑_{j∈I}(⟨p_{j}, z) + c_{j})g_{j} is the quotient Kähler metric on Ŝ at z, and ω_{Ŝ,z}, Scal_{Ŝ,z} and Δ_{Ŝ,z} denote its Kähler form, scalar curvature and laplacian, respectively; Substituting back in (17), we obtain

\begin{matrix}L(f) = & L(f_{s}) + L_{Ŝ,z}(f_{z}) + Δ_{Ŝ,z}((Δ_{g_{0}}f_{s})_{z}) + Δ_{g_{0}}((Δ_{Ŝ,z}f_{z})_{s}) \\ + ∑_{j∈I} R_{j}(z)Δ_{g_{j}}(f_{z}), \end{matrix}

where L_{Ŝ,z} is the Lichnerowicz operator of g_{Ŝ,z}, and R_{j}(z) are coefficients (that can be found explicitly from the above formulae) depending only on z, and such that p(z)R_{j}(z) are smooth on Δ.

If we integrate the above expression for L(f) against u(z) (by using (8)) we get that ∫_{M} L(f)u ω_{0}^{m} is a non-zero constant multiple of

∫Ŝ(∫_{Δ^{0}} L(f_{s})u(z)p(z)dv)ωd ; Ŝ + ∫Ŝ(∫_{Δ^{0}} Δ_{g_{0}}((Δ_{Ŝ,z}f_{z})_{s})u(z)p(z)dv)ωd ; Ŝ

+ ∫_{Δ^{0}}(∫Ŝ L_{Ŝ,z}(f_{z})ω_{Ŝ,z}^{d})u(z)dv + ∫_{Δ^{0}}(∫Ŝ Δ_{Ŝ,z}((Δ_{g_{0}}f_{s})_{z})ω_{Ŝ,z}^{d})u(z)dv

+ ∑_{j∈I} ∫_{Δ^{0}}(∫Ŝ Δ_{g_{j}}(f_{z}) ωd ; Ŝ)p(z)R_{j}(z)u(z)dv.

To see that all the terms vanish, note that the first term is zero by (15); the third and fourth terms are zero because L_{Ŝ,z} and Δ_{Ŝ,z} are self-adjoint (with respect to ω_{Ŝ,z}) and therefore their images are L^{2}-orthogonal to constants on Ŝ. The fifth term is also zero because Δ_{g_{j}}(f) is L^{2}-orthogonal to constants on Ŝ with respect to ωŜ: this follows easily from the local product structure of gŜ. For the second term one uses that Δ_{g_{0}} defines a self-adjoint operator on C^{∞}(Δ) with respect to the measure p(z)dv: thus, for any smooth function ϕ(z) on Δ,

∫Ŝ(∫_{Δ^{0}} Δ_{g_{0}}((Δ_{Ŝ,z}f_{z})_{s})ϕ(z)p(z)dv)ωd ; Ŝ = ∫_{Δ^{0}}(∫Ŝ Δ_{Ŝ,z}(f_{z})ω_{Ŝ,z}^{d})(Δ_{g_{0}}ϕ)dv = 0

because Δ_{Ŝ,z}f_{z} is L^{2}-orthogonal to constants on Ŝ; as u is in the closure in L^{2} of pullbacks of smooth functions on Δ, the second term vanishes too.

This concludes the proof of the lemma. □

An immediate consequence of Proposition 2 is the following

Corollary 1. The existence of a compatible extremal Kähler metric is an open condition on the set of admissible Kähler classes on M.

Proof. As we have already observed, the admissible Kähler classes are parametrized by the real constants c_{a} for a ∈ A. We thus apply Proposition 2 by taking λ = (c_{a}, p_{a}^{0}, Scal_{a}^{0}). □

3.2. Proof of Theorem 3. To deduce Theorem 3 from Proposition 2, we observe that the differential operators (11) satisfy P_{tλ} = P_{λ} for any real number t ≠ 0.

On any Kähler manifold (M, g, ω) obtained by the generalized Calabi construction with data λ = (c_{a}, p_{a}, Scal_{a}), we can consider the sequence of differential operators P_{λ_{k}} where λ_{k} = (c_{a} + k, p_{a}, Scal_{a}). The differential operator P_{λ_{k}} is the same as P_{λ_{k}/k} and \frac{1}{k}λ_{k} converges when k → ∞ to the data corresponding to the extremal Kähler metric equation for a compatible Kähler metrics on W. Theorem 3 follows at once.

Remark 6. As any invariant Kähler metric on a toric manifold is compatible, Theorem 3 implies the existence of (compatible) extremal metrics on a rigid semisimple toric bundles M over a CSC locally product Kähler manifold, in the case when there are no blowdowns and W = V is a toric extremal Kähler manifold.

Remark 7. An interesting class of rigid toric bundles comes from the multiplicity-free manifolds recently discussed in [25]. A typical example is obtained by taking a compact connected semisimple Lie group G and a maximal torus 𝕋 ⊂ G with Lie algebra 𝔱; if we pick a positive Weyl chamber 𝔱_{+} ⊂ 𝔱 (and identify 𝔱 with its dual space 𝔱^{∗} via the Killing form), for any Delzant polytope Δ contained in the interior of 𝔱_{+}, one can consider the manifold M = G×_{𝕋}V →^{p} S = G/𝕋, where V is the toric manifold with Delzant polytope Δ. Note that G is a principal 𝕋-bundle over the flag manifold S = G/𝕋 with a connection 1-form θ ∈ Ω^{1}(G, 𝔱) whose curvature ω(z) = ⟨dθ, z⟩ defines a family of symplectic forms on S (the Kirillov–Kostant–Souriau forms); identifying S ≅ G^{c}/B, where B is a Borel subgroup of the complexification G^{c} of G, each ω(z) defines a homogeneous Kähler metric g(z) on the complex manifold S (which is therefore of constant scalar curvature); the Ricci form ω_{S} of ω(z) is independent of z, giving rise to the normal (Kähler–Einstein) metric g_{S} on S. Now, for any toric Kähler metric on V, corresponding to a symplectic potential U ∈ S(Δ), one considers the Kähler metric on M

g = p^{∗}(g(z) + kg_{S}) + ⟨dz, G, dz⟩ + ⟨θ, H, θ⟩, ω = p^{∗}(ω(z) + kω_{S}) + ⟨dz ∧ dθ⟩,

where G = Hess(U), H = G^{−1}, z ∈ Δ and k > 0. In this case, G → S = G/𝕋 is not necessarily a diagonalizable principal 𝕋-bundle over S = G/𝕋 (in other words, M = G ×_{𝕋} V → S = G/𝕋 is a rigid but not in general semisimple toric bundle). However, most parts of the discussion in Sect. 2 do extend to this case too (see also [3]), with some obvious modifications. The key points are that (a) the volume form of g(z) + kg_{S} is a multiple p(z) (depending only on z) of Vol_{g_{S}}: this allows to extend the curvature computations (see [3, Prop. 7]) and formula (8) to this case, (b) for any z ∈ Δ, g(z)+kg_{S} is a CSC Kähler metric on S: this allows to extend the results in §2.4, and (c) there is a similar formula to (7) for the scalar curvature of g, found by Raza [59], which allows to reduce the extremal equation for the Kähler metrics in the above form to (10) with p_{a} being essentially the positive roots of G, c_{a} = k and Scal_{a} positive constants. Proposition 2 and its corollaries (Corollary 1 and Theorem 3) extend to this setting too. We thus get both openness and existence of extremal Kähler metrics of the above form when V is an extremal toric Kähler variety and k ≫ 0.

3.3. Proof of Theorem 4. Theorem 4 (from the introduction) may be derived as the special case of Theorem 3, in which V = ℂP^{ℓ} and W = ℂP^{r}, r ≥ ℓ ≥ 1 and M = P (E_{0} ⊕ ⋯ ⊕ E_{ℓ}) → S (see §2.2). It follows from the general theory of hamiltonian 2-forms [3, 4] that any Fubini–Study metric on ℂP^{r} admits a rigid semisimple isometric action of an ℓ-dimensional torus 𝕋, for any 1 ≤ ℓ ≤ r (see in particular [3, Prop. 17] and [4, Thm. 5]): thus, W = ℂP^{r} admits a compatible extremal Kähler metric.

Let ω be a compatible Kähler on M; as the fibre is ℂP^{r}, by re-scaling, we can assume without loss of generality that [ω] = 2πc_{1}(O(1)_{E}) + p^{∗}α, where α is a cohomology class on S. The form (6) of ω and the assumption on the first Chern classes c_{1}(E_{i}) imply that α is diagonal with respect to the product structure of S, in the sense that it pulls back to the covering product space as α = ∑_{a∈A} q_{a}[ω_{a}] for some real constants q_{a}. Therefore, Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{S}] = [ω] + ∑_{a∈A}(k − q_{a})p^{∗}[ω_{a}]. If we choose q with q > q_{a}, then ω̃ = ω + ∑_{a∈A}(q − q_{a})p^{∗}ω_{a} is clearly a compatible Kähler metric too. Thus, Ω_{k} = [ω̃] + (k − q)p^{∗}[ω_{S}] with [ω̃] compatible, and we conclude using Theorem 3.

3.4. Proof of Theorem 2. The sufficiency of the condition for the existence of an extremal Kähler metric follows from Theorem 4.

For the necessity, suppose (g, ω) is an extremal Kähler metric in Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{Σ}] on (M, J) = P (E_{0} ⊕ … ⊕ E_{ℓ}) → Σ, where E_{i} are indecomposable holomorphic vector bundles over a compact curve Σ of genus g ≥ 2. We can assume without loss of generality that ω_{Σ} is the Kähler form of a constant curvature metric on Σ and, by virtue of Theorem 1, that the scalar curvature of g is not constant. In particular, ℓ ≥ 1.

We have seen in Lemma 1 that the ℓ-dimensional torus 𝕋 acting by scalar multiplication on each E_{i} is maximal in the reduced automorphism group Ãut_{0}(M, J) ≅ H^{0}(Σ, P GL(E)). By a well-known result of Calabi [14] the identity component of the group of Kähler isometries of an extremal Kähler metric is a maximal compact subgroup of Aut_{0}(M, J), so we can assume without loss that (g, ω) is 𝕋-invariant.

By considering small stable deformations E_{i}(t) and applying Lemma 4, we can find a smooth family of extremal 𝕋-invariant Kähler metrics (J_{t}, g_{t}, ω_{t}), converging to (J, ω) in any C^{k}(M), such that (M, J_{t}) ≅ P (⨁_{i=1}^{ℓ} E_{i}(t)), and [ω_{t}] = [ω] in H_{dR}^{2}(M). By the equivariant Moser lemma, we can assume without loss of generality that ω_{t} = ω.

It is not difficult to see that any Kähler class on (M, J_{t}) (for t ≠ 0) is compatible: this follows from the fact that the cohomology H^{2}(M) ≅ H^{1,1}(M, J_{t}) is generated by any compatible Kähler class on (M, J_{t}) and the pullback p^{∗}[ω_{Σ}]. By Theorem 4 and the uniqueness of the extremal Kähler metrics up to automorphisms [15], for any t ≠ 0 we can take k ≫ 0 such that the extremal Kähler metric (g_{t}, ω) on (M, J_{t}) is compatible with respect to the rigid semisimple action of the maximal torus 𝕋. Strictly speaking, Theorem 4 produces a lower bound k_{0} for such k, depending on J_{t}. However, in our case |A| = 1, the simplex Δ, the moment map z and the metric on Σ are fixed, and the parameter λ = (c, p, Scal_{Σ}) defining the corresponding extremal equation (10) for a compatible metric on (M, J_{t}, [ω]) is independent of t: indeed, the constants p ∈ 𝔱 and c ∈ ℝ are determined by the first Chern classes c_{1}(E_{i}) and the cohomology class Ω_{k} = [ω] ∈ H_{dR}^{2}(M). Thus, the deformation argument used in §3.2 produces a lower bound k_{0} independent of t, such that for any k > k_{0} and t ≠ 0, (g_{t}, ω) is an extremal Kähler metric in Ω_{k} with respect to which the maximal torus 𝕋 acts in a rigid and semisimple way.

Take a regular value z_{0} of the momentum map z associated to the hamiltonian action of 𝕋 on (M, ω) and consider the family of Kähler quotient metrics (ĝ_{t}, Ĵ_{t}) on the symplectic quotient Ŝ. By identifying the symplectic quotient with the stable quotient, we see that (Ŝ, Ĵ_{t}) ≅ P (E_{0}(t)) ×_{Σ} ⋯ ×_{Σ} P (E_{ℓ}(t)) → Σ (see §2.2). As for t ≠ 0 the action of 𝕋 is rigid and semisimple and g_{t} is compatible, the quotient Kähler metric (ĝ_{t}, Ĵ_{t}) must be locally a product of CSC Kähler metrics. By the de Rham decomposition theorem ĝ_{t} must be a locally-symmetric metric modelled on the hermitian-symmetric space ℂP^{d_{0}} × ⋯ × ℂP^{d_{ℓ}} × ℍ, where d_{i} + 1 = rk(E_{i}) (so that ℂP^{d_{i}} is a point if d_{i} = 1) and ℍ is the hyperbolic plane. By continuity, (ĝ_{0}, Ĵ_{0}) is a locally-symmetric Kähler metric on Ŝ of the same type. By the de Rham decomposition theorem and considering the form of the covering transformations we obtain representations ρ_{i}: π_{1}(Σ) → P U(d_{i} + 1), and therefore E_{i} must be stable by the standard theory [56].

Finally, consider the case when ℓ = 1, i.e., (M, J) = P (E_{0} ⊕ E_{1}). If E_{0} and E_{1} have equal slopes, any extremal Kähler metric on (M, J) must be CSC by Lemma 3 and our claim follows from Theorem 1.

If the slopes of E_{0} and E_{1} are different and (g, ω) is some extremal Kähler metric on (M, J) = P (E_{0} ⊕ E_{1}), Lemma 4 shows that ω is the smooth limit of extremal Kähler metrics ω_{t} on (M, J_{t}) = P (E_{0}(t) ⊕ E_{1}(t)) where E_{i}(t) are stable for t ≠ 0. Now, [5, Thm. 1] affirms that for any t ≠ 0, ω_{t} is compatible with respect to the natural S^{1}-action, so we conclude as above that E_{0} and E_{1} must be stable. □

4. Further observations

4.1. Relative K-energy and the main conjecture. Leaving aside the specific motivation of this paper to study projective bundles over a curve, the theory of rigid semisimple toric bundles which we reviewed in Sect. 2 extends the theory of extremal Kähler metrics on toric manifolds [21, 22, 24, 67, 76, 77] to this more general context.

To recast the leading conjectures [21, 67] in the toric case to this setting, recall from [21] that if we parametrize compatible Kähler metrics g by their symplectic potentials U ∈ S(Δ), then the relative (Mabuchi–Guan–Simanca) K-energy E^{Ω} on this space satisfies the functional equation

(dE^{Ω})_{g}(U̇) = ∫_{Δ}(Scal_{g}^{⊥})U̇(z)p(z)dv

= ∫_{Δ}((⟨A, z⟩ + B + ∑_{j∈I} \frac{Scal_{j}}{⟨p_{j}, z⟩ + c_{j}} )p(z) − \frac{∂^{2}}{∂z_{r}∂z_{s}} (p(z)H_{rs}))U̇(z)dv

= 2 ∫_{∂Δ} U̇(z)p(z)dσ + ∫_{Δ}(⟨A, z⟩ + B + ∑_{j∈I} \frac{Scal_{j}}{⟨p_{j}, z⟩ + c_{j}} )U̇(z)p(z)dv

− ∫_{Δ}⟨H, Hess U̇(z)⟩p(z)dv,

where we have used (10) and integration by parts by taking into account (4). Following [21, 67, 76], let us introduce the linear functional

(18) F^{Ω}(f) := ∫_{∂Δ} f(z)p(z)dσ + \frac{1}{2} ∫_{Δ}(⟨A, z⟩ + B + ∑_{j} \frac{Scal_{j}}{⟨p_{j}, z⟩ + c_{j}} )f(z)p(z)dv.

The above calculation of dE_{g}^{Ω} shows that F^{Ω}(f) = 0 if f is an affine function of z. Furthermore, using the fact that the derivative of log det H is tr H^{−1}dH, we obtain the following generalization of Donaldson’s formula for E^{Ω}:

(19)  E^{Ω}(U) = 2F^{Ω}(U) − ∫_{Δ}(log det Hess U(z))p(z)dv.

(In case of doubt about the convergence of the integrals, one can introduce a reference potential U_{c} and a relative version E_{g_{c}}^{Ω} of E^{Ω}, but in fact, as Donaldson shows, the convexity of U ensures that the positive part of log det Hess U(z) is integrable, hence − log det Hess U(z) has a well defined integral in (−∞, ∞].)

According to [21, 67], the existence of a solution U ∈ S(Δ) to (10) should be entirely governed by properties of the linear functional (18):

Conjecture 2. Let Ω be a compatible class on M. Then the following conditions should be equivalent:

(1) Ω admits an extremal Kähler metric.

(2) Ω admits a compatible extremal Kähler metric (i.e., (10) has a solution in S(Δ)).

(3) F^{Ω}(f) ≥ 0 for any piecewise linear convex function f on Δ, and is equal to zero if and only if f is affine.^{11}

Of course, by the proof of Theorem 2, Conjecture 2 would imply Conjecture 1.

Our formula (19) can be used to show as in [21, Prop. 7.1.3] that F^{Ω}(f) ≥ 0 if the relative K-energy is bounded from below. However, according to Chen–Tian [15], the boundedness from below of E^{Ω} is a necessary condition for the existence of an extremal Kähler metric.

11Generalizing computations in [21, 67, 76], one can show that the value of F^{Ω} at a rational piecewise linear convex function computes the relative Futaki invariant introduced in [67] of a ‘compatible’ toric test configuration on (M, Ω); in general, one might need positivity of F^{Ω} on a larger space of convex functions [21, Conjecture 7.2.2.] in order to solve (10) but in the case when ℓ = 2 and the base Ŝ is a point Donaldson shows in [21] that the space of piecewise linear convex functions will do.

If Ω admits a compatible extremal Kähler metric with symplectic potential U and inverse hessian H, one can use (10) and integration by parts (taking into account (4)) in order to re-write (18) as

(20)  F^{Ω}(f) = ∫_{Δ}⟨H, Hessf⟩p(z)dv.

This formula makes sense for smooth functions f(z), but can also be used to calculate F^{Ω}(f) in distributional sense for any piecewise linear convex function as in [77]: using the fact that H is positive definite, we obtain the analogue of a result in [77], showing that the second statement of Conjecture 2 implies the third.

We thus have the following partial result.

Proposition 3. If Ω admits an extremal Kähler metric then F^{Ω}(f) ≥ 0 for any convex piecewise linear function. If Ω admits a compatible extremal metric then, furthermore, F^{Ω}(f) = 0 if and only if f is an affine function on Δ.

Of course, the most difficult part of Conjecture 2 is to prove (3) ⇒ (2). So far the Conjecture 2 has been fully established in the cases when ℓ = 1 [5] and when M is a toric surface (i.e., ℓ = 2 and Ŝ is a point) with vanishing extremal vector field [24].

4.2. Computing F^{Ω}. It is natural to consider (following Donaldson [21]) the space of S^{2}𝔱^{∗}-valued functions H on Δ satisfying just the boundary conditions (4). If such a function satisfies the (underdetermined, linear) equation (10), then formula (20) holds, and it can be used to compute the action of F^{Ω} (in distributional sense) on piecewise linear functions.

Note that if a solution to (10) exists, then so do many because the double divergence is underdetermined.

If a solution H of (10) happens to be positive definite on each face of Δ, i.e., if it verifies the positivity condition in §2.3, then formulae (6) introduce an almost Kähler metric on M (see e.g. [4]) and one can show that (7) computes its hermitian scalar curvature (see Appendix A). Thus, positive definite solutions of (10) correspond to compatible extremal almost Kähler metrics. If such extremal almost Kähler metrics exist, it then follows from (20) (see [77] and Proposition 3 above) that the condition (3) of Conjecture 2 is verified. Thus, the existence of a positive definite solution H of (10) (and verifying the boundary conditions (4)) is conjecturally equivalent to the existence of a compatible extremal Kähler metric (corresponding to another positive definite function H^{Ω} with inverse equal to the hessian of a function U_{Ω}). In fact, following [21], as log det is strictly convex on positive definite matrices, the functional ∫_{Δ}(log det H)p(z)dv is strictly convex on the space of positive definite solutions of (10), and therefore has at most one minimum H^{Ω}. Such a minimum would automatically have its inverse equal to the hessian of a function U_{Ω} (see [21]). Thus, H^{Ω} would then give the extremal Kähler metric in the compatible Kähler class Ω.

Thus motivated, it is natural to wonder if on the manifolds we consider in this paper a (not necessarily positive definite) solution H of (10) exists, thus generalizing the extremal polynomial introduced in [5] on M = P (E_{0}⊕E_{1}) → S (in fact P(z) = p(z)H(z) would be the precise generalization).

4.3. Example: projective plane bundles over a curve. We illustrate the above discussion by explicit calculations on the manifold M = P (O ⊕ L_{1} ⊕ L_{2}) → Σ, where L_{1} and L_{2} are holomorphic line bundles over a compact complex curve Σ of genus g. We put p_{i} = deg(L_{i}) and assume without loss of generality that p_{2} ≥ p_{1} ≥ 0. Note that in the case p_{1} = p_{2} = 0, the vector bundle E = O ⊕ L_{1} ⊕ L_{2} is polystable, and therefore the existence of extremal Kähler metrics is given by Theorem 1. The cases p_{1} = p_{2} > 0 and p_{2} > p_{1} = 0, on the other hand, are solved in [5]. We thus assume furthermore that p_{2} > p_{1} > 0.

To recast our example in the set up of Sect. 2, we take a riemannian metric g_{Σ} of constant scalar curvature 4(1−g) on Σ. To ease the notation, we put C = 4(g−1). Let z_{i} be the momentum map of the natural S^{1}-action by multiplication on L_{i}. Thus, without loss of generality, for a compatible Kähler metric on M, the momentum coordinate z = (z_{1}, z_{2}) takes values in the simplex Δ = {(z_{1}, z_{2}) ∈ ℝ^{2} | z_{1} ≥ 0, z_{2} ≥ 0, 1−z_{1}−z_{2} ≥ 0} (which is the Delzant polytope of the fibre ℂP^{2} viewed as a toric variety).

It is shown in [5, App. A2] that in this case there are no extremal compatible Kähler metrics with a hamiltonian 2-form of order 2 while Theorem 2 does imply existence of compatible extremal Kähler metrics in small Kähler classes. Therefore, we do not have an explicit construction of these extremal Kähler metrics. Instead, we will now attempt to find explicit extremal almost Kähler metrics (see the preceding section and the Appendix below). We thus want to find a smooth matrix function H(z) = (H_{rs}(z)) satisfying the boundary conditions (4) and which solves the linear equation (10). Motivated by the explicit form of such a matrix in the case when a hamiltonian 2-form does exist [3], we look for solutions of a ‘polynomial’ form H_{rs} = \frac{P_{rs}}{(c+p_{1}z_{1}+p_{2}z_{2})}, where P_{rs}(z) are fourth degree polynomials in z_{1} and z_{2}, and the constant c is such that c + p_{1}z_{1} + p_{2}z_{2} > 0 on Δ, i.e., c ∈ (0, ∞): recall that c parametrizes the compatible Kähler classes on M. The boundary conditions are then solved by

P_{11} = 2(c + p_{1}z_{1} + p_{2}z_{2})z_{1}(1 − z_{1})
+ z_{1}^{2}(x_{0}z_{2}^{2} + x_{2}(1 − z_{1} − z_{2})^{2} + 2y_{1}z_{2}(1 − z_{1} − z_{2})),
P_{12} = −2(c + p_{1}z_{1} + p_{2}z_{2})z_{1}z_{2}
+ z_{1}z_{2}(y_{0}(1 − z_{1} − z_{2})^{2} − x_{0}z_{1}z_{2} − (1 − z_{1} − z_{2})(y_{1}z_{1} + y_{2}z_{2})),
P_{22} = 2(c + p_{1}z_{1} + p_{2}z_{2})z_{2}(1 − z_{2})
+ z_{2}^{2}(x_{0}z_{1}^{2} + x_{1}(1 − z_{1} − z_{2})^{2} + 2y_{2}z_{1}(1 − z_{1} − z_{2})),

where x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2} are free parameters. The extremal condition (10) corresponds to the linear equations

\begin{aligned}y_{0} & = −x_{1} − x_{2} + v_{0}, \\ y_{1} & = −x_{0} − x_{2} + v_{1}, \\ y_{2} & = −x_{0} − x_{1} + v_{2}, \end{aligned} ; (21)

with

v_{0} = \frac{−(12c+C+4p_{1}+4p_{2})(5cp_{1}^{2}+p_{1}^{3}+5cp_{1}p_{2}+5p_{1}^{2}p_{2}+5cp_{2}^{2}+5p_{1}p_{2}^{2}+p_{2}^{3}))}{(2(50c^{3}+50c^{2}p_{1}+13cp_{1}^{2}+p_{1}^{3}+50c^{2}p_{2}+37cp_{1}p_{2}+5p_{1}^{2}p_{2}+13cp_{2}^{2}+5p_{1}p_{2}^{2}+p_{2}^{3})}
v_{1} = \frac{−(12c+C+4p_{1}+4p_{2})(15cp_{1}^{2}+3p_{1}^{3}−15cp_{1}p_{2}+3p_{1}^{2}p_{2}+5cp_{2}^{2}−3p_{1}p_{2}^{2}+p_{2}^{3}))}{(2(50c^{3}+50c^{2}p_{1}+13cp_{1}^{2}+p_{1}^{3}+50c^{2}p_{2}+37cp_{1}p_{2}+5p_{1}^{2}p_{2}+13cp_{2}^{2}+5p_{1}p_{2}^{2}+p_{2}^{3})}
v_{2} = \frac{−(12c+C+4p_{1}+4p_{2})(5cp_{1}^{2}+p_{1}^{3}−15cp_{1}p_{2}−3p_{1}^{2}p_{2}+15cp_{2}^{2}+3p_{1}p_{2}^{2}+3p_{2}^{3}))}{(2(50c^{3}+50c^{2}p_{1}+13cp_{1}^{2}+p_{1}^{3}+50c^{2}p_{2}+37cp_{1}p_{2}+5p_{1}^{2}p_{2}+13cp_{2}^{2}+5p_{1}p_{2}^{2}+p_{2}^{3})}.

Thus, given a compatible Kähler class on M, we have a 3-parameter family of smooth ‘polynomial’ solutions H(z) to (10), which verify the boundary conditions (4) on Δ. Now, investigating the integrability condition (that H^{−1} be a hessian of a smooth function on Δ^{0}, see the previous section), we find out that it is equivalent to the following five algebraic equations on the parameters (x_{0}, x_{1}, x_{2}, y_{0}, y_{1}.y_{2})

x_{0} = y_{1} + y_{2},  x_{1} = y_{2} + y_{0},  x_{2} = y_{0} + y_{1}, (22)

\begin{aligned}2(p_{2} − p_{1})y_{0} + 2p_{2}y_{1} − y_{0}y_{1} & = 0 \\ 2(p_{1} − p_{2})y_{0} + 2p_{1}y_{2} − y_{0}y_{2} & = 0. \end{aligned} ; (23)

The problem is over-determined, but there is a unique solution H_{0}^{Ω} satisfying the linear system (22) (additionally to (21)): we compute that this solution is given by

x_{0} = \frac{1}{10}(−2v_{0} + 3v_{1} + 3v_{2}),
x_{1} = \frac{1}{10}(3v_{0} − 2v_{1} + 3v_{2}),
x_{2} = \frac{1}{10}(3v_{0} + 3v_{1} − 2v_{2}),
y_{0} = \frac{1}{10}(4v_{0} − v_{1} − v_{2}),
y_{1} = \frac{1}{10}(−v_{0} + 4v_{1} − v_{2}),
y_{2} = \frac{1}{10}(−v_{0} − v_{1} + 4v_{2}).

Substituting back in (23), one sees that the full integrability conditions can be solved if 12c+C +4p_{1} +4p_{2} = 0 (a constraint that is never satisfied for p_{2} > p_{1} ≥ 1, C = 4(g −1) and c > 0); this observation is consistent with the non-existence result in [5, App. A2].

We now investigate the positivity condition for our distinguished solution H_{0}^{Ω} of (4) and (10). First of all, when c → ∞, the v_{i}’s tend to 0, so H_{0}^{Ω} tends to the matrix associated to a Fubini–Study metric on ℂP^{2}. It follows that H_{0}^{Ω} becomes positivedefinite on each face for sufficiently small Kähler classes, and therefore H_{0}^{Ω} defines an explicit extremal (non-Kähler) almost Kähler metric in Ω (see Appendix A below). This is of course consistent (via Conjecture 2) with the existence of a (non-explicit) extremal Kähler metric in Ω, given by Theorem 2. Furthermore, if g = 0, 1 (i.e., C < 0), a computer assisted verification shows that, in fact, H_{0}^{Ω} is positive definite on each face of Δ for all Kähler classes. We thus obtain the following result.

Proposition 4. Let M = P (E) ^{p} Σ with E = O ⊕ L_{1} ⊕ L_{2}, where L_{1} and L_{2} are holomorphic line bundles of degrees 1 ≤ p_{1} < p_{2} over a compact complex curve Σ of genus g.

If g = 0, 1, then M admits a compatible extremal almost Kähler metric in any Kähler class Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{Σ}]. In particular, for every Kähler class on M the condition (3) of Conjecture 2 is verified.

If g ≥ 2, then the same conclusion holds for the Kähler classes Ω_{k} = 2πc_{1}(O(1)_{E}) + kp^{∗}[ω_{Σ}] with k ≫ 0, i.e., the Kähler classes sufficiently far from the boundary of the Kähler cone.

As speculated in the previous section, the explicit solution H_{0}^{Ω} of (4) and (10) can be used to compute the action of the functional F^{Ω} on piece-wise linear convex functions (by extending formula (20) in a distributional sense, after integrating by parts and using (4)). As a simple illustration of this, let us take a simple crease function f_{a} with crease along the segment S_{a} = {(t, a − t), 0 < t < a} for some a ∈ (0, 1) (thus as a → 0, the crease moves to the lower left corner of the simplex Δ). A normal of the crease is u = (1, 1) and one easily finds that

\begin{aligned}F^{Ω}(f_{a}) & = ∫_{S_{a}} H_{0}^{Ω}(u, u)dσ \\ & = ∫_{0}^{a}((H_{11} + 2H_{12} + H_{22})(t, a − t))(c + p_{1}t + p_{2}(a − t)) dt, \end{aligned} ; (24)

where dσ is the contraction of the euclidian volume dv on ℝ^{2} by u. Note that the integrand (being a rational function of c with a non-vanishing denominator at c = 0), and hence the integral, is continuous near c = 0; for c = 0 the integral equals

\frac{1}{6}(1 − a)a^{3}(−C + 2(p_{1} + p_{2}) + a(C + 4(p_{1} + p_{2}))),

which is clearly negative for a ∈ (0, 1) sufficiently small as long as C = 4(g − 1) > 2(p_{1} + p_{2}). If we take g > 2, such p_{1} and p_{2} do exist. By Proposition 3, this implies a non-existence result of extremal Kähler metrics when p_{1} and p_{2} satisfy the above inequality and c is small enough. (As a special case, for p_{1} = p_{2} we have recast the non-existence part of [5, Thm. 6].)

Proposition 5. Let M be as in Proposition 4, with g > 2 and p_{1}, p_{2} satisfying 2(g−1) > p_{1} + p_{2}. Then the Kähler classes sufficiently close to the boundary of the Kähler cone do not admit any extremal Kähler metric.

Appendix A. Compatible extremal almost Kähler metrics

In this appendix, we calculate the hermitian scalar curvature of a compatible almost Kähler metric and extend the notion of extremal Kähler metrics to the more general almost Kähler case.

Recall that on a general almost Kähler manifold (M^{2m}, g, J, ω), the canonical hermitian connection ∇ is defined by

(25)  ∇_{X}Y = D_{X}Y − \frac{1}{2}J(D_{X}J)(Y),

where D is the Levi–Civita connection of g. Note that

(26)  g((D_{X}J)Y, Z) = \frac{1}{2}g(N(X, Y), JZ)

where N(X, Y) = [JX, JY] − J[JX, Y] − J[X, JY] − [X, Y] is the Nijenhuis tensor of J. The Ricci form, ρ^{∇}, of ∇ represents 2πc_{1}(M, J) and its trace s^{∇} (given by 2mρ^{∇} ∧ ω^{m−1} = s^{∇}ω^{m}) is called hermitian scalar curvature of (g, J, ω).

The hermitian scalar curvature plays an important role in a setting described by Donaldson [19] (see also [32]), in which s^{∇} is identified with the momentum map of the action of the group Ham(M, ω) of hamiltonian symplectomorphisms of a compact symplectic manifold (M, ω) on the (formal) Kähler Fréchet space of ω-compatible almost Kähler metrics AK_{ω}. It immediately follows from this formal picture [26, 47] that the critical points of the functional on AK_{ω}

g ⟼ ∫_{M}(s^{∇})^{2}ω^{m}

are precisely the ω-compatible almost Kähler metrics for which grad_{ω}s^{∇} is a Killing vector field. This provides a natural extension of the notion of an extremal Kähler metric to the more general almost Kähler context.

Definition 5. An almost Kähler metric (g, ω) for which grad_{ω}s^{∇} is a Killing vector field is called extremal.

Now let M be a manifold obtained by the generalized Calabi construction of §2.3. In the notation of this section, for any S^{2}𝔱^{∗}-valued function H on Δ, satisfying the boundary and positivity conditions, formulae (6) introduce a pair (g, ω) of a smooth metric g and a symplectic form ω on M, such that the field of endomorphisms J defined by ω(·, ·) = g(J·, ·) is an almost complex structure, i.e., (g, ω) is an almost Kähler structure on M.^{12} We shall refer to such pairs (g, ω) as compatible almost Kähler metrics on M.

12It is easily seen as in [1] that J is integrable, i.e., (g, ω) defines a Kähler metric, if and only if H^{−1} is the hessian of a smooth function on Δ^{0}.

Lemma 9. The hermitian scalar curvature s^{∇} of a compatible almost Kähler metric corresponding to H = (H_{rs}) is given by

s^{∇} = ∑_{j∈I} \frac{Scal_{j}}{c_{j} + ⟨p_{j}, z⟩} − \frac{1}{p(z)} ∑_{r,s=1}^{ℓ} \frac{∂^{2}}{∂z_{r}∂z_{s}} (p(z)H_{rs}).

Proof. The result is local and we work on the open dense subset M^{0} where the ℓ-torus 𝕋 acts freely. Recall that M^{0} is a principal 𝕋^{c} bundle over Ŝ. Let V be the foliation defined by the 𝕋^{c} fibres and K_{r} = Jgrad_{g}z_{r} be the Killing vector fields generating 𝕋; then T V is spanned by K_{r}, JK_{r} at each point of M^{0} and, by construction,

(27)  L_{K_{r}}J = 0, K_{r}^{♭} = ∑_{s} H_{rs}θ̂_{s}, JK_{r}^{♭} = −dz_{r}.

In order to compute the hermitian Ricci tensor, we take a local non-vanishing holomorphic section ΦŜ of the anti-canonical bundle K−1 ; Ŝ = ∧^{d,0}(Ŝ) of Ŝ (which pulls back to a (d, 0)-form on M) and wedge it with the (ℓ, 0)-form Φ_{V} = (K_{1}^{♭} − \sqrt{−1}JK_{1}^{♭}) ∧ ⋯ ∧ (K_{ℓ}^{♭} − \sqrt{−1}JK_{ℓ}^{♭}). Thus, Φ = ΦŜ ∧ Φ_{V} is a non-vanishing section of K_{M^{0}}^{−1} and the hermitian Ricci form ρ^{∇} is then given by

ρ^{∇} = −dℑm(α),

where ∇Φ = α ⊗ Φ.

Denote by H the g-orthogonal complement of T V; the spaces H and T V then define the decomposition of T M^{0} as the sum of horizontal and vertical spaces and, therefore

(28)  ∇_{X}Φ = (∇_{X}^{H}ΦŜ) ∧ Φ_{V} + ΦŜ ∧ ∇_{X}^{V}Φ_{V},

where ∇_{X}Y = ∇_{X}^{H}Y + ∇_{X}^{V}Y denotes the decomposition into horizontal and vertical parts.

Our first observation is that [3, Prop. 8] generalizes in the non-integrable case in the following sense: The foliation V is totally-geodesic with respect to both the Levi–Civita and hermitian connections. Indeed, with respect to the Levi–Civita connection D we have ⟨D_{K_{r}}K_{s}, X⟩ = ⟨D_{JK_{r}}K_{s}, X⟩ = 0 for any X ∈ H; using [K_{r}, JK_{s}] = 0, our claim reduces to check that ⟨D_{JK_{r}}JK_{s}, X⟩ = 0. We take X be the horizontal lift of a basic vector field and use the Koszul formula

2⟨D_{JK_{r}}JK_{s}, X⟩ = L_{JK_{r}}⟨JK_{s}, X⟩ + L_{JK_{s}}⟨JK_{r}, X⟩ − L_{X}⟨JK_{r}, JK_{s}⟩
+ ⟨[JK_{r}, JK_{s}], X⟩ + ⟨L_{X}JK_{r}, JK_{s}⟩ + ⟨L_{X}JK_{s}, JK_{r}⟩
= ⟨L_{X}JK_{r}, JK_{s}⟩ + ⟨L_{X}JK_{s}, JK_{r}⟩
= (L_{X}g)(JK_{r}, JK_{s}) = (L_{X} ĝ(z))(JK_{r}, JK_{s}) = 0,

where (ĝ = ĝ(z), ω̂ = ω̂(z)) denote the Kähler quotient structure on Ŝ (also identified with the horizontal part of (g, ω)). Considering the hermitian connection ∇, by (25) and (26), our claim reduces to showing that N(K_{r}, X) is horizontal for any X ∈ H; using (26) and the fact that V is totally-geodesic with respect to D, we get ⟨N(K_{r}, X), JU⟩ = 2⟨(D_{U}J)(K_{r}), X⟩ = 0, for any U ∈ T V.

The observation that V is totally geodesic with respect to D shows that formulae (42)–(46) in [3, Prop. 9] hold true in the non-integrable case too, i.e., we have

\begin{aligned}D_{X}Y & = D_{X}^{H}Y − C(X, Y) \\ D_{X}U & = ⟨C(X, ·), U⟩ + [X, U]^{V} \\ D_{U}X & = [U, X]^{H} + ⟨C(X, ·), U⟩ \\ D_{U}V & = D_{U}^{V}V, \end{aligned} ; (29)

where X, Y ∈ H, U, V ∈ T V and C(·, ·) is the O’Neill tensor given by

2C(X, Y) = ∑_{r=1}^{ℓ}(Ω_{r}(X, Y)K_{r} + Ω_{r}(JX, Y)JK_{r})

with Ω_{r} = dθ̂_{r} = ∑_{j∈I} p_{jr} ⊗ ω_{j}. Using (25), it follows that the horizontal lift of ∇Ŝ coincides with the projections of both D and ∇ to horizontal vectors. In particular, for any horizontal lift X, ∇_{X}^{H}ΦŜ = \frac{1}{2}((dŜ − \sqrt{−1}dc ; Ŝ) log ||ΦŜ||2 ; ĝ)(X)ΦŜ. On the other hand,

as K_{r} are Killing and V is totally geodesic, ∇_{X}^{V}Φ_{V} = 0, so that we get from (28)

(30)  α(X) = \frac{1}{2}((dŜ − \sqrt{−1}dc ; Ŝ) log ||ΦŜ||2 ; ĝ)(X), ∀X ∈ H.

To compute α(U) for U ∈ T V, consider first ∇_{U}Φ_{V}. As V is totally geodesic, we can write ∇_{U}Φ_{V} = (a(U) − \sqrt{−1}b(U))Φ_{V}. It follows from the very definition of Φ_{V} (and the fact that span(K_{1}, ⋯, K_{ℓ}) is ω-Lagrangian) that Φ_{V}(K_{1}, K_{2}, ⋯, K_{ℓ}) = det g(K_{r}, K_{s}) = det H, and therefore

(∇_{U}Φ_{V})(K_{1}, K_{2}, ⋯, K_{ℓ}) = (a(U) − \sqrt{−1}b(U)) det H.

Using the definition of Φ_{V} again, we obtain

b(U) = trace(H^{−1} ∘ A_{U}), (A_{U})_{rs} = −⟨∇_{U}K_{r}, JK_{s}⟩.

Using that K_{r} is Killing, (26) and (27) we further calculate

(A_{U})_{rs} = −⟨D_{U}K_{r}, JK_{s}⟩ + \frac{1}{2}⟨(D_{U}J)(K_{r}), K_{s}⟩

= \frac{1}{2}(dK_{r}^{♭}(JK_{s}, U) − \frac{1}{2}ω(N(K_{r}, K_{s}), U))

= \frac{1}{2}(∑_{p,k} H_{rk,p}dz_{p}(JK_{s})θ̂_{k}(U) − \frac{1}{2} ∑_{k} dz_{k}([JK_{r}, JK_{s}])θ̂_{k}(U))

= \frac{1}{2}(− ∑_{k,p} H_{rk,p}H_{ps}θ̂_{k}(U) − \frac{1}{2} ∑_{k} dz_{k}([JK_{r}, JK_{s}])θ̂_{k}(U))

= −\frac{1}{4} ∑_{k,p}(H_{rk,p}H_{ps} + H_{sk,p}H_{pr})t̂heta_{k}(U),

so that

(31)  b(U) = ∑_{r,s} H^{rs}(A_{U})_{rs} = −\frac{1}{2} ∑_{r,k} H_{rk,r}θ̂_{k}(U).

Finally, in order to compute ∇_{U}ΦŜ, note that L_{U}ΦŜ = 0, and therefore

\begin{aligned}(∇_{U}ΦŜ)(X_{1}, ⋯, X_{d}) & = ∑_{k=1}^{d}(\frac{1}{2}ΦŜ(X_{1}, ⋯, X_{k−1}, J(D_{U}J)(X_{k}), X_{k+1}, ⋯, X_{d}) \\ & − ΦŜ(X_{1}, ⋯, X_{k−1}, (D_{X_{k}}^{H}U), X_{k+1}, ⋯, X_{d})), \end{aligned}

where X_{k} ∈ H. Now, using (29), we further specify

\begin{matrix}(D_{X_{k}}^{H}U) = \frac{1}{2} ∑_{r=1}^{ℓ} ∑_{j∈I}(K_{r}^{♭}(U)p_{jr}(J_{j}X_{k}^{j}) − JK_{r}^{♭}(U)p_{jr}X_{k}^{j}), \\ ((D_{U}J)(X_{k}))^{H} = 0, \end{matrix}

where X_{k}^{j} (resp. J_{j}) denote the gŜ-orthogonal projection (resp. restriction) of X_{k} (resp. J) to the subspace T S_{j} ⊂ T Ŝ (recall that the universal cover of (Ŝ, gŜ) is the Kähler product of (S_{j}, g_{j}, ω_{j}), so that the projection of T S_{j} to T Ŝ is a well-defined D-parallel subbundle of T Ŝ). Using (27), and the expressions (30) and (31), we eventually find that

ℑm(α) = −\frac{1}{2}dc ; Ŝ log ||ΦŜ||2 ; ĝ + \frac{1}{2}d^{c} log p(z) + \frac{1}{2} ∑_{k,r} H_{kr,k}θ̂_{r}

= −\frac{1}{2}dc ; Ŝ log ||ΦŜ||2 ; ĝ + \frac{1}{2p(z)} ∑_{k,r}((\frac{∂p}{∂z_{k}})H_{kr} + p(z)\frac{∂H_{kr}}{∂z_{k}} )θ̂_{r}

ρ^{∇} = ∑_{j∈I} ρ_{j} − ∑_{i,r,k} \frac{∂}{∂z_{k}} (\frac{1}{2p(z)} \frac{∂(p(z)H_{ir})}{∂z_{i}})dz_{k} ∧ θ_{r}

− \frac{1}{2p(z)} ∑_{i,r} \frac{∂(p(z)H_{ir})}{∂z_{i}} \frac{∂ω̂}{∂z_{r}} ,

where, we recall, ρ_{j} is the Ricci form of (S_{j}, g_{j}, ω_{j}), ω̂(z) = ∑_{j∈I}(∑_{r=1}^{ℓ}(p_{jr}z_{r} + c_{j})ω_{j}),

and p(z) = ∏_{j∈I}(∑_{r=1}^{ℓ} p_{jr}z_{r} + c_{j})^{d_{j}}. The formula for s^{∇} follows easily. □

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Vestislav Apostolov, Département de Mathématiques, UQAM, C.P. 8888, Succ. Centreville, Montréal (Québec), H3C 3P8, Canada

E-mail address: apostolov.vestislav@uqam.ca

David M. J. Calderbank, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

E-mail address: D.M.J.Calderbank@bath.ac.uk

Paul Gauduchon, Centre de Mathématiques, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France

E-mail address: pg@math.polytechnique.fr

Christina W. Tønnesen-Friedman, Department of Mathematics, Union College, Schenectady, New York 12308, USA

E-mail address: tonnesec@union.edu