THE GEOMETRY OF WEAKLY SELFDUAL KÄHLER SURFACES

V. APOSTOLOV, D. M. J. CALDERBANK AND P. GAUDUCHON

Abstract. We study Kähler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kähler metrics, including Kähler–Einstein metrics and conformally Einstein Kähler metrics. We also extend some of our results to almost Kähler 4-manifolds, providing new examples of Ricci-flat almost Kähler metrics which are not Kähler.

Introduction

Selfdual Kähler surfaces have been considered in several recent works, in particular in a paper by R. Bryant [13], where selfdual Kähler surfaces appear as the four-dimensional case of a comprehensive study of Bochner-flat Kähler manifolds in all dimensions, and in a paper by two of the authors [8], where a generic equivalence has been established between selfdual Kähler surfaces and selfdual Hermitian Einstein metrics and where an explicit local description of the latter is provided.

Whereas selfdual surfaces are easily proved to be extremal, i.e., admitting a hamiltonian Killing vector field whose momentum map is the scalar curvature, it was an a priori unexpected fact, independently discovered in the above works, that they actually admit a second hamiltonian Killing vector field; moreover, a crucial observation of R. Bryant [13], is that the momentum map of the latter is the pfaffian of the normalized Ricci form. Since these Killing vector fields commute, this also provides a link with the work of H. Pedersen and the second author [17], where an explicit local classification of selfdual Einstein metrics with two commuting Killing vector fields is obtained, without the hypothesis that they are Hermitian.

In this paper, we show that we can relax the assumption of selfduality, and establish the same bi-hamiltonian structure for weakly selfdual Kähler surfaces, i.e., Kähler surfaces whose anti-selfdual Weyl tensor W^{−} is harmonic; this fact has its origins in the basic Matsumoto–Tanno identity for such surfaces, recently re-discovered by W. Jelonek [32], and leads to a surprisingly simple explicit expression, generalizing an expression found by Bryant in the selfdual case. On the other hand, we also observe that in Calabi’s family of extremal Kähler metrics on the first Hirzebruch surface F_{1}, there is a unique (and completely explicit) weakly selfdual metric up to homothety. This is in contrast to the selfdual case, where the compact (smooth) examples are all locally symmetric [20].

Date: April 2001.

The first author was supported in part by a PAFARC-UQAM grant, and by an NSERC grant, the second author by the Leverhulme Trust and the William Gordon Seggie Brown Trust.

All three authors are members of EDGE, Research Training Network HPRN-CT-2000-00101, supported by the European Human Potential Programme.

Our results concerning weakly selfdual Kähler surfaces may be summarized as follows (definitions and more precise statements are given in the body of the paper).

Theorem. Let (M, g, J, ω) be a weakly selfdual Kähler surface. Then (g, J) is a bi-extremal Kähler metric in the sense that the scalar curvature and the pfaffian of the normalized Ricci form of (g, J) are Poisson-commuting momentum maps for hamiltonian Killing vector fields K_{1} and K_{2} respectively. Furthermore, on each connected component of M one of the following holds.

(i) K_{1} and K_{2} are linearly independent on a dense open set. Then (g, J, ω) has the explicit local form (41)–(45), depending on an arbitrary polynomial of degree 4 and an arbitrary constant which is zero if and only if g is selfdual, cf. Theorem 2.

(ii) K_{1} is non-vanishing on a dense open set, but K_{1} ∧ K_{2} is identically zero. Then (g, J, ω) is locally of cohomogeneity one and is given explicitly by the Calabi construction, cf. Theorem 4.

(iii) K_{1} and K_{2} vanish identically. Then g has parallel Ricci curvature (hence is either Kähler–Einstein or locally a Kähler product of two Riemann surfaces of constant curvatures).

If (M, g, J, ω) is compact and connected, then it necessarily belongs to case (ii) or (iii) above, and in case (ii) (M, g, J, ω) is isomorphic to the weakly selfdual Calabi extremal metric on F_{1} (see Theorem 5).

The path to proving this result touches upon various important themes in Kähler geometry, and along the way we introduce further ideas, results and examples. A key aspect of our approach is to study weakly selfdual Kähler surfaces within a more general setting. The Matsumoto–Tanno identity for weakly selfdual Kähler surfaces is equivalent to the fact that the primitive part ρ_{0} of the Ricci form ρ of (g, J) satisfies an overdetermined linear differential equation. On the open set where ρ_{0} is nonvanishing, the equation means that ρ_{0} defines a conformally Kähler Hermitian structure I inducing the opposite orientation to J. Many of the properties of weakly selfdual Kähler surfaces are simple consequences of the fact that the ρ is a closed J-invariant 2-form, whose primitive part satisfies this equation. In particular, we prove in Proposition 2 that two of the algebraic invariants (essentially the trace and the pfaffian) of any such 2-form φ are Poisson-commuting momentum maps for hamiltonian Killing vector fields. Therefore, throughout the work, we develop the theory of Kähler surfaces with such ‘hamiltonian’ 2-forms φ, which include other interesting examples in addition to weakly selfdual Kähler surfaces.

First of all we study the generic case that the hamiltonian Killing vector fields are linearly independent. This means that the Kähler structure (g, J, ω) is toric, and in Theorem 1 we characterize the class of toric Kähler structures arising in this way from hamiltonian 2-forms. Whereas toric Kähler surfaces in general depend essentially on an arbitrary function of two variables [1, 29], our toric surfaces, which we call ‘ortho-toric’, have an explicit form—given in Proposition 8— depending only on two arbitrary functions of one variable. This has the practical advantage that curvature conditions lead to ordinary differential equations for these functions. In particular, we are able to obtain explicitly all of the extremal toric Kähler structures in our class, including some new examples of Kähler metrics which are conformally Einstein, but neither selfdual nor anti-selfdual, and also some explicit Kähler–Einstein metrics. The weakly selfdual metrics in this family are classified in Theorem 2.

The case that the hamiltonian Killing vector fields associated to a hamiltonian 2-form φ are linearly dependent, but not both zero, is closely related to the Calabi construction of Kähler metrics on line bundles over a Riemann surface [14]. We provide, in Theorem 3, an geometric local characterization of these Kähler metrics: they are the Kähler metrics (g, J), with a Killing vector field K such that the almost Hermitian pair (g, I), where I is equal to J on span of {K, JK} but −J on the orthogonal distribution, is conformally Kähler. Over a fixed Riemann surface Σ, the general form of these Kähler metrics ‘of Calabi type’ again depends essentially only on functions of one variable from which it is easy to recover the Calabi extremal metrics. We present these in Proposition 14: the Riemann surface Σ has constant curvature, and the metrics have local cohomogeneity one under U(2), U(1, 1) or a central extension of the Heisenberg group Nil. The weakly selfdual Calabi extremal metrics are classified in Theorem 4: there is a four parameter family, one of which is globally defined on the first Hirzebruch surface.

The proof of the above theorem is completed by classifying the compact weakly selfdual Kähler surfaces. A partial classification for real analytic Kähler surfaces has been recently obtained by Jelonek [32], but we improve it in two respects: first, as speculated by Jelonek, we are able to remove the assumption of real-analyticity; second we prove that the only (non-product non-Kähler–Einstein) weakly selfdual Kähler metric on a ruled surface, is the Calabi extremal example on the first Hirzebruch surfaces F_{1}.

The paper is organized as follows. In the first section, we establish some basic facts concerning weakly selfdual Kähler surfaces; in particular, using general properties of hamiltonian 2-forms, we show that weakly selfdual Kähler surfaces are bi-extremal and that their anti-selfdual Weyl tensor is degenerate (some facts proved in this section also appear in Jelonek’s paper [32]). We also present the rough classification, Proposition 6, that allows us to deduce the above theorem from Theorems 2, 4 and 5. The generic, toric case, and Theorem 2, are described in the second section, whereas the third section treats the Calabi examples and Theorem 4. The classification of compact weakly selfdual Kähler surfaces is given in section 4.

In the final section we show that some of our results generalize to the class of almost Kähler 4-manifolds (M, g, J, ω) whose Ricci tensor is J-invariant. We first observe that the Calabi construction gives rise to new (local) examples of selfdual, Ricci-flat almost Kähler 4-manifolds (see Example 2), which provide further local counterexamples (cf. [9, 43]) to the still open Goldberg conjecture which states that a compact Einstein almost Kähler manifold must be Kähler–Einstein. Next we consider compact almost Kähler 4-manifolds with J-invariant Ricci tensor which are weakly selfdual, i.e., have harmonic anti-selfdual Weyl tensor. We show in Theorem 6 that weak-selfduality has strong consequences for the integrability of the corresponding almost complex structure, providing another interesting link with the Goldberg conjecture. As an application of this global result, we prove that a compact almost Kähler 4-manifold has constant sectional curvatures on the Lagrangian 2-planes if and only if it is a selfdual Kähler surface (see Corollary 1).

1. Weakly selfdual Kähler surfaces

1.1. The Matsumoto–Tanno identity. A Kähler surface (M, g, J) is an oriented Riemannian four-dimensional manifold equipped with a selfdual complex structure J, such that ∇J = 0, where ∇ denotes the Levi-Civita connection of g. The Kähler form is the J-invariant selfdual 2-form ω(·, ·) = ⟨J·, ·⟩; ω is closed and (M, ω) is a symplectic manifold.

The vector bundle Λ^{+}M of selfdual 2-forms is the orthogonal direct sum of the trivial bundle generated by ω and of the bundle of J-anti-invariant 2-forms, whereas the bundle Λ^{−}M coincides with the bundle of primitive—or trace-free— J-invariant 2-forms.

We denote by R, Ric, Ric_{0}, Scal, W = W^{+} + W^{−}, the curvature, the Ricci tensor, the trace-free part of Ric, the scalar curvature (i.e., the trace of Ric), and the Weyl tensor, expressed as the sum of its ±-selfdual components W^{±}.

The Ricci form, ρ, of a Kähler surface is the J-invariant 2-form defined by ρ(·, ·) = Ric(J·, ·); ρ is closed and, up to a factor 2π, is a representative of the first Chern class of (M, J) in de Rham cohomology; the trace-free part of ρ is denoted by ρ_{0}.

Definition 1. A Kähler surface (M, g, J) is weakly selfdual if its anti-selfdual Weyl tensor W^{−} is harmonic, i.e., satisfies:

δ^{g}W^{−} = 0.

where the codifferential δ^{g} acts on W^{−} as on a 2-form with values in Λ^{−}M.

Because of the Bianchi identity, the weak selfduality condition can also be defined in terms of the Ricci tensor; from this point of view, it can also be considered as a weak Einstein condition, in the sense that every Einstein metric is weakly selfdual. The link is provided by the Cotton–York tensor C_{X,Y} (Z) of the Riemannian metric g. Recall that the Cotton–York tensor is defined by C_{X,Y} (Z) = −(∇_{X}h)(Y, Z) + (∇_{Y} h)(X, Z), where h = \frac{1}{2}Ric_{0} + \frac{1}{24}Scal g denotes the normalized Ricci tensor of g.

The normalized Ricci tensor is the form in which the Ricci tensor appears in the well-known decomposition of the Riemannian curvature R = h ∧ Id + W, cf. e.g. [12]; therefore, via the (differential) Bianchi identity, the ±-selfdual components, C^{±}, of the Cotton–York tensor are linked to the ±-selfdual components of the Weyl tensor by δW^{+} = C^{+} and δW^{−} = C^{−}.

Definition 1 can thus be rephrased as follows:

Definition 2. A Kähler surface is weakly selfdual if its Cotton–York tensor is selfdual.

The normalized Ricci tensor plays a natural role throughout this work. For this reason, to simplify formulae, we write s = \frac{1}{6}Scal for the normalized scalar curvature, which is the trace of h.

Lemma 1. ([42, 32]) For any Kähler surface (M, g, J, ω) we have

(1)  ∇_{X}ρ_{0} = −2C^{−}(JX) − \frac{1}{2}ds(X)ω + \frac{1}{2}(ds ∧ JX^{♭} − Jds ∧ X^{♭}).

In particular, the Kähler surface (M, g, J, ω) is weakly selfdual if and only if the following Matsumoto–Tanno identity

(2)  ∇_{X}ρ_{0} = −\frac{1}{2}ds(X)ω + \frac{1}{2}(ds ∧ JX^{♭} − Jds ∧ X^{♭})

is satisfied (for any vector field X).

Proof. Since 2h = Ric − s g, the Cotton–York tensor of any Kähler surface can be written as follows:

2C_{X,Y} (Z) = −2(∇_{X}h)(Y, Z) + 2(∇_{Y} h)(X, Z)

= −(∇_{X}Ric)(Y, Z) + (∇_{Y} Ric)(X, Z) + ds(X)⟨Y, Z⟩−ds(Y)⟨X, Z⟩ = −(∇_{X}ρ)(Y, JZ) + (∇_{Y} ρ)(X, JZ) + ds(X)⟨Y, Z⟩ − ds(Y)⟨X, Z⟩

= (∇_{JZ}ρ)(X, Y) + ds(X)⟨Y, Z⟩ − ds(Y)⟨X, Z⟩.

(In order to obtain the last line from the preceding one, we use the fact that dρ = 0.) We then have:

(∇_{Z}ρ)(X, Y) = −2C_{X,Y} (JZ) − ds(X)⟨JY, Z⟩ + ds(Y)⟨JX, Z⟩,

or, equivalently

\begin{aligned}(∇_{Z}ρ_{0})(X, Y) = & − 2C_{X,Y} (JZ) \\ & − \frac{3}{2}ds(Z)⟨JX, Y ⟩ − ds(X)⟨JY, Z⟩ + ds(Y)⟨JX, Z⟩. \end{aligned} ; (3)

The anti-selfdual component of (3) gives identity (1); the last statement follows immediately. □

1.2. Twistor 2-forms and Kähler metrics. On any Riemannian manifold (M, g), if Φ is an anti-selfdual 2-form, i.e., a section of the vector bundle Λ^{−}M, then ∇Φ is a section of the vector bundle T^{∗}M ⊗ Λ^{−}M. This bundle has an orthogonal direct sum decomposition

(4)  T^{∗}M ⊗ Λ^{−}M = V^{(0)}M ⊕ V^{(1)}M,

in accordance with the algebraic decomposition ℝ^{4} ⊗ Λ^{−}ℝ^{4} = ℝ^{4} ⊕ (Σ_{+} ⊗ Σ_{−}^{3}) into irreducible sub-representations under the action of the orthogonal group. In (4), V^{(0)}M corresponds to the factor ℝ^{4}, hence is isomorphic to T^{∗}M, whereas V^{(1)}M corresponds to the factor Σ_{+} ⊗ Σ_{−}^{3}, so it is the kernel of the natural contraction T^{∗}M ⊗ Λ^{−}M → T^{∗}M. The projection of the connection to V^{(0)}M may be identified with the exterior derivative or divergence on anti-selfdual 2forms (which are related by the Hodge ∗ operator), while the projection to V^{(1)}M is often called the twistor or Penrose operator on anti-selfdual 2-forms.

Definition 3. An anti-selfdual 2-form Φ is called a twistor 2-form if ∇Φ is a section of the sub-bundle V^{(0)}M of T^{∗}M ⊗ Λ^{−}M.

Any non-vanishing section Φ of Λ^{−}M can be written uniquely as Φ = λ ω_{I}, where λ = \frac{|Φ|}{\sqrt{2}} is a positive function and ω_{I} is the Kähler form of an anti-selfdual almost-complex structure I on (M, g).

Lemma 2. [44] If Φ = λ ω_{I} is a non-vanishing section of Λ^{−}M, then Φ is a twistor 2-form if and only if the almost-Hermitian pair (ḡ = λ^{−2}g, I) is Kähler, with Kähler form ω̄ = λ^{−2}ω_{I} = λ^{−3}Φ.

Proof. Φ is a twistor 2-form if and only if there exists a 1-form γ such that, at each point of M, ∇Φ = ∑_{i=1}^{3} I_{i}γ ⊗ ω_{i}, where the triple (I_{1}, I_{2}, I_{3} = I_{1}I_{2}) is any positively oriented, orthonormal frame of (anti-selfdual) almost-complex structures at that point, and ω_{i} is the Kähler form of I_{i}.

If Φ = λ ω_{I} is a non-vanishing twistor 2-form on M, then, by choosing I_{1} = I, we have

λ∇I = (Iγ − dλ) ⊗ I + I_{2}γ ⊗ I_{2} + I_{3}γ ⊗ I_{3}.

Since the norm of I is constant, this equality implies that Iγ = dλ. Now observe that the equation

(5)  λ∇I = I_{3}dλ ⊗ I_{2} − I_{2}dλ ⊗ I_{3}

is equivalent to I being parallel with respect to the Levi-Civita connection of ḡ = λ^{−2}g. Hence if Φ is a twistor 2-form (ḡ, I) is Kähler. Conversely, if (ḡ, I) is Kähler then (5) holds, from which it follows that ∇Φ = ∇(λ ω_{I}) is the section of V^{(0)}M corresponding to the 1-form γ = −Idλ. □

If (M, g, J, ω) is a Kähler surface then a 2-form is anti-selfdual if and only if it is trace-free and J-invariant, and there is the following reformulation of Definition 3.

Lemma 3. Let (M, g, J, ω) be a Kähler surface and let φ_{0} be an anti-selfdual 2-form . Then φ_{0} is a twistor 2-form if and only if there is a 1-form β such that

(6)  ∇_{X}φ_{0} = −β(X)ω + β ∧ JX^{♭} − Jβ ∧ X^{♭}

for any vector field X.

Proof. It is easily checked that the right hand side of (6) is (the contraction with X of) the general form of a section of V^{(0)}M ≅ T^{∗}M. □

By the contracted Bianchi identity, C^{−} is a section of V^{(1)}M, and so the last statement of Lemma 1 can be rephrased as follows.

Lemma 4. A Kähler surface is weakly selfdual if and only if the trace-free part ρ_{0} of the Ricci form ρ is a twistor 2-form.

Together with Lemma 2, this implies:

Proposition 1. On the open set M_{0} where ρ_{0} = λ ω_{I} does not vanish, a Kähler surface (M, g, J, ω) is weakly selfdual if and only if the pair (ḡ = λ^{−2}g, I) is Kähler.

In particular it follows that on M_{0}, the selfdual Weyl tensor of ḡ, with the orientation induced by I, is degenerate, and equal to \frac{1}{2} s̄ω̄ ⊗_{0} ω̄, where s̄ is the (normalized) scalar curvature of ḡ, ω̄ = λ^{−2}ω_{I}, and ω̄ ⊗_{0} ω̄ stands for the traceless part of ω̄ ⊗ ω̄ viewed as an endomorphism of Λ^{+}M. By the conformal covariance of the Weyl tensor, it follows that on M_{0}, the anti-selfdual Weyl tensor of g (using the orientation induced by J) is given by

(7)  W^{−} = κ ω_{I} ⊗_{0} ω_{I}

where

(8)  κ = s̄λ^{−2}

is the conformal scalar curvature of the Hermitian pair (g, I), which is related to the Riemannian scalar curvature of g by

(9)  κ − s = δθ − |θ|^{2};

here θ denotes the Lee form of the pair (g, I), defined by dω_{I} = −2θ ∧ ω_{I}, see [21, 7]. Note that we have normalized the conformal scalar curvature by a factor \frac{1}{6} to be consistent with our normalization of the scalar curvature s.

Notice that this only uses the fact that M_{0} admits a non-vanishing twistor 2form, namely ρ_{0}. On the other hand, ρ_{0} is not an arbitrary twistor 2-form: by Lemma 1, the 1-form β defined by ∇ρ_{0} using (6), is equal to \frac{1}{2}ds, and so is exact. This fact, which is equivalent to the fact that the Ricci form ρ = ρ_{0} + \frac{3}{2}s ω is closed, will be exploited in the next subsection.

1.3. Hamiltonian 2-forms.

Definition 4. A hamiltonian 2-form on a Kähler surface (M, g, J, ω) is a closed J-invariant 2-form φ whose trace-free (i.e., anti-selfdual) part φ_{0} is a twistor 2-form.

On a weakly selfdual Kähler surface the Ricci form ρ is hamiltonian. In general, hamiltonian 2-forms are characterized by an analogue of the Matsumoto–Tanno identity (2).

Lemma 5. Let (M, g, J, ω) be a Kähler surface. Then a J-invariant 2-form φ = φ_{0} + \frac{3}{2}σ ω is hamiltonian if and only if

(10)  ∇_{X}φ_{0} = −\frac{1}{2}dσ(X)ω + \frac{1}{2}(dσ ∧ JX^{♭} − Jdσ ∧ X^{♭})

for any vector field X.

Proof. This is immediate from Lemma 3: in (6), dφ_{0} = −\frac{3}{2}dσ ∧ ω if and only if β = \frac{1}{2}dσ. □

In order to explain the use of the term ‘hamiltonian’, we recall the following definition.

Definition 5. A real function f on a Kähler manifold (M, g, J, ω) is a (real) holomorphic potential if the gradient grad_{g}f is a holomorphic vector field, i.e., preserves J; equivalently, f is a holomorphic potential if Jgrad_{g}f is a Killing vector field with respect to g.

A holomorphic potential f is therefore a momentum map for a hamiltonian Killing vector field with respect to the symplectic form ω.

To any J-invariant 2-form φ = φ_{0} + \frac{3}{2}σ ω, we may associate a normalized 2-form φ̃ = \frac{1}{2}φ_{0} + \frac{1}{4}σ ω. For example, if φ is the Ricci form ρ, then φ̃ is the 2-form ρ̃ associated to the normalized Ricci tensor: ρ̃(·, ·) = h(J·, ·).

We are going to show that if φ is hamiltonian, then the trace and pfaffian of φ̃ are holomorphic potentials.

Recall that, in general, the pfaffian pf(ψ) of a 2-form ψ is defined by \frac{1}{2}pf(ψ) = ∗(ψ ∧ ψ), where ∗ is the Hodge operator; alternatively, ψ ∧ ψ = \frac{1}{4}pf(ψ) ω ∧ ω.

Since φ̃ = \frac{1}{2}φ_{0} + \frac{1}{4}σ ω, its pfaffian π is given by

(11)  π = \frac{1}{4}σ^{2} − \frac{1}{2}|φ_{0}|^{2} = (\frac{σ}{2} + λ)(\frac{σ}{2} − λ),

where λ = \frac{|φ_{0}|}{\sqrt{2}}. We write this product as π = ξη so that σ = ξ+η and λ = \frac{1}{2}(ξ−η).

Proposition 2. Let (M, g, J, ω) be a Kähler surface and let φ = φ_{0} + \frac{3}{2}σ ω be a hamiltonian 2-form.

Then the trace σ and the pfaffian π of φ̃ = \frac{1}{2}φ_{0} + \frac{1}{4}σ ω are Poisson-commuting holomorphic potentials.

Proof. (i) Identity (10) can be written in terms of φ̃ as

(12)  ∇_{X} φ̃ = \frac{1}{4}(dσ ∧ JX^{♭} − Jdσ ∧ X^{♭}).

Differentiating again and skew-symmetrizing, we get

\begin{aligned}R_{X,Y} · φ̃ = [R_{X,Y} , φ̃] & = \frac{1}{4}(∇_{Y} dσ ∧ JX^{♭} − ∇_{X}dσ ∧ JY^{♭} \\ & − J∇_{Y} dσ ∧ X^{♭} + J∇_{X}dσ ∧ Y^{♭}). \end{aligned}

Since [R_{X,Y} , φ̃] is J-invariant in X, Y, it follows that

∇_{Y} dσ ∧ JX − J∇_{Y} dσ ∧ X − ∇_{X}dσ ∧ JY + J∇_{X}dσ ∧ Y = − ∇_{JY} dσ ∧ X − J∇_{JY} dσ ∧ JX + ∇_{JX}dσ ∧ Y + J∇_{JX}dσ ∧ JY,

hence

(13)  S(Y) ∧ JX − JS(Y) ∧ X − S(X) ∧ JY + JS(X) ∧ Y = 0,

where

S(X) = ∇_{X}dσ + J∇_{JX}dσ.

As an algebraic object, S is a symmetric, J-anti-commuting, endomorphism of T M; hence by contracting (13) with a vector field Z and taking the trace over Y and Z, we see that S = 0, and therefore, σ is a holomorphic potential.

(ii) From (12) we derive

(∇_{X} φ̃) ∧ φ̃ = \frac{1}{2}(X ∧ Jdσ ∧ φ̃),

hence

(14)  dπ = \frac{1}{2}∗(Jdσ ∧ φ̃).

From this, we infer (again using (12)):

\begin{aligned}∇_{X}dπ & = \frac{1}{2} ∗(J∇_{X}dσ ∧ φ̃ + Jdσ ∧ ∇_{X} φ̃) \\ & = \frac{1}{2} ∗(J∇_{X}dσ ∧ φ̃ + \frac{1}{4}Jdσ ∧ dσ ∧ JX). \end{aligned} ; (15)

The second term of the right hand side of (15) is clearly J-invariant; the first term is J-invariant as well since σ is a holomorphic potential and φ̃ is J-invariant. Hence π is also a holomorphic potential.

(iii) By contracting equation (14) with Jdσ, we see that σ and π Poissoncommute. □

Since σ and π Poisson-commute, the Killing vector fields K_{1} := Jgrad σ and K_{2} := Jgrad π commute. Also ω(K_{1}, K_{2}) = 0.

Remark 1. The Killing vector fields K_{1} and K_{2} need not be non-zero or independent in general. In particular K_{1} and K_{2} both vanish if σ is constant, since φ is then parallel by (10).

Notice, however, that K_{1}^{1,0}, K_{2}^{1,0}, and hence K_{1}^{1,0} ∧ K_{2}^{1,0}, are holomorphic, so that on each connected component of M there are three possibilities:

(i) K_{1} ∧ K_{2} is non-vanishing on a dense open set.

(ii) K_{1} is non-vanishing on a dense open set, but K_{1} ∧K_{2} vanishes identically; (iii) K_{1} and K_{2} vanish identically;

One consequence of this remark is the following lemma.

Lemma 6. If φ is a hamiltonian 2-form on a Kähler surface M, then the open set M_{0}, where φ_{0} is non-zero, is empty or dense in each connected component of M.

Proof. On each component of M where K_{1} = Jgrad_{g}σ is non-zero, the set U where dσ is non-vanishing is dense, hence ∇φ_{0} is non-vanishing on U and so the zero set of φ_{0} is dense in this connected component. On the other hand if K_{1} is identically zero on a component, then φ_{0} is parallel on that component, hence identically zero or everywhere non-zero. □

When K_{1} and K_{2} vanish identically, φ does not contain much information about the geometry of M (it could be just a constant multiple of ω, even zero). In the other two cases, however, we shall obtain an explicit classification of Kähler surfaces with a hamiltonian 2-form. The keys to these classifications are Proposition 2 and the following observation.

Proposition 3. Let φ be a hamiltonian 2-form on a Kähler surface M and write σ = ξ + η and π = ξη for the trace and pfaffian of φ̃.

Then on each connected component of M where φ_{0} is not identically zero, dξ and dη are orthogonal.

Proof. The contraction of (10) with φ_{0} yields

⟨∇_{X}φ_{0}, φ_{0}⟩ = \frac{1}{2}(φ_{0}(dσ, JX) − φ_{0}(Jdσ, X))
= −φ_{0}(Jdσ, X)

and hence

(16)  2λ dλ = d(λ^{2}) = \frac{1}{2}d(|φ_{0}|^{2}) = −φ_{0}(Jdσ).

Since (φ_{0} ∘ J)^{2} = λ^{2}Id, we deduce that φ_{0}(Jdλ) = −\frac{1}{2}λ dσ and therefore

φ_{0} ∘ J(\frac{dσ}{2} + dλ) = −λ (\frac{dσ}{2} + dλ), φ_{0} ∘ J(\frac{dσ}{2} − dλ) = λ (\frac{dσ}{2} − dλ).

This means that the 1-forms dξ and dη, wherever they are non-zero, are eigenforms for the symmetric endomorphism −φ_{0} ∘ J, corresponding to the eigenvalues λ and −λ, respectively; in particular, they are orthogonal on the open set M_{0} where λ (i.e., φ_{0}) is non-zero. However, by Lemma 6 this open set is empty or dense in each connected component of M, and the result follows. □

Remark 2. On the open set M_{0} where λ is non-zero, so that φ_{0} = λ ω_{I}, observe that equation (16) may be rewritten

(17)  Idσ = 2Jdλ.

Indeed, supposing only that φ_{0} is a twistor 2-form, ω̄ = λ^{−3}φ_{0} is closed, and so, if φ = φ_{0} + \frac{3}{2}σ ω,

dφ = d(φ_{0} + \frac{3}{2}σ ω) = 3λ^{2}dλ ∧ ω̄ + \frac{3}{2}dσ ∧ ω
= 3(dλ ∧ ω_{I} + \frac{1}{2}dσ ∧ ω).

Hence equation (17) holds if and only if φ is closed.

1.4. Bi-extremal Kähler surfaces. In the case that (M, g, J, ω) is weakly selfdual and φ_{0} = ρ_{0}, we can take φ = ρ, so that σ is the (normalized) scalar curvature s, and π, the pfaffian p of the normalized Ricci form ρ̃(·, ·) = h(J·, ·); evidently, ρ̃ = \frac{1}{2}ρ_{0} + \frac{1}{4}s ω.

Recall that a Kähler metric is said to be extremal if the scalar curvature is a holomorphic potential.

Definition 6. A Kähler metric is called bi-extremal if both the the (normalized) scalar curvature s = tr_{ω}ρ̃ and the pfaffian p = pf(ρ̃) of the normalized Ricci form ρ̃ are holomorphic potentials.

Note that the potential function p appearing in the above definition is not the pfaffian of the usual Ricci form ρ = ρ_{0}+ \frac{3}{2}s ω; thus, our definition for bi-extremality differs from the one given in [41, 31] (compare Theorem 5 below and [31, Th.1.1 & Prop.3.8]).

Proposition 2 immediately implies:

Proposition 4. A weakly selfdual Kähler metric is bi-extremal.

On a bi-extremal Kähler surface, the holomorphic potentials s and p automatically Poisson-commute, since K_{2} preserves g, hence s, so that ds(K_{2}) = 0.

For a weakly selfdual Kähler surface, Proposition 3 generically implies that dξ and dη are orthogonal, where s = ξ + η and p = ξη. We shall see in Sections 2 and 3 that a bi-extremal Kähler surface satisfying this orthogonality condition is weakly selfdual.

1.5. The Bach tensor. The Bach tensor, B, of an n-dimensional Riemannian manifold (M, g) is defined by

(18)  B_{X,Y} = ∑_{i=1}^{n} ( − (∇_{e_{i}}C)_{e_{i},X}(Y) + (W_{e_{i},X}h(e_{i}), Y)),

where, we recall, C is the Cotton–York tensor and h is the normalized Ricci tensor (here, {e_{i}} is an arbitrary g-orthonormal frame). When n = 4, the Bach tensor is conformal covariant of weight −2, i.e., B^{ϕ^{−2}g} = ϕ^{2}B^{g}, and B can be indifferently expressed in terms of W^{+} or of W^{−}. Specifically

\begin{aligned}B_{X,Y} & = 2 ∑_{i=1}^{n}(−(∇_{e_{i}}C^{+})_{e_{i},X}(Y) + (W_{e_{i},X}^{+}h(e_{i}), Y)) \\ & = 2 ∑_{i=1}^{n}(−(∇_{e_{i}}C^{−})_{e_{i},X}(Y) + (W_{e_{i},X}^{−}h(e_{i}), Y)); \end{aligned} ; (19)

in particular, the Bach tensor vanishes whenever W^{+}, W^{−} or Ric_{0} vanishes.

If (M, g, J) is a Kähler surface, the Bach tensor is easily computed by using the above identity and the fact that W^{+} = \frac{1}{2}s ω ⊗_{0} ω. Indeed, if B^{+} and B^{−}, denote the J-invariant and J-anti-invariant parts of B, we get

(20)  B^{+} = s Ric_{0} + 2(∇ds)_{0}^{+},  B^{−} = −(∇ds)^{−},

where (∇ds)_{0}^{+} is the J-invariant trace-free part of the Hessian and (∇ds)^{−} is the Janti-invariant part. (This formula is due to Derdziński [21], where it was obtained by a different argument.)

It follows that B is J-invariant if and only if (M, g, J) is an extremal Kähler surface; moreover, if this holds, we get B = s Ric_{0} + 2(∇ds)_{0}. On the open set U where s has no zero, the vanishing of B then means that the conformally related metric g̃ = s^{−2}g is Einstein. Therefore, on U, the following two statements are thus equivalent (see also [21]):

(i) The Bach tensor of the Kähler surface (M, g, J) vanishes; (ii) (M, g, J) is extremal and the conformally related metric g̃ = s^{−2}g is

Einstein. Note also that when B is J-invariant, it is determined by the associated antiselfdual 2-form B̃(·, ·) = B(J·, ·), which is also given by

(21)  B̃ = (dJds)_{0} + sρ_{0}.

On a weakly selfdual Kähler surface, C^{−} = 0, while W^{−} = \frac{1}{2}κ ω_{I} ⊗_{0} ω_{I}. It follows that B̃ is a multiple of κρ_{0}, which vanishes if and only if W^{−} = 0 or ρ_{0} = 0

Proposition 5. A weakly selfdual Kähler surface is Bach-flat (i.e., has vanishing Bach tensor) if and only if it is selfdual or Kähler–Einstein.

1.6. Rough classification of weakly selfdual Kähler surfaces. We have seen in Lemma 6 that the open set M_{0}, on which the trace-free part φ_{0} of a hamiltonian 2-form is non-zero, is empty or dense in each connected component of M. For a weakly selfdual Kähler surface (M, g, J, ω), the Ricci form ρ is hamiltonian, and M_{0} is the set of points at which g is not Kähler–Einstein. Because ρ is closely linked to the anti-selfdual Weyl tensor of M, we can obtain more information about M_{0} except when g is selfdual.

We first recall the general fact, first observed by A. Derdziński in [21], that for any Kähler surface (M, g, J) with non-vanishing scalar curvature s, the conformally related metric g̃ = s^{−2}g satisfies δg̃W^{+} = 0; moreover, up to rescaling, g̃ is the unique metric in the conformal class [g] that satisfies this property. This follows from the fact that the selfdual Cotton–York tensor C^{+} of any Kähler surface can be written as

(22)  C^{+}(X) = −W^{+}(\frac{ds}{s} ∧ X);

on the other hand, the selfdual Cotton–York tensors of two conformally related metrics g and f^{−2}g are related by

C^{+,f^{−2}g}(X) = C^{+,g}(X) + W^{+}(\frac{df}{f} ∧ X);

it follows from (22) that the selfdual Cotton–York tensor of the metric s^{−2}g vanishes identically; moreover, as W^{+} has no kernel (for s non-vanishing), the latter property characterizes s^{−2}g up to a constant multiple.

Lemma 7. On each connected component of the open set M_{0} where ρ_{0} does not vanish, the scalar curvature s̄ of ḡ is a constant multiple of λ^{−1}, i.e.,

(23)  s̄ = cλ^{−1},

where λ is the positive eigenvalue of Ric_{0} and c is a constant.

Proof. We apply the preceding argument to the Kähler pair (ḡ, I) on M_{0} and to g = λ^{2}ḡ; by hypothesis, g satisfies δ^{g}W^{−} = C^{−} = 0, where W^{−} is actually the selfdual Weyl tensor of g for the orientation induced by I; from the above mentioned uniqueness property, it follows that, wherever s̄ is non-zero, g coincides with s̄^{−2}ḡ up to rescaling, i.e., that s̄ is a locally constant multiple of λ^{−1}. However, the same holds on the interior of the zero set of s̄. Hence by the continuity of s̄ on M_{0}, s̄ = cλ^{−1} for some constant c on each connected component of M_{0}. □

A more global statement may be obtained using the conformal scalar curvature κ = s̄λ^{−2}. Since the anti-selfdual Weyl tensor of g is given by W^{−} = \frac{1}{2}κ ω_{I} ⊗_{0} ω_{I}, it follows that κ^{2} is equal to |W^{−}|^{2} on M_{0}, up to a numerical factor, and hence we may extend κ continuously to the closure of M_{0}. Also λ is globally defined and continuous.

Therefore, using the fact that the closure of M_{0} is a union of connected components by Lemma 6, we can rewrite Lemma 7.

Lemma 8. Let (M, g, J, ω) be a weakly selfdual Kähler surface. Then, on each component of M where ρ_{0} is not identically zero, the conformal scalar curvature κ of (g, I) is linked to λ by

(24)  κλ^{3} = c,

where c is the constant of Lemma 7. Moreover, c = 0 if and only if W^{−} = 0 on that component.

This lemma yields the following rough classification of weakly selfdual Kähler surfaces (see also [32]).

Proposition 6. Let (M, g, J, ω) be a weakly selfdual, connected, Kähler surface. Then either:

(i) ρ_{0} is identically zero so (g, J) is Kähler–Einstein; or (ii) the scalar curvature s of g is constant, but ρ_{0} is not identically zero; then, (g, J) is locally the Kähler product of two Riemann surfaces of constant curvatures; or

(iii) s is not constant and g is selfdual; or (iv) W^{−} and ρ_{0} have no zero: then, the Kähler metric (ḡ = λ^{−2}g, I) of Proposition 1 is extremal and globally defined on M; in particular, W^{−} is degenerate everywhere.

Proof. If s is constant, then by (2) the Ricci form is parallel. Hence either g is locally irreducible, and is Einstein, or (g, J) is locally the Kähler product of two Riemann surfaces of constant curvatures.

If s is not constant, then by Lemma 6 the open set M_{0} where ρ_{0} does not vanish is an open dense subset of M. However, by Lemma 8, κλ^{3} is constant. If this constant is zero, then κ must vanish identically and M is selfdual; otherwise κ and λ have no zero on M, so M_{0} = M, the Kähler pair (ḡ = λ^{−2}g, I) is defined on M and W^{−} is degenerate, but nonzero everywhere. As observed in Section 1.5, for a weakly selfdual Kähler surface the Bach form B̃ is a multiple of ω_{I}. Since B is a conformally covariant tensor, it follows that Bach tensor of ḡ is I-invariant, showing that (ḡ, I) is an extremal Kähler metric. □

Kähler–Einstein metrics and Kähler products of Riemann surfaces clearly are weakly selfdual. Since these are well studied, we henceforth assume that s is not constant (on any component), i.e., K_{1} is non-vanishing on a dense open set. In Section 2, we analyse the generic case that K_{1} and K_{2} are independent, while Section 3 is devoted to the case that K_{1} ∧ K_{2} vanishes identically (but K_{1} is nonzero). In both sections, we obtain an explicit local classification within the more general framework of Kähler surfaces with a hamiltonian 2-form.

2. Ortho-toric Kähler surfaces

2.1. Toric Kähler surfaces. We have seen in Section 1 that on a Kähler surface with a hamiltonian 2-form φ—in particular on a weakly selfdual Kähler surface—the trace σ and the pfaffian π of the associated normalized 2-form φ̃ are holomorphic potentials for hamiltonian Killing vector fields K_{1} = Jgrad σ and K_{2} = Jgrad π. Furthermore, σ and π Poisson-commute, i.e., ω(K_{1}, K_{2}) = 0.

A (usually compact) Kähler surface (M, g, J, ω), with holomorphic Killing vector fields K_{1} and K_{2} which are independent on a dense open set and satisfy ω(K_{1}, K_{2}) = 0, is said to be toric. We begin this section by recalling the local theory of such surfaces, and we therefore assume that K_{1} and K_{2} are everywhere independent and that x_{1} and x_{2} are globally defined momentum maps for K_{1} and K_{2}.

The condition ω(K_{1}, K_{2}) = 0 is equivalent to the fact that x_{1} and x_{2} commute for the Poisson bracket determined by ω. Hence also [K_{1}, K_{2}] = 0, and since K_{1}, K_{2}, JK_{1} and JK_{2} are all holomorphic, they all commute. In particular the rank 2 distributions Π, generated by K_{1} and K_{2}, and JΠ, generated by JK_{1} and JK_{2}, are integrable.

These distributions Π, JΠ are also orthogonal, since ⟨JK_{1}, K_{2}⟩ = 0. It follows that K_{1}, K_{2}, JK_{1} and JK_{2} form a frame. Since they commute, the 1-forms in the dual coframe are closed, and may be written dt_{1}, dt_{2}, Jdt_{1}, Jdt_{2}, where t_{1} and t_{2} are only given locally and up to an additive constant. Now observe that

\begin{aligned}Jdt_{1} & = \frac{|K_{2}|^{2}dx_{1} − ⟨K_{1}, K_{2}⟩dx_{2}}{|K_{1} ∧ K_{2}|^{2}} \\ Jdt_{2} & = \frac{|K_{1}|^{2}dx_{2} − ⟨K_{1}, K_{2}⟩dx_{1}}{|K_{1} ∧ K_{2}|^{2}} \end{aligned}

and so

Jdt_{i} = ∑_{j=1,2} G_{ij}dx_{j} (i = 1, 2),

where G_{ij} is a positive definite symmetric matrix of functions of x_{1} and x_{2} (note that K_{i} = ∂/∂t_{i}). These 1-forms are closed if and only if G_{ij} is the Hessian of a function of x_{1} and x_{2}. The following well-known explicit classification is then readily obtained (see [29], [1]).

Proposition 7. Let G_{ij} be a positive definite 2×2 symmetric matrix of functions of 2-variables x_{1}, x_{2} with inverse G^{ij}. Then the metric

∑_{i,j}(G_{ij}dx_{i}dx_{j} + G^{ij}dt_{i}dt_{j})

is almost-Kähler with Kähler form

ω = dx_{1} ∧ dt_{1} + dx_{2} ∧ dt_{2}

and has independent hamiltonian Killing vector fields ∂/∂t_{1}, ∂/∂t_{2} with Poissoncommuting momentum maps x_{1} and x_{2}. Any almost Kähler structure with such a pair of Killing vector fields is of this form (where the t_{i} are locally defined up to an additive constant), and is Kähler if and only if G_{ij} is the Hessian of a function of x_{1} and x_{2}.

2.2. The ortho-toric case. Propositions 2 and 3 motivate the following definition.

Definition 7. A Kähler surface (M, g, J, ω) is ortho-toric if it admits two independent hamiltonian Killing vector fields with Poisson-commuting momentum maps ξη and ξ + η such that dξ and dη are orthogonal.

An explicit classification of ortho-toric Kähler metrics follows from Proposition 7 by changing variables and imposing the orthogonality of dξ and dη. However, since the coordinate change is awkward, and we have not spelt out the proof of Proposition 7, we give a self-contained proof of this classification.

Proposition 8. The almost-Hermitian structure (g, J, ω) defined by

\begin{aligned}g & = (ξ − η) (\frac{dξ^{2}}{F (ξ)} − \frac{dη^{2}}{G(η)}) \\ & + \frac{1}{ξ − η} (F (ξ)(dt + ηdz)^{2} − G(η)(dt + ξdz)^{2}), \end{aligned} (25)

\begin{aligned}Jdξ & = \frac{F (ξ)}{ξ − η} (dt + η dz),  Jdt = −\frac{ξ dξ}{F (ξ)} − \frac{η dη}{G(η)}, \\ Jdη & = \frac{G(η)}{η − ξ} (dt + ξ dz),  Jdz = \frac{dξ}{F (ξ)} + \frac{dη}{G(η)}, \end{aligned} (26)

ω = dξ ∧ (dt + η dz) + dη ∧ (dt + ξ dz) (27) is an ortho-toric Kähler structure for any functions F, G of one variable. Every ortho-toric Kähler surface is of this form, where t, z are locally defined up to an additive constant.

Proof. (i) The Kähler form may be written

ω = d(ξ + η) ∧ dt + d(ξη) ∧ dz

which is certainly closed. If is also immediate that ∂/∂t and ∂/∂z are hamiltonian Killing vector fields with Poisson-commuting momentum maps ξ +η and ξη. Since dt+iJdt and dz +iJdz are closed, J is integrable, and the Kähler surface is clearly ortho-toric.

(ii) Conversely, suppose that (g, J, ω) is an ortho-toric Kähler surface with Killing vector fields K_{1}, K_{2}. Since the dual frame to K_{1}, K_{2}, JK_{1}, JK_{2} consists of closed 1-forms, we may write it as dt, dz, Jdt, Jdz, where t and z are locally defined up to an additive constant. Note also that dξ, dη, dt, dz are linearly independent 1-forms—where ξ + η and ξη are the momentum maps of K_{1} and K_{2}—so we may use (ξ, η, t, z) as a coordinate system.

Since (Jdz)(K_{1}) = 0 and (Jdz)(K_{2}) = 0 we may write

Jdz = \frac{dξ}{F} + \frac{dη}{G}

for some functions F and G (of ξ and η). The equations

0 = (Jdz)(JK_{1}) = −⟨Jdz, dξ + dη⟩
and  1 = (Jdz)(JK_{2}) = −⟨Jdz, η dξ + ξ dη⟩

give F = |dξ|^{2}(ξ − η) and G = |dη|^{2}(η − ξ), using the fact that dξ and dη are orthogonal. A similar argument tells us that

Jdt = \frac{ξ dξ}{F} + \frac{η dη}{G} .

for the same functions F and G. Now since Jdt and Jdz are closed, we obtain (ξ − η)F_{η} = 0 and (η − ξ)G_{ξ} = 0, so that F = F (ξ) and G = G(η). The Kähler form ω is evidently given by (27), and since we know Jdt and Jdz, we readily obtain (26), and hence the metric (25). □

Any ortho-toric Kähler surface (M, g, J, ω) comes equipped with an anti-selfdual almost-complex structure, I, whose Kähler form, ω_{I} is defined by

\begin{aligned}ω_{I} & = \frac{dξ ∧ Jdξ}{|dξ|^{2}} − \frac{dη ∧ Jdη}{|dη|^{2}} \\ & = dξ ∧ (dt + ηdz) − dη ∧ (dt + ξdz); \end{aligned} ; (28)

equivalently

\begin{aligned}Idξ & = Jdξ = \frac{F (ξ)}{ξ − η} (dt + ηdz),  Idt = −\frac{ξdξ}{F (ξ)} + \frac{ηdη}{G(η)}, \\ Idη & = −Jdη = −\frac{G(η)}{η − ξ} (dt + ξdz),  Idz = \frac{dξ}{F (ξ)} − \frac{dη}{G(η)}. \end{aligned} ; (29)

Proposition 9. For any ortho-toric Kähler surface, the almost-Hermitian pair (ḡ = (ξ − η)^{−2}g, I) is Kähler.

Proof. Clearly Idt and Idz are closed, so I is integrable. From (28), we easily infer that the Lee form θ of the Hermitian pair (g, I), defined by dω_{I} = −2θ ∧ ω_{I}, is

(30)  θ = −d log |ξ − η|.

It follows that ω̄ := (ξ − η)^{−2}ω_{I} is closed, i.e., the pair (ḡ = (ξ − η)^{−2}g, I) is Kähler. □

In particular, on any ortho-toric Kähler surface, the anti-selfdual Weyl tensor— which is the selfdual Weyl tensor of g for the orientation induced by I—is degenerate: W^{−} = κ ω_{I} ⊗_{0} ω_{I}, where κ is the conformal scalar curvature of the Hermitian pair (g, I).

Remark 3. The vector fields K_{1} and K_{2} are still Killing with respect to ḡ and hamiltonian with respect to ω̄, with momentum maps −\frac{1}{ξ−η} and −\frac{ξ+η}{2(ξ−η)} respectively. However, the Kähler metric (ḡ, I) is not ortho-toric in general, as it can be checked using Lemma 9 below.

Combining Propositions 2, 3 with 8 and 9, we obtain the following theorem.

Theorem 1. A Kähler surface is ortho-toric if and only if it admits a hamiltonian 2-form whose associated Killing vector fields are independent. The Kähler structure is then given explicitly in terms of two arbitrary functions F, G of one variable by (25)–(27).

Indeed, using (28)–(29), notice that, by definition, Jd(ξ + η) = Id(ξ − η), cf. Remark 2, so the hamiltonian 2-form φ is \frac{1}{2}(ξ − η)ω_{I} + (ξ + η)ω.

2.3. Ortho-toric weakly selfdual Kähler surfaces. The curvature of an orthotoric Kähler surface is entirely determined by the scalar curvature s of g, the conformal scalar curvature κ of the Hermitian pair (g, I), and the trace-free part ρ_{0} of the Ricci form of (g, J).

Lemma 9. For any ortho-toric Kähler surface (M, g, J, ω), ρ_{0} is a multiple μ of the Kähler form ω_{I} of the Hermitian pair (g, I), and μ, s, κ are given by

(31)  μ = \frac{F′(ξ) − G′(η)}{2(ξ − η)^{2}} − \frac{F′′(ξ) + G′′(η)}{4(ξ − η)},
(32)  s = −\frac{F′′(ξ) − G′′(η)}{6(ξ − η)},
(33)  κ = −\frac{F′′(ξ) − G′′(η)}{6(ξ − η)} + \frac{F′(ξ) + G′(η)}{(ξ − η)^{2}} − \frac{2(F (ξ) − G(η))}{(ξ − η)^{3}}.

In particular, on the open subset of M where μ has no zero, the anti-selfdual almost-complex structure determined by ρ_{0} is equal to I.

Proof. From (27), we infer that the volume-form v_{g} = \frac{1}{2}ω ∧ ω of g is given by

v_{g} = −(ξ − η) dξ ∧ dη ∧ dt ∧ dz,

since t and z are the real parts of J-holomorphic coordinates. By putting v_{0} = dt ∧ Jdt ∧ dz ∧ Jdz, we have

ρ = −\frac{1}{2}dJd log \frac{v_{g}}{v_{0}} .

Now according to (26),

v_{0} = −\frac{ξ − η}{F (ξ)G(η)}dξ ∧ dη ∧ dt ∧ dz,

and hence

\frac{v_{g}}{v_{0}} = F (ξ)G(η);

this implies

ρ = −\frac{1}{2}dJd log |F (ξ)| − \frac{1}{2}dJd log |G(η)|,

from which (31) and (32) follow easily.

From (9) and (30), we get

κ = s − 2 \frac{|dξ|^{2} + |dη|^{2}}{(ξ − η)^{2}} − \frac{Δ(ξ − η)}{ξ − η};

on the other hand, we compute that

Δξ = −\frac{F′(ξ)}{ξ − η}, Δη = \frac{G′(η)}{ξ − η};

and we obtain (33). □

Proposition 10. An ortho-toric Kähler surface M is extremal if and only if F and G are of the form

\begin{aligned}F (x) & = kx^{4} + ℓx^{3} + Ax^{2} + B_{1}x + C_{1}, \\ G(x) & = kx^{4} + ℓx^{3} + Ax^{2} + B_{2}x + C_{2}, \end{aligned} ; (34)

in which case

(35)  s = −2k(ξ + η) − ℓ,

and (ḡ = (ξ − η)^{−2}g, I) is an extremal Kähler metric as well.

Moreover, M is • Bach-flat if and only if 4k(C_{1} − C_{2}) = (B_{1} − B_{2})ℓ;

• of constant scalar curvature if and only if k = 0;

• scalar-flat (i.e., anti-selfdual) if and only if k = ℓ = 0.

Proof. Since the scalar curvature s is a function of ξ and η, Jgrad_{g}s belongs to the span of the Killing vector fields K_{1} and K_{2} and commutes with them; if it is itself a Killing vector field, it has to be a linear combination of K_{1} and K_{2} with constant coefficients, i.e. s = a(ξ + η) + bξη + c, where a, b, c are constants. By (32), this implies (34). Finally, using (21), we easily compute that the anti-selfdual 2-form associated to the Bach tensor of an ortho-toric extremal Kähler surface is

B̃ = \frac{4k(C_{1} − C_{2}) − (B_{1} − B_{2})ℓ}{2(ξ − η)^{2}} ω_{I}.

Since B is also I-invariant, the Kähler metric (ḡ, I) is extremal as well (see Section 1.5). □

Example 1. For k ≠ 0 and 4(C_{1} −C_{2}) = (B_{1} −B_{2})\frac{ℓ}{k}, we obtain explicit Bach-flat Kähler surfaces with non-constant scalar curvature. These metrics are therefore not anti-selfdual, and for B_{1} ≠ B_{2} they are not selfdual either (note that a selfdual Kähler surface is bi-extremal and see the next Proposition). According to Section 1.5, the metric

g̃ = (2k(ξ + η) + ℓ)^{−2}g,

which is defined on the open subset where 2k(ξ +η)+ℓ ≠ 0, is Einstein, Hermitian (but non-Kähler) with a locally defined toric isometric action.

Proposition 11. An ortho-toric Kähler surface M is bi-extremal if and only if F and G are of the form

\begin{aligned}F (x) & = kx^{4} + ℓx^{3} + Ax^{2} + Bx + C_{1}, \\ G(x) & = kx^{4} + ℓx^{3} + Ax^{2} + Bx + C_{2}, \end{aligned} ; (36)

in which case the Ricci form is given by ρ = −2kφ − ℓω. Hence M is weakly selfdual and is

• selfdual if and only if C_{1} = C_{2};

• Kähler–Einstein if and only if k = 0;

• Ricci-flat if and only if k = ℓ = 0.

Proof. Since a bi-extremal surface is extremal, we may apply Proposition 10. We then compute

(37)  μ = −k(ξ − η) + \frac{B_{1} − B_{2}}{2(ξ − η)^{2}} ,
(38)  p = 4k^{2}ξη + kℓ(ξ + η) + \frac{ℓ^{2}}{4} − k \frac{B_{1} − B_{2}}{ξ − η} + \frac{(B_{1} − B_{2})^{2}}{4(ξ − η)^{4}} .

As in the proof of Proposition 10, p cannot be a J-holomorphic potential unless it is a linear combination of ξ + η and ξη with constant coefficients; this in turn is equivalent to the condition B_{1} = B_{2}. Since μ = −2kλ and s = −2kσ − ℓ it follows that ρ = −2kφ − ℓω. Hence ρ is a hamiltonian and so M is weakly selfdual by (4).

By substituting in the expression of κ given by (33) we get κ = −\frac{2(C_{1}−C_{2})}{(ξ−η)^{3}}; since the condition W^{−} = 0 is equivalent to κ = 0, the characterization of the selfdual case follows. Also M is Einstein if and only if μ = 0, and so the last two assertions are immediate. □

Remark 4. For k ≠ 0, we can set k = −\frac{1}{2} and ℓ = 0 by a simultaneous affine change of ξ and η. In this case ρ = φ, and so the ortho-toric reduction is defined by ρ. However, not all weakly selfdual Kähler surfaces can be put in orthotoric form; for example weakly selfdual metrics belonging to the general family of cohomogeneity-one extremal metrics considered by E. Calabi [14] are not in general ortho-toric, since K_{1} and K_{2} are then collinear. We discuss this case in Section 3.

On the other hand, the examples with k = 0 in the above Proposition show that among Kähler–Einstein metrics (which are weakly selfdual), there are some which can be put into ortho-toric form, because even though ρ does not define an ortho-toric reduction, there happens to be another hamiltonian 2-form φ. If additionally A = ℓ = 0 or C_{1} = C_{2}, these examples in fact have cohomogeneity one and their ortho-toric form arises from a choice of maximal torus inside the isometry group. However, the other examples do not have additional symmetries and therefore appear to be new.

Proposition 12. On an ortho-toric extremal Kähler surface M, the space of infinitesimal symmetries of the Kähler structure is generated by K_{1} and K_{2} (and infinitesimal rotations in this plane if they are globally defined), except in the following two cases:

(i) M is locally a complex space form, i.e., Kähler–Einstein and selfdual; or (ii) M is Ricci-flat (hence anti-selfdual) and, locally, of cohomogeneity one. In terms of F and G, these two cases are respectively described by

(39)  F (x) = G(x) = ℓx^{3} + Ax^{2} + Bx + C;

and

\begin{matrix}F (x) = Bx + C_{1}, \\ G(x) = Bx + C_{2}. \end{matrix} ; (40)

Proof. Suppose there exist a third infinitesimal symmetry of M, say K_{3}, which does not lie in the plane spanned by K_{1} and K_{2}; then, we must have ds(K_{i}) = 0 and dμ(K_{i}) = 0, for i = 1, 2, 3; this implies that ds and dμ are colinear; we then infer from (35) and (37) that we have k = 0 and B_{1} = B_{2} in (34), so that M is Kähler–Einstein by Proposition 11.

If, in addition, W^{−} = 0, we obtain (39) and g is then a selfdual Kähler–Einstein surface, i.e., a complex space form.

If W^{−} does not vanish identically, on the open set where W^{−} ≠ 0, the Hermitian pair (g, I) is invariant under the action of the Killing vector fields K_{i}’s, as I is determined by the eigenform of W^{−} corresponding to its simple eigenvalue, cf. [21, 7]; then, κ (a constant multiple of the simple eigenvalue of W^{−}), the square-norm |θ|^{2} of the Lee form θ of the pair (g, I) as well as δθ are also invariant under the action of K_{i}’s, i = 1, 2, 3; we then have d|θ|^{2} ∧dκ = 0; by using (30) and Lemma 9, we can check that this implies that F and G satisfy (40); by Proposition 10, the corresponding ortho-toric Kähler surface is Ricci-flat and κ = −\frac{2(C_{1}−C_{2})}{(ξ−η)^{3}}; it then follows from (30) that θ = \frac{1}{3} d ln |κ|; in particular, d|θ|^{2} ∧ θ = 0; by [8, Theorem 1], this implies that the metric is of cohomogeneity one. Since κ is non-zero, it is not constant, and so the metric is not homogeneous.

Evidently the two cases overlap when C_{1} = C_{2} and A = ℓ = 0, in which case g is flat. □

Remark 5. In fact we do not need to assume a priori that M is extremal in the above proof, as long as we assume that the additional symmetry preserves φ, hence ξ − η. Then ds and dξ − dη are collinear, from which it is easy to deduce that M is extremal.

Let us now collect the results we have established so far about weakly selfdual Kähler surfaces.

Theorem 2. Let (M, g, J, ω) be a weakly selfdual Kähler surface. Denote by s, λ, and p = (\frac{s}{2} + λ)(\frac{s}{2} − λ), the (normalized) scalar curvature, the positive eigenvalue of the trace-free Ricci tensor Ric_{0}, and the pfaffian of the normalized Ricci tensor, respectively. Then:

(i) K_{1} := Jgrad_{g}s and K_{2} := Jgrad_{g}p are commuting Killing vector fields, and on any simply connected open subset where K_{1} and K_{2} are linearly independent, the functions ξ := \frac{s}{2} + λ, η := \frac{s}{2} − λ, t, z form a globally defined coordinate system with respect to which the Kähler structure (g, J, ω) is

\begin{aligned}g & = (ξ − η) (\frac{dξ^{2}}{F (ξ)} − \frac{dη^{2}}{G(η)}) \\ & + \frac{1}{ξ − η} (F (ξ)(dt + ηdz)^{2} − G(η)(dt + ξdz)^{2}), \end{aligned} (41)

\begin{aligned}Jdξ & = \frac{F (ξ)}{ξ − η} (dt + η dz),  Jdt = −\frac{ξ dξ}{F (ξ)} − \frac{η dη}{G(η)}, \\ Jdη & = \frac{G(η)}{η − ξ} (dt + ξ dz),  Jdz = \frac{dξ}{F (ξ)} + \frac{dη}{G(η)}, \end{aligned} (42)

ω = dξ ∧ (dt + η dz) + dη ∧ (dt + ξ dz), (43)

where

(44)  F (x) = kx^{4} + ℓx^{3} + Ax^{2} + Bx + C_{1},
(45)  G(x) = kx^{4} + ℓx^{3} + Ax^{2} + Bx + C_{2}.

(ii) Conversely, each almost Kähler structure (g, J, ω) described by (41)–(45) is Kähler and weakly selfdual with

s = −2k(ξ + η) − ℓ, p = 4k^{2}ξη + kℓ(ξ + η) + \frac{ℓ^{2}}{4} ,

so that K_{1} = \frac{∂}{∂t} and K_{2} = \frac{∂}{∂z}. (iii) The Kähler structure described by (41)–(45) is selfdual if and only if C_{1} = C_{2}.

In the selfdual case, we recover the general expression found by Bryant [13, Section 4.3.2] depending on an arbitrary polynomial of degree 4.

3. Kähler surfaces of Calabi type

3.1. Hamiltonian 2-forms and the Calabi construction. In this section we classify weakly selfdual Kähler surfaces of nowhere constant scalar curvature s, but for which p and s are not independent. As in the previous section (when we assumed p and s were independent) we do this by finding an explicit formula for a Kähler surface (M, g, J, ω) with a hamiltonian 2-form φ = φ_{0} + \frac{3}{2}σ ω such that K_{1} := Jgrad_{g}σ has no zero, but K_{2} := Jgrad_{g}π = bK_{1}, where π is the pfaffian of φ̃ and b is (necessarily) constant.

The general theory of Kähler surfaces with a hamiltonian 2-form, described in Section 1, applies equally to this case. In particular, since K_{1} has no zero, we may still write π = ξη and σ = ξ + η for the trace and pfaffian of φ̃, and dξ and dη are orthogonal by Proposition 3.

On the other hand, the constructions of Section 2 definitely fail, because K_{1} and K_{2} are no longer independent: π is an affine function of σ, so ξ and η are not independent functions. Therefore, dξ and dη, in addition to being orthogonal, are collinear! This is not a contradiction: we deduce that either ξ or η is constant. Since π = ξ(σ − ξ) = η(σ − η), this constant is the constant b above, so that π = b(σ − b) and λ = ±\frac{1}{2}(σ − 2b).

We observe that K_{1} is an eigenvector of −φ_{0} ∘ J, for the eigenvalue λ, and the conformally Kähler anti-selfdual complex structure I is characterized as follows: I coincides with J on the distribution generated by K_{1} and JK_{1}, but with −J on the orthogonal distribution. Hence we are in the following situation.

Definition 8. A Kähler surface (M, g, J, ω) is said to be of Calabi type if it admits a non-vanishing hamiltonian Killing vector field K such the almost-Hermitian pair (g, I)—with I equal to J on the distribution spanned by K and JK, but −J on the orthogonal distribution—is conformally Kähler.

It is straightforward to obtain an explicit formula for Kähler metrics of Calabi type, using the LeBrun form of a Kähler metric with a hamiltonian Killing vector field [36].

Proposition 13. Let (M, g, J, ω) be a Kähler surface of Calabi type. Then the Kähler structure is given locally by

(46)  g = (az − b)g_{Σ} + w(z)dz^{2} + w(z)^{−1}(dt + α)^{2},
(47)  ω = (az − b) ω_{Σ} + dz ∧ (dt + α),

where z is the momentum map of the Killing vector field K, t is a function on M with dt(K) = 1, w is a function of one variable, g_{Σ} is a metric on 2-manifold Σ with area form ω_{Σ}, α is a 1-form on Σ with dα = a ω_{Σ}, and a, b are constant. Conversely equations (46)–(47) define a Kähler structure of Calabi type with K = ∂/∂t, for any g_{Σ} and V.

Proof. The proof follows LeBrun’s description [36] of Kähler metrics with a hamiltonian Killing vector field K. Supposing first that (g, J, ω) is only almost Hermitian, note that K −iJK is a holomorphic vector field, so that the complex quotient is locally a Riemann surface Σ. Introducing a local holomorphic coordinate x + iy on Σ, we may write

g = e^{u}w(dx^{2} + dy^{2}) + w dz^{2} + w^{−1}(dt + α)^{2},
Jdx = dy, Jdz = w^{−1}(dt + α),
ω = e^{u}w dx ∧ dy + dz ∧ (dt + α).

where dt(K) = 1, α is an invariant 1-form with α(K) = 0, and u, w are functions of x, y, z. The almost Hermitian structure I is given by

Idx = −dy, Idz = w^{−1}(dt + α),

with Kähler form

ω_{I} = −e^{u}w dx ∧ dy + dz ∧ (dt + α).

We now impose the condition that (g, J, ω) and (ḡ = λ^{−2}g, I, ω̄ = λ^{−2}ω_{I}) are Kähler for some non-vanishing function λ. Now dω = 0 if and only if

(48)  (e^{u}w)_{z}dz ∧ dx ∧ dy = dz ∧ dα,

while λ^{−2}ω_{I} is closed if and only if

(49) λ dz ∧ dα + ((e^{u}w)_{z}λ − 2λ_{z}e^{u}w)dz ∧ dx ∧ dy + 2λ_{x}dx ∧ dz ∧ (dt + α) + 2λ_{y}dy ∧ dz ∧ (dt + α) + 2λ_{t}dt ∧ ω_{I} = 0.

In the presence of (48), (49) is equivalent to

\begin{matrix}(e^{u}w)_{z}λ = λ_{z}e^{u}w, \\ λ_{x} = λ_{y} = λ_{t} = 0, \end{matrix} ; (50)

which holds if and only if λ = λ(z) and e^{u}w = hλ for some function h(x, y).

Let θ be the complex 1-form w dz +i(dt+α). Then, since dx±idy is closed, the complex structures I and J are integrable if and only if dθ belongs to the ideals generated by \{θ, dx − idy\} and by \{θ, dx + idy\} respectively. Since dθ(\frac{∂}{∂t}, ·) = 0, these conditions force (dx − idy) ∧ dθ and (dx + idy) ∧ dθ to vanish. Hence I is integrable if and only if

(51)  dα = −w_{x}dy ∧ dz + w_{y}dx ∧ dz + fdx ∧ dy

while J is integrable if and only if

(52)  dα = w_{x}dy ∧ dz − w_{y}dx ∧ dz + fdx ∧ dy;

here f is an arbitrary function. Hence I and J are both integrable if and only if

(53)  w_{x} = w_{y} = 0  and  dα = fdx ∧ dy

and f is necessarily a function of x, y only.

Putting together (48), (50), and (53), we see that (g, J, ω) is of Calabi type, with Killing vector field K, if and only if e^{u}w = h(x, y)λ(z), with dα = h(x, y)λ_{z}, λ_{zz} = 0 and w_{x} = w_{y} = 0, so that λ = az − b for constants a, b and w = w(z).

Using the freedom in the choice of t, we may then assume α is a 1-form on Σ, while g_{Σ} = h(x, y)(dx^{2} + dy^{2}) is a metric on Σ, and the result follows. □

This Proposition shows that Kähler metrics of Calabi type are essentially the same as metrics arising from the well-known Calabi construction [14] of metrics on the total space of a Hermitian line bundle over a Riemann surface. In this interpretation, the Killing vector field K generates the natural circle action on the line bundle and the Kähler form is

(54)  ω_{Σ} + dJdf

for a function f of the fibre norm r. Since −dJd log r is the curvature of the line bundle, which is basic, the momentum map z of K is also a function of r. Hence we may locally view f as a function of z, so that, if we write Jdz = dt + α where dt(K) = 1 and α is basic,

(55)  dJdf = (\frac{f′(z)}{w(z)})_{z}dz ∧ (dt + α) + \frac{f′(z)}{w(z)}dα.

Therefore, in Proposition 13, setting b = −1 without loss of generality, we have f′(z) = zw(z).

The metrics of Proposition 13 certainly admit a hamiltonian 2-form, namely φ = (az − b)ω_{I} + 3azω. Hence σ = 2az and λ = az − b, so that ξ = 2az − b and η = b. The hamiltonian Killing vector fields associated to φ both vanish when a = 0. On the other hand, for a nonzero, we can use the freedom in the choice of z to set a = 1 and b = 0.

Theorem 3. A Kähler surface is of Calabi type if and only if either:

(i) it is locally a Kähler product of two Riemann surfaces, one of which admits a Killing vector field; or

(ii) it admits a hamiltonian 2-form whose associated Killing vector fields are

dependent but not both zero. The Kähler structure is then given explicitly by the Calabi construction (46)–(47): in case (i) a = 0, while in case (ii) we may take a = 1, b = 0 without loss of generality.

3.2. Weakly selfdual Kähler surfaces of Calabi type. We begin this section by computing the curvature of a Kähler surface of Calabi type which is not a local Kähler product. Therefore we set a = 1, b = 0, and write w(z) = z/V (z), so that the Kähler structure is

(56)  g = zg_{Σ} + \frac{z}{V (z)}dz^{2} + \frac{V (z)}{z}(dt + α)^{2},
(57)  ω = z ω_{Σ} + dz ∧ (dt + α),

As with ortho-toric Kähler surfaces, the curvature is entirely determined by the scalar curvature s of g, the conformal scalar curvature κ of the Hermitian pair (g, I), and the trace-free part ρ_{0} of the Ricci form of (g, J).

Lemma 10. For a non-product Kähler surface (M, g, J, ω) of Calabi type, given by (56)–(57), ρ_{0} is a multiple μ of the Kähler form ω_{I} of the Hermitian pair (g, I), and μ, s, κ are given by

(58)  μ = −\frac{1}{4z} (s_{Σ} + (\frac{V_{z}}{z^{2}} )_{z}z^{2}),
(59)  s = \frac{s_{Σ} − V_{zz}}{6z} ,
(60)  κ = \frac{1}{6z} (s_{Σ} − z^{2}(z^{2}(\frac{V}{z^{4}} )_{z})_{z}),

where s_{Σ} is the scalar curvature of Σ. In particular, on the open subset of M where μ has no zero, the anti-selfdual almost-complex structure determined by ρ_{0} is equal to I.

Proof. The Ricci form ρ is given by ρ = ρ_{Σ} − \frac{1}{2}dJd log V. The first term is \frac{1}{2}s_{Σ}ω_{Σ}, and we compute the second term as follows:

dJd log V = d(\frac{V_{z}}{V} Jdz) = d(\frac{V_{z}}{z} (dt + α))

= \frac{V_{zz}}{z} dz ∧ (dt + α) + V_{z}(\frac{1}{z}ω_{Σ} − \frac{1}{z^{2}} dz ∧ (dt + α)).

Evidently we may write this as a linear combination of ω and ω_{I}, and we readily obtain (58) and (59).

The conformal scalar curvature is most easily computed by noticing that the conformal Kähler metric ḡ = z^{−2}g is also of Calabi type, with z̄ = 1/z and V̄ (z̄) = z̄^{4}V (1/z̄) = V (z)/z^{4}. Hence its scalar curvature is

\frac{s_{Σ} − V̄_{z̄z̄}}{6z̄} = \frac{z}{6}(s_{Σ} − z^{2}(z^{2}(\frac{V}{z^{4}} )_{z})_{z})

from which (60) follows, since κ = z^{−2}s̄. □

Proposition 14. Let M be a non-product Kähler surface M of Calabi type, with Killing vector field K. Then the scalar curvature of M is a momentum map for a multiple of K if and only if g_{Σ} has constant curvature k and V is of the form

(61)  V (z) = A_{1}z^{4} + A_{2}z^{3} + kz^{2} + A_{3}z + A_{4}.

Any Kähler surface given by (56)–(57) is extremal, with Ricci form μ ω_{I} + \frac{3}{2}s ω, where

(62)  μ = −A_{1}z + \frac{A_{3}}{2z^{2}}
(63)  s = −2A_{1}z − A_{2}.;

also the conformal scalar curvature of (g < I) is

(64)  κ = −\frac{A_{3}}{z^{2}} − \frac{2A_{4}}{z^{3}} .

Hence:

(i) g has constant scalar curvature if and only if A_{1} = 0; (ii) g is scalar-flat (i.e., anti-selfdual) if and only if A_{1} = A_{2} = 0. (iii) (g, J) is Kähler–Einstein if and only if A_{1} = A_{3} = 0; (iv) g is weakly selfdual if and only if A_{3} = 0.

(v) g is selfdual if and only if A_{3} = A_{4} = 0.

(vi) (g, J) is bi-extremal if and only if g is weakly selfdual; (vii) the Bach tensor of g vanishes if and only if 4A_{1}A_{4} − A_{2}A_{3} = 0.

Proof. From (59), we have

6zs + V_{z}z = s_{Σ}.

If s is an affine function of z, then both sides of this equation must be constant. Hence s_{Σ} = 2k and V must be a quartic in z with quadratic term kz^{2}. The formulae for μ, s and κ are immediate, as are (i)–(v). (For (iv) we use the fact that I is integrable and z^{−2}ω_{I} is closed, so that weak selfduality is equivalent to the equation ds = 2dμ.)

(vi) The pfaffian p of the normalized Ricci form is given by

(65)  p = (−2A_{1}z − \frac{1}{2}A_{2} + \frac{A_{3}}{2z^{2}} )(−\frac{1}{2}A_{2} − \frac{A_{3}}{2z^{2}} )

in particular, p is a rational function of z; since z is a holomorphic potential, p is a holomorphic potential if and only if it is an affine function of z, if and only if A_{3} = 0; p is then equal to −\frac{1}{2}A_{2}(s + \frac{1}{2}A_{2}).

(vii) For any extremal Kähler surface the Bach tensor is J-invariant and the corresponding anti-selfdual 2-form, is expressed by (21) which yields

B̃ = \frac{(4A_{1}A_{4} − A_{2}A_{3})}{z^{2}} ω_{I}.

□

This family of extremal Kähler metrics has been considered in many places. In particular, it includes the extremal Kähler metrics of cohomogeneity one under U(2) constructed by Calabi in [14]; more generally, it turns out that these metrics all have cohomogeneity one under a (local) action of a four-dimensional Lie group, locally isomorphic to a central extension of the isometry group of a surface of constant curvature k. We refer to these metrics as extremal Kähler surfaces of Calabi type and briefly recall how they may be realized as diagonal Bianchi metrics, of class IX, VIII or II, according to whether k is positive, negative or zero.

Up to rescaling, we can—and will—assume that k = ε, with ε = 1, 0 or −1. As is well known, we can now write dt + α = σ_{3} and introduce t-dependent 1-forms σ_{1}, σ_{2} on Σ with g_{Σ} = σ_{1}^{2} + σ_{2}^{2}, σ_{1} ∧ σ_{2} = ω_{Σ} and

(66)  dσ_{1} = εσ_{2} ∧ σ_{3}, dσ_{2} = εσ_{3} ∧ σ_{1}, dσ_{3} = σ_{1} ∧ σ_{2}.

By substituting σ_{1}, σ_{2}, σ_{3} in (56)

(67)  g = \frac{z}{V (z)}dz^{2} + z(σ_{1}^{2} + σ_{2}^{2}) + \frac{V (z)}{z} σ_{3}^{2};

the complex structure J is determined by

(68)  Jσ_{1} = σ_{2},  Jdz = \frac{V (z)}{z} σ_{3}

while the Kähler form ω is

(69)  ω = dz ∧ σ_{3} + zσ_{1} ∧ σ_{2} = d(zσ_{3}).

We here recognize bi-axial diagonal Bianchi metrics of class IX, VIII or II, according as ε is equal 1, −1 or 0; these admit a cohomogeneity one local action of SU(2) if ε = 1, of SU(1, 1) if ε = −1, of the Heisenberg group Nil if ε = 0, and the orbits are level sets of z. This can be seen as follows: denote by (Z_{1}, Z_{2}, Z_{3} = K_{1}) the triplet of vector fields determined by σ_{i}(Z_{j}) = δ_{ij} and dz(Z_{i}) = 0, i, j = 1, 2, 3, where δ_{ij} is the Kronecker symbol. Then, for each value of z, Z_{1}, Z_{2}, Z_{3} are tangent to the corresponding orbit M_{z} and generate a Lie algebra isomorphic to 𝔰𝔲(2), 𝔰𝔲(1, 1) or 𝔫𝔦𝔩^{3} according as ε is equal to 1, −1 or 0; therefore, each orbit can be locally identified to the corresponding Lie group and we can locally construct a new triple of independent vector fields, (Z̃_{1}, Z̃_{2}, Z̃_{3}), such that [Z̃_{i}, Z_{j}] = 0 and dz(Z̃_{i}) = 0, i, j = 1, 2, 3 (for each orbit, if (Z_{1}, Z_{2}, Z_{3}) is a basis of left-invariant vector fields, (Z̃_{1}, Z̃_{2}, Z̃_{3}) is a basis of right-invariant vector fields); Z̃_{1}, Z̃_{2}, Z̃_{3} are clearly Killing with respect to g and all commute with K_{1}.

If ε ≠ 0, K_{1}, Z̃_{1}, Z̃_{2}, Z̃_{3} generate a 4-dimensional Lie algebra, corresponding to a local action of U(2) if ε = 1, of U(1, 1) if ε = −1. If ε = 0, Z̃_{3} equals K_{1}, up to a constant factor; on the other hand, we get an additional Killing vector field K̃_{1} generated by the rotations around the origin in the Euclidean 2-plane 𝔼^{2} of x, y; then, K̃_{1}, K_{1}, Z̃_{1}, Z̃_{2} generate a 4-dimensional Lie algebra, say 𝔤, corresponding to a local action of the group, G, obtained by forming the semi-direct product of Nil by S^{1} for the natural action of S^{1} on Nil by (outer) automorphisms; the centre of G = Nil ⋉ S^{1} coincides with the centre of Nil; the latter is one dimensional again and the quotient of G by its center is isomorphic to Isom(𝔼^{2}); in other words, G is isomorphic to a (non-trivial) one-dimensional central extension of Isom(𝔼^{2}).

If we concentrate our attention on weakly selfdual Kähler surfaces, we readily infer from the foregoing:

Theorem 4. Let (M, g, J) be a weakly selfdual Kähler surfaces. Denote by s the scalar curvature and by p the pfaffian of the normalized Ricci form; let K_{1} = Jgrad_{g}s and K_{2} = Jgrad_{g}p be the associated Killing vector fields and assume that K_{1} has no zero and that K_{2} = bK_{1}, where b is a real constant (possibly zero).

(i) Then (M, g, J) admits a local action of cohomogeneity one of G = U(2), U(1, 1) or Nil ⋉ S^{1} and is locally isomorphic to a diagonal Bianchi metric of class IX, VIII or II respectively.

More precisely, let σ_{1}, σ_{2}, σ_{3} denote the (local) 1-forms on M induced by this action, corresponding to a triple of G-invariant 1-forms on G, so that dσ_{3} = σ_{1} ∧ σ_{2}, dσ_{2} = εσ_{3} ∧ σ_{1}, dσ_{1} = εσ_{2} ∧ σ_{3}, where ε = 1, −1 or 0 according as G = U(2), U(1, 1) or Nil ⋉ S^{1}; then, the Kähler structure (g, J) can be put in the form (67)–(68), where z is an affine function of s and V (z) is of the form

(70)  V (z) = A_{1}z^{4} + A_{2}z^{3} + εz^{2} + A_{4}.

(ii) Conversely, each Kähler surface of the form (67)–(68), with V given by (70) is weakly selfdual. (iii) This Kähler structure is selfdual if and only if A_{4} = 0.

Remark 6. By substituting A_{3} = 0 in (62) and (64), we readily infer that that A_{4} and the constant c appearing in (23) and (24) are linked together by

(71)  c = 2A_{1}^{3}A_{4};

moreover, A_{1} ≠ 0, as s is non-constant, so that A_{4} = 0 if and only if c = 0; as we already know, both conditions are equivalent to g being selfdual.

3.3. The weakly selfdual Kähler metric on the first Hirzebruch surface. We close this section by providing an example of a compact weakly selfdual Kähler surface (M, g, J); this belongs to the family of extremal metrics constructed in [14] by E. Calabi on the first Hirzebruch surface F_{1}, viewed as the compactification of the total space of the tautological line bundle L = O(−1) over the complex projective line ℂP^{1} = ℙ(ℂ^{2}) obtained by adding a section at infinity, say C_{∞}; then, the zero section, C_{0}, has self-intersection −1 and F_{1} can also be considered as a blown up of the projective plane ℂP^{2} at some point, with exceptional divisor C_{0}, and F_{1} − C_{0} = ℂ^{2} − {(0, 0)}.

We here recall the main features of Calabi’s construction from [14]. Let r be the usual norm in ℂ^{2} and, for convenience, introduce the function t defined on ℂ^{2} − {(0, 0)} by e^{t} = r^{2}; the group U(2) naturally acts on F_{1}, by preserving C_{0} and C_{∞}; by [15], the connected group of isometries, Isom_{0}(M, g), of any compact extremal Kähler surface (M, g, J) is a maximal compact subgroup of the connected group of holomorphic transformations Aut_{0}(M, J); it follows that any extremal metric on F_{1} is isometric to a U(2)-invariant metric, in particular has a globally defined potential on the open set M_{0} = F_{1} − C_{0} = ℂ^{2} − {(0, 0)} of the form u = u(t); the Kähler form is thus given by ω = ψ(t)dd^{c}t + ψ′(t)dt ∧ d^{c}t, where ψ(t) := u′(t); conversely, any such Kähler metric extends to F_{1} if and only if ψ(t) extends to a C^{∞} function of e^{t} in the neighbourhood of t = −∞ and to a C^{∞} function of e^{−t} in the neighbourhood of t = ∞; in particular, ψ has a limit a when t → −∞ and a limit, b, when t → ∞,with 0 < a < b; the corresponding Kähler class [ω] is then equal to 4π(−a[C_{0}] + b[C_{∞}]), where [C_{0}] and [C_{∞}] denotes the Poincaré-dual of C_{0} and similarly for [C_{∞}]; conversely, each Kähler class on F_{1} is of this form,for some pair 0 < a < b; it is easily checked that the Ricci for is given by ρ = dd^{c}v, with v = t − \frac{1}{2} log ψ − \frac{1}{2} log ψ′, so that s = 2(\frac{v′}{ψ} + \frac{v′′}{ψ′} ); on the other hand, such a metric is extremal if and only if s is an affine function of ψ; by easy successive partial integrations, we infer that this metric is extremal if and only if ψ satisfies the following differential relation: ψψ′ = V (ψ), where V is a polynomial of the form (61); moreover, the extremal metric is actually defined on F_{1} and has its Kähler class parameterized by the pair (a, b) as above if and only if the coefficients of V are given by

\begin{matrix}A_{1} = \frac{−2a}{(b − a)(a^{2} + 4ab + b^{2})}, \\ A_{2} = \frac{(3a^{2} − b^{2})}{(b − a)(a^{2} + 4ab + b^{2})}, \\ A_{3} = \frac{ab(3a^{2} − b^{2})}{(b − a)(a^{2} + 4ab + b^{2})}, \\ A_{4} = \frac{−2a^{3}b^{2}}{(b − a)(a^{2} + 4ab + b^{2})}; \end{matrix} ; (72)

we thus get an extremal Kähler metric, say g_{(a,b)}, in each Kähler class of F_{1} (we have a similar construction for the other Hirzebruch surfaces F_{k} [14]; the coefficients of V are then given by (76) below).

All these metrics are of cohomogeneity one under the action of U(2) and can also be put in the form (67)–(68), with the same parameter ψ, the same polynomial V (ψ), d^{c}t = σ_{3} and dd^{c}t = σ_{1} ∧ σ_{2}; in particular, according to Proposition 14, g_{(a,b)} if weakly selfdual if and only if A_{3} = 0, and this happens if and only if a and b are related by

(73)  b^{2} = 3a^{2};

in this case, A_{2} = 0 as well, meaning that p = 0, and V (ψ) = A_{1}ψ^{4} + ψ^{2} + A_{4} is a function of ψ^{2}; in particular, ψψ′ = V (ψ) is easily integrated into

(74)  ψ = a(\frac{1 + 3e^{t+t_{0}}}{1 + e^{t+t_{0}}} )^{\frac{1}{2}} = a(\frac{1 + 3e^{t_{0}}r^{2}}{1 + e^{t_{0}}r^{2}} )^{\frac{1}{2}} ,

where t_{0} is a constant; the latter can be made equal to zero without loss of generality by a mere translation of the parameter t, i.e. a rescaling of r; then, up to rescaling, the Kähler form is given by

(75)  ω = \frac{(1 + 3e^{t})^{\frac{1}{2}}}{(1 + e^{t})^{\frac{1}{2}}} dd^{c}t + \frac{e^{t}}{(1 + e^{t})^{\frac{3}{2}} (1 + 3e^{t})^{\frac{1}{2}}} dt ∧ d^{c}t.

In the sequel, the Kähler metric given by (75) will be referred to as the Calabi weakly selfdual Kähler metric of F_{1}. It may be noticed that, according to (72), the scalar curvature s = −2A_{1}ψ is non-constant and (strictly) positive.

In the next section we show that, conversely, each compact weakly selfdual Kähler surface with non-constant scalar curvature is, up to rescaling, isomorphic to the first Hirzebruch surface equipped with the Calabi weakly selfdual Kähler metric.

4. Compact weakly selfdual Kähler surfaces

Compact selfdual Kähler surfaces have been described by B.-Y. Chen in [20]: these are locally symmetric, hence of constant holomorphic sectional curvature or, locally the product of Riemann surfaces of opposite constant curvature (see [13] for a higher dimensional generalization).

In [32], W. Jelonek proved that compact real analytic weakly selfdual Kähler surfaces are either Kähler–Einstein, or locally the product of two Riemann surfaces of constant Gauss curvatures, or biholomorphic to a ruled surface.

We show that the hypothesis of real analyticity can actually be removed and that, except in the case when the scalar curvature is constant, the only weakly selfdual Kähler ruled surface is the first Hirzebruch surface F_{1} equipped with the Calabi weakly selfdual Kähler metric, as described in the preceding section (up to rescaling).

More precisely, we have the following

Theorem 5. Let (M, g, J) be a compact weakly selfdual Kähler surface. Then (M, g, J) is either

(i) Kähler–Einstein; or (ii) locally isomorphic to the product of two Riemann surfaces of constant

Gauss curvatures, or (iii) up to rescaling, isomorphic to the first Hirzebruch surface F_{1} equipped

with a Calabi weakly selfdual Kähler metric.

Proof. By Proposition 6, we know that g is either selfdual, hence, by the above mentioned result of B.-Y. Chen, described by (i) or (ii), or of constant scalar curvature, hence, again, described by (i) or (ii), or of non constant scalar curvature. In the latter case, the (negative) Kähler structure (ḡ, I) is globally defined; it then follows from a result of Kotschick [35] that the signature of M is zero [35]; moreover, since the (real) holomorphic field K_{1} = Jgrad_{g}s has non-empty zero set, we know by [19] that the Kodaira dimension of the Kähler surface (M, J) is −∞, hence (M, J) is a ruled surface which is the projectivization ℙ(E) of a rank 2 holomorphic vector bundle E over a compact complex curve Σ [11].

If Σ = ℂP^{1}, (M, J) is a Hirzebruch surface F_{k} = ℙ(O ⊕ O(−k)), where k is a positive integer, or the product ℂP^{1} × ℂP^{1}; the only extremal Kähler metrics of ℂP^{1} ×ℂP^{1} are the (symmetric) product metrics, which are of constant scalar curvature; on the other hand, any maximal compact subgroup of Aut_{0}(M, J) is conjugate to U(2), and therefore any extremal Kähler metric must be a cohomogeneityone U(2) metric [15], hence, locally of the form (67) with ε = 1 (cf. the end of the preceding section for the case when k = 1); as shown by E. Calabi in [14], for each k > 0, any Kähler class of F_{k} carries a unique extremal Kähler metric (up to a re-parameterization); each one can be put in the form (67), where the polynomial V, in the notation of (61), is determined by

\begin{matrix}A_{1} = \frac{(k + 1)a + (k − 1)b}{(b − a)(a^{2} + b^{2} + 4ab)} \\ A_{2} = \frac{(2 − k)b^{2} − (k + 2)a^{2}}{(b − a)(a^{2} + b^{2} + 4ab)}, \\ A_{3} = \frac{ab((2 − k)b^{2} − (k + 2)a^{2})}{(b − a)(a^{2} + b^{2} + 4ab)}, \\ A_{4} = \frac{a^{2}b^{2}((k + 1)a + (k − 1)b)}{(b − a)(a^{2} + b^{2} + 4ab)}, \end{matrix} ; (76)

where 0 < a < b are the parameters of the Kähler class; according to Proposition 14, g is weakly selfdual precisely when A_{3} = 0; in the present situation, this is equivalent to A_{2} = 0 and happens if and only if k = 1 and μ := \frac{a}{b} = \frac{1}{\sqrt{3}}, i.e. if (M, g, J) is the first Hirzebruch surface equipped with a Calabi weakly selfdual Kähler metric.

We now show that a compact ruled surface (M, J) whose base Σ is a compact complex curve of genus g(Σ) at least 1 does not carry weakly selfdual Kähler metrics of non-constant scalar curvature. We thus assume that (M, J) = ℙ(E) carries a weakly selfdual Kähler metric of non-constant scalar curvature to get a contradiction.

Using an argument from [37, 48], we first observe that the rank two vector bundle E splits as E = O ⊕ L, where O stands for the trivial holomorphic line bundle and L is a holomorphic line bundle L of degree deg(L) > 0. Indeed, recall the already mentioned result of E. Calabi [15] that the connected component of the isometry group Isom_{0}(M, g) is a maximal compact subgroup in Aut_{0}(M, J); according to M. Maruyama [40], the group of automorphisms of ruled surfaces can be described as follows: If g(Σ) ≥ 1, there exists an exact sequence

(77)  {1} → Aut_{Σ}(ℙ(E)) → Aut(ℙ(E)) → Aut(Σ),

where Aut_{Σ}(ℙ(E)) denotes the group of relative automorphisms of the bundle ℙ(E) → Σ, and Aut(Σ) is the group of automorphisms of Σ (of course, Aut(Σ) is finite if g(Σ) ≥ 2); on the other hand, the (non-trivial) homomorphic vector field Ξ_{1} = K_{1}−iJK_{1} whose real part is the Killing vector field K_{1} = Jgrad_{g}s has a nonempty zero set; since Ξ_{1} preserves the (unique) ruling M = ℙ(E) → Σ, it projects onto a holomorphic vector field on the base Σ; since Ξ_{1} has at least one zero, the induced vector field on Σ vanishes; it follows that Ξ_{1} is tangent to the ℂP^{1} fibers (equivalently, Ξ_{1} belongs to the Lie algebra of Aut_{Σ}(ℙ(E))); this shows that the kernel of the group homomorphism f : Isom_{0}(M, g) → Aut_{0}(Σ) induced by the exact sequence (77) is a non-trivial compact subgroup of Aut_{Σ}(ℙ(E)); one can therefore find an S^{1} in the connected component of the identity of Aut_{Σ}(ℙ(E)); denote by Ξ_{0} the induced holomorphic vector field, such that the imaginary part K_{0} = Im(Ξ_{0}) generates the S^{1}-action, whereas Ξ_{0} itself generates a ℂ^{∗}-action; as a matter of fact, Ξ_{0} can be identified to a traceless holomorphic section of End(E), say s; note that s is of constant determinant; since K_{0} = Im(Ξ_{0}) generates a periodic S^{1}-action, s must be diagonalizable; this shows that we have a holomorphic splitting of E into eigensubbundles of s; by twisting by a line bundle, we obtain the splitting E = O ⊕ L, where deg(L) ≥ 0; then, Ξ_{0} is nothing but the Euler vector field of L. If the degree of L is zero, then any Kähler class contains a locally symmetric Kähler metric [49], so that any extremal Kähler metric on (M, J) is of constant scalar curvature [14], a contradiction; we thus obtain a splitting E = O ⊕ L where L is a holomorphic line bundle of deg(L) > 0.

As Isom_{0}(M, g) is a maximal compact subgroup in Aut_{0}(M, J), we may assume [48] that (up to a biholomorphism) the metric g is invariant under the fixed S^{1} action generated by K_{0} = Im(Ξ_{0}). For any non-trivial Killing vector field, K, which arises from a real holomorphic potential, the argument already used above shows that Ξ = K − iJK must be tangent to the fibers, and therefore Ξ ∧ Ξ_{0} = 0. In particular, we get that K_{0} = fK_{1} + hJK_{1}, where f, h are smooth functions defined on an open dense subset of M where K_{1} = Jgrad_{g}s ≠ 0. But ⟨K_{0}, JK_{1}⟩ = −ds(K_{0}) = −L_{K_{0}}s = 0, i.e. h = 0, and therefore f is a constant. By rescaling the metric if necessary we may assume therefore K_{1} = K_{0} = Im(Ξ_{0}). Similarly, K_{2} = Jgrad_{g}p must be a constant multiple of K_{1} and by Theorem 4, g must be locally of cohomogeneity one, i.e., g can be written of the form (67) on an open dense subset of M.

Note that M contains exactly two curves fixed by the ℂ^{∗}-action generated by Ξ_{0}, corresponding to the zero and infinity sections, C_{0} and C_{∞}, of M = ℙ(O ⊕ L); moreover, the function z appearing in (67) makes sense on the whole of M as being a momentum map of the corresponding S^{1}-action (up to multiplication by a non-zero constant); it then follows that z : M → ℝ maps M onto an interval [a, b], such that z is regular on M −(z^{−1}(a)∪z^{−1}(b)); therefore, for any t_{0} ∈ (a, b), Σ = z^{−1}(t_{0})/S^{1}, whereas C_{0} = z^{−1}(a) and C_{∞} = z^{−1}(b) (see [37]). By using an argument from [37, p.42], it is shown that q = |K_{1}|^{2} is a smooth function on Σ × [a, b], which satisfies the boundary conditions

(78)  q(., a) = q(., b) = 0; (\frac{∂}{∂z} q)(., a) = −(\frac{∂}{∂z} q)(., b) = k,

where k is a real constant; for Calabi’s metrics (67) one has q = \frac{V (z)}{z}, so that the equations (78) read

V (a) = V (b) = 0; V′(a) = ka; V′(b) = −kb;

we thus obtain the following values for the coefficients A_{1}, A_{2}, A_{3}, A_{4} of V (notations of (61)):

\begin{matrix}A_{1} = \frac{k(a + b) + ε(a − b)}{(b − a)(a^{2} + 4ab + b^{2})}, \\ A_{2} = \frac{−k(a^{2} + b^{2}) + 2ε(b^{2} − a^{2})}{(b − a)(a^{2} + 4ab + b^{2})}, \\ A_{3} = \frac{ab(−k(a^{2} + b^{2}) + 2ε(b^{2} − a^{2}))}{(b − a)(a^{2} + 4ab + b^{2})}, \\ A_{4} = \frac{a^{2}b^{2}(k(a + b) + ε(a − b))}{(b − a)(a^{2} + 4ab + b^{2})}; \end{matrix} ; (79)

according to Proposition 14 we have A_{3} = 0, and from (79) we also get A_{2} = 0; by (63) and (62) it follows that the Ricci tensor of g has two distinct eigenvalues, equal to \frac{s}{6} and \frac{s}{3} respectively; by Proposition 6, Ric_{0} nowhere vanishes on M, meaning that the scalar curvature s nowhere vanishes as well; since K_{1} is a nontrivial Killing vector field, s must be everywhere positive; this shows that the first Chern class c_{1}(M) of (M, J) is positive, and therefore c_{1}^{2}(M) > 0, a contradiction [11]. □

Remark 7. The case (ii) of Theorem 5 includes in particular ruled surfaces ℙ(E) over a Riemann surface Σ of genus g ≥ 1 when the holomorphic vector bundle E is stable or polystable, cf. e.g. [49].

5. Weakly selfdual almost Kähler manifolds

5.1. The Matsumoto-Tanno identity for almost Kähler 4-manifolds. Recall that an almost Kähler manifold is an almost Hermitian manifold (M, g, J, ω) for which the Kähler 2-form ω is closed. The almost complex structure J of an almost Kähler manifold is not integrable in general; if it is, we obtain a Kähler manifold.

We would like to identify the Ricci tensor Ric of an almost Kähler manifold with a 2-form ρ, the Ricci form, as in the Kähler case. However, only the Jinvariant part of Ric defines a 2-form, whereas on a (non-integrable) almost Kähler manifold, the Ricci tensor is not in general J-invariant. We shall therefore impose J-invariance as an extra requirement.

Throughout this section we will always assume that (M, g, J) is an almost Kähler 4-manifold whose Ricci tensor is J-invariant, i.e., Ric(J·, J·) = Ric(·, ·). We then adopt the notations of Section 1, and, in analogy with the Kähler case, we consider the type (1, 1) Ricci form, ρ, of (M, g, J) defined by ρ(·, ·) = Ric(J·, ·); the anti-selfdual part of ρ is denoted by ρ_{0}. It is a remarkable fact [22] that even though J is not integrable, ρ is still a closed (1,1)-form (although it is no longer a representative of \frac{1}{2π} c_{1}^{ℝ}). Using this observation, the proof of Lemma 1 easily extends to the case of almost Kähler 4-manifolds with J-invariant Ricci tensor.

Lemma 11. For any almost Kähler 4-manifold with J-invariant Ricci tensor the identity (1) is satisfied. In particular, the anti-selfdual Weyl tensor W^{−} of such a manifold is harmonic if and only if the Matsumoto-Tanno identity (2) is satisfied.

Proof. The proof follows the one given in the Kähler case, with slight modifications in places where the non-integrability of J must be taken into account: the Cotton-York tensor is now written as

\begin{aligned}C_{X,Y} (Z) = & − \frac{1}{2}((∇_{X}ρ)(Y, JZ) − (∇_{Y} ρ)(X, JZ)) \\ & − \frac{1}{2}(ρ(Y, (∇_{X}J)(Z)) − ρ(X, (∇_{Y} J)(Z))) \\ & + \frac{1}{2}(ds(X)⟨Y, Z⟩ − ds(Y)⟨X, Z⟩). \end{aligned} ; (80)

Since ρ closed [22], we have

(81)  (∇_{X}ρ)(Y, JZ) − (∇_{Y} ρ)(X, JZ) = −(∇_{JZ}ρ)(X, Y).

As an algebraic object, ∇_{X}J is a skew-symmetric endomorphism of T M, associated (by g-duality) to the section ∇_{X}ω of the bundle of J-anti-invariant 2-forms; it then anti-commutes with J, and commutes with any skew-symmetric endomorphism associated to a section of Λ^{−}M; in particular, ∇_{X}J commutes with the endomorphism corresponding to ρ_{0} via the metric (which will be still denoted by ρ_{0}). We thus obtain

\begin{aligned}ρ(Y, (∇_{X}J)(Z)) − ρ(X, (∇_{Y} J)(Z)) & = \frac{3s}{2}((∇_{Y} ω)(X, JZ) − (∇_{X}ω)(Y, JZ)) \\ & + (∇_{X}ω)(Y, ρ_{0}(Z)) − (∇_{Y} ω)(X, ρ_{0}(Z)). \end{aligned}

By using the closedness of ω we derive

\begin{aligned}ρ(Y, (∇_{X}J)(Z)) − ρ(X, (∇_{Y} J)(Z)) = & \frac{3s}{2}(∇_{JZ}ω)(X, Y) \\ & − (∇_{ρ_{0}(Z)}ω)(X, Y). \end{aligned} ; (82)

Substituting (81) and (82) in (80), we finally get

(83)  ∇_{Z}ρ_{0} = −\frac{3}{2}ds(Z)ω − 2C(JZ) − ∇_{Ric_{0}(Z)}ω + ds ∧ JZ^{♭},

The Λ^{−}M-component of (83) gives the identity (1); the last part of the lemma is immediate. □

It follows that Lemma 4 and hence Proposition 1 remain true for weakly selfdual Kähler surfaces, so that on the open set M_{0} where ρ_{0} ≠ 0 the almost Hermitian structure (ḡ = λ^{−2}g, I) defined on M_{0} by ρ_{0} = λω_{I} (see Lemma 2) is Kähler.

The theory of hamiltonian 2-forms φ does not extend automatically to the almost Kähler case: there is no reason, in general, to suppose that the trace and pfaffian of φ are Poisson-commuting holomorphic potentials, nor can we appeal to the open mapping theorem when J is not integrable. On the other hand Proposition 3 does generalize in the following sense: if we write σ = ξ + η and π = ξη, then dξ and dη are orthogonal and Jdσ = 2Idλ on the closure of M_{0}.

Fortunately, when φ is the Ricci form, we can show more.

Lemma 12. On a weakly selfdual almost Kähler 4-manifold with J-invariant Ricci tensor, K = Jgrad_{g}s is a Killing vector field.

Proof. On M_{0}, the Ricci tensors of both ḡ = λ^{−2}g and g are I-invariant, and therefore Igrad_{g}λ is Killing vector field (with respect to both metrics) [7]. Since Ids = 2Jdλ, Jds = 2Idλ and K is a Killing vector field on M_{0}. On the other hand if λ vanishes identically on an open set U then g is Einstein on U, so that s is constant, and K is a trivial Killing vector field. Hence by continuity K is a Killing vector field everywhere. □

Because of this observation, it is natural to strengthen the definition of hamiltonian in the almost Kähler case: we say that a closed J-invariant 2-form φ on an almost Kähler 4-manifold is hamiltonian if its trace-free part φ_{0} is a twistor 2-form and its trace σ is a momentum map for a Killing vector field.

Note that for any Killing vector field K, ∇_{X}(∇K) = R_{K,X}^{g} and so the 1-jet {K, ∇K} is parallel with respect to a globally defined connection, cf. [34]. Hence if K vanishes on an open set so does ∇K, and therefore K vanishes on any connected component meeting that open set. It follows that if φ is hamiltonian, the open set M_{0} where φ_{0} ≠ 0 is dense or empty in each connected component.

In particular, we obtain the following generalization of Proposition 6.

Proposition 15. Let (M, g, J, ω) be a connected weakly selfdual almost Kähler 4-manifold with J-invariant Ricci tensor. Then one of the following holds:

(i) ρ_{0} is identically zero; then, (g, J, ω) is an Einstein almost Kähler 4manifold; or

(ii) the scalar curvature s of g is constant, but ρ_{0} is not identically zero; then, (g, J) is obtained from a Kähler surface (g, I) with two distinct constant principal Ricci curvatures, λ and μ, in the following manner: J equals to I on the λ-eigenspace of the Ricci tensor, but to −I on the μ-eigenspace; hence I is compatible with the opposite orientation of (M, J); or

(iii) s is not constant and g is selfdual; or (iv) W^{−} and ρ_{0} have no zero; then, (ḡ = λ^{−2}g, I) is a globally defined extremal Kähler metric of non-constant scalar curvature, which is compatible with the opposite orientation of (M, J).

5.2. Weak selfduality, hamiltonian 2-forms, and a conjecture of Goldberg. The existence of non-integrable almost Kähler 4-manifolds listed in (i)-(iv) of Proposition 15 appears to be a non-trivial problem. We collect below some remarks and known results on this issue:

• A long-standing conjecture of Goldberg [28, 46] states that a compact Einstein almost Kähler manifold must be a Kähler-Einstein manifold. The first local examples of non-integrable Einstein almost Kähler 4-manifolds have been recently discovered in [43, 9]; we shall provide new examples (see Example 2 below), but for all these examples the Ricci tensor and the anti-selfdual Weyl tensor identically vanish.

• The almost Kähler 4-manifolds described in Proposition 15(ii) have been recently studied in [6]. It is known that there are essentially two examples of homogeneous Kähler surfaces (M, g, I) which give rise to homogeneous nonintegrable almost Kähler 4-manifolds described in (ii) of Proposition 15; specifically, (M, g, I) is either isomorphic to (SU(2)⋉Sol_{2})/U(1) (in the case when the signature of the hermitian form is (2,2), cf. [47]), or to (Isom(𝔼^{2})⋉Sol_{2})/SO(2) (in the case when the signature of the hermitian form is (0,2), cf. [49]); see also Example 3 below. However, there are also many non-homogeneous examples [4].

• We believe that there should exist non-integrable examples of almost Kähler 4-manifolds described in Proposition 15(iii) and (iv). However, as we discuss below, it is not clear how to generalize the constructions of Sections 2 and 3 to obtain such examples. According to Proposition 4, for a weakly selfdual Kähler surface, not only the

scalar curvature, but also the pfaffian of the normalized Ricci form is a momentum map for a Killing vector field. Indeed, this much holds for the trace and the pfaffian of the normalized 2-form associated to any hamiltonian 2-form, by Proposition 2.

Hence one approach to generalize the constructions of Sections 2 and 3 to the almost Kähler case is to study hamiltonian 2-forms such that the pfaffian is also a momentum map for a Killing vector field. The following Lemma shows that there are no non-integrable examples with linearly independent Killing vector fields, which means that there is no direct generalization of the constructions of Section 2.

Lemma 13. For an almost Kähler 4-manifold with a hamiltonian 2-form φ, the pfaffian π of associated normalized 2-form φ̃ is a momentum map for a Killing vector field K̃ = Jgrad_{g}π if and only if either (g, J) is a Kähler or K̃ is a constant multiple of K = Jgrad_{g}σ.

Proof. Since π = \frac{1}{4}σ^{2} − λ^{2} we have dπ = \frac{1}{2}σdσ + φ_{0}(Jdσ). Straightforward calculation gives

∇(Jdπ) = dσ ∧ Jdσ − \frac{1}{2}|dσ|^{2}ω + \frac{1}{2}σ∇(Jdσ) − φ_{0} ∘ ∇dσ.

Since K = Jgrad_{g}σ is Killing by assumption, it follows that Jgrad_{g}π is Killing if and only if φ_{0} ∘∇dσ is skew. This is automatic on the open set where φ_{0} vanishes, where dσ = 0 and hence dπ = 0. Therefore we can assume φ_{0} is nonvanishing and write φ_{0} = λω_{I}, where I is a complex structure of the opposite orientation to J.

Now I ∘ ∇dσ is skew if and only if ∇dσ is I-invariant; since J and I commute, this means that the J-anti-invariant part of ∇dσ must be I-invariant. But K = Jgrad_{g}σ is Killing, so the J-anti-invariant part of (∇_{X}dσ)(Y) is equal to

2(∇_{X}ω)(K, Y) + (∇_{K}ω)(X, Y).

The latter is I-invariant if and only if for any vector field X we have

(84)  (∇_{IX}J)(K) = I(∇_{X}J)(K).

Suppose now that ∇_{X}J = A ≠ 0 for a vector X at some point. Since ∇_{IX}J, like A, is a J-anti-invariant endomorphism and we can write ∇_{IX}J = bA + cJA for some b, c ∈ ℝ. Equation (84) now reads

bA(K) + cJA(K) = IA(K).

Since A commutes with I and anticommutes with J, by applying A to the both sides we obtain: −bK + cJK = −IK. However K is orthogonal to both JK and IK, and so b = 0 and c = ±1. Thus, on the open set where J is non-integrable and φ_{0} ≠ 0, we have Jdσ = ±Idσ = ±2Jdλ, so dπ and dσ are linearly dependent. □

The Calabi construction in Section 3 does generalize to the almost Kähler case and generates some new examples of selfdual Ricci-flat almost Kähler 4-manifolds. However, we shall see that there are no non-integrable examples of non-constant scalar curvature.

Proposition 16. Let (M, g, J, ω) be an almost Kähler 4-manifold with J-invariant Ricci tensor and a non-vanishing hamiltonian Killing vector field K. Suppose that the pair (ḡ = λ^{−2}g, I) is Kähler, where λ is a momentum map for a nonzero multiple of K, and I is equal to J on span(K, JK), but to −J on the orthogonal complement of span(K, JK).

Then either J is integrable, or (g, ω) is given explicitly by

(85)  g = \frac{W}{z} (z^{2}g_{Σ} + dz^{2}) + \frac{z}{W} (dt + \frac{V}{z} dz + β)^{2},
(86)  ω = zW ω_{Σ} + dz ∧ (dt + \frac{V}{z} dz + β),

where g_{Σ} is a metric on 2-manifold Σ with area form ω_{Σ}, β is a 1-form on Σ with dβ = W ω_{Σ}, and V + iW is an arbitrary holomorphic function on Σ.

Conversely any such metric satisfies the above hypotheses, and J is integrable if and only if the function W is constant.

Proof. Without loss of generality, we take λ = z to be a momentum map for K (with respect to ω). As in Proposition 13, cf. LeBrun [36], we may introduce coordinates such that

g = e^{u}w(dx^{2} + dy^{2}) + w dz^{2} + w^{−1}(dt + α)^{2}
ω = e^{u}w dx ∧ dy + dz ∧ (dt + α)
ω̄ = z^{−2}(−e^{u}w dx ∧ dy + dz ∧ (dt + α))

where ω̄ is the Kähler form of I with respect to ḡ = z^{−2}g. The integrability of I together with the closedness of ω and ω̄ yields

e^{u}w = h(x, y)z,
dα = −w_{x}dy ∧ dz + w_{y}dx ∧ dz + h(x, y)dx ∧ dy

from which we obtain the integrability condition

(87)  w_{xx} + w_{yy} = 0.

The Ricci tensor of g is J-invariant if and only if

(\frac{zu_{z} − 2}{zw})_{x} = 0 and (\frac{zu_{z} − 2}{zw})_{y} = 0

so that we can write

zu_{z} = f(z) zw − 2.

Since e^{u}w = h(x, y)z, we obtain

(zw)_{z} + \frac{f(z)}{z}(zw)^{2} = 0.

The latter is explicitly integrated, and we get

z w(x, y, z) = \frac{1}{F (z) + G(x, y)}

for some functions F (z) and G(x, y). By substituting into (87) we discover that either F or G must be constant.

If G is constant, then w_{x} = w_{y} = 0, i.e., g is of Calabi type and J is integrable. Consider now the case when F is a constant; then,

w = \frac{W}{z}  and  e^{u} = z^{2}e^{U},

where W (x, y) is a positive harmonic function and U(x, y) is an arbitrary function of (x, y); the almost Kähler structure (g, ω) takes the form (85)–(86) where:

• g_{Σ} = e^{U}(dx^{2} + dy^{2});

• ω_{Σ} = e^{U}dx ∧ dy;

• W is a positive harmonic function on Σ;

• α satisfies dα = −W_{y}dx ∧ dz/z + W_{x}dy ∧ dz/z + W ω_{Σ} and we can locally choose t so that α = V dz/z + β, where V is a harmonic conjugate of W and dβ = W σ_{Σ}.

This almost Hermitian structure (g, J, ω) is almost Kähler with J-invariant Ricci tensor, since w, e^{u} and α solve the required equations. □

One directly calculates the normalized scalar curvature s of the metric (85): it is given by

s = \frac{1}{6zW e^{U}} (U_{xx} + U_{yy} + 2e^{U}).

On a weakly selfdual almost Kähler 4-manifold with J-invariant Ricci tensor, we have seen that s is a momentum map for a Killing vector field. However, s cannot be a multiple of z unless it vanishes. Hence this construction does not yield any non-integrable weakly selfdual almost Kähler metrics of Calabi type with nonconstant scalar curvature. However, it does provide the following new examples of selfdual Ricci-flat strictly almost Kähler 4-manifolds. Example 2. Let (g, J, ω) be given by (85) and suppose that s = 0; this means that U is a solution of the Liouville equation, i.e., that g_{Σ} is the standard metric on an open subset of S^{2}, while H = W + iV is a non-constant holomorphic function on Σ with positive real part. If we write z = r, we see that the metric

g = \frac{W}{r} (dr^{2} + r^{2}g_{S^{2}}) + \frac{r}{W} (dt + α)^{2}

is given by applying the Gibbons–Hawking Ansatz using the harmonic function W/r, which is invariant under dilation with weight −1 in the sense that

r \frac{∂}{∂r} (\frac{W}{r} ) = −\frac{W}{r} .

(In fact this is the natural scaling weight for W/r, since it is the Higgs field of an abelian monopole on ℝ^{3}.)

This class of Gibbons–Hawking metrics has been studied before in [16] and [18]. In addition to the triholomorphic Killing vector field \frac{∂}{∂t}, these metrics also admit a triholomorphic homothetic vector field r \frac{∂}{∂r}. Therefore, by [27], the local quotient by r \frac{∂}{∂r} is a hyperCR Einstein–Weyl space. In this case the quotient Einstein– Weyl structure was obtained explicitly in [16] and is an Einstein-Weyl space with a geodesic symmetry.

The reader is referred to these references for more information. However, to the best of our knowledge, the observation that these metrics are almost Kähler is new. Note that the Kähler form is not an eigenform of the Weyl tensor, showing that the solutions are different from the previously known examples of Nurowski– Przanovski [43] and Tod, which were obtained by applying the Gibbons–Hawking Ansatz to a translation-invariant harmonic function.

We next use the rough classification given by Proposition 15 to obtain the following partial result which motivates the further study of compact weakly selfdual almost Kähler 4-manifolds with J-invariant Ricci tensor:

Theorem 6. Suppose that there exists a compact weakly selfdual almost Kähler 4manifold (M, g, J, ω) with J-invariant Ricci tensor, for which the almost complex structure J is not integrable. Then one of the following two alternatives holds:

(i) The scalar curvature of g is a negative constant; then M admits an Einstein, non-integrable almost Kähler structure; or

(ii) (M, g, J, ω) belongs to case (iv) of Proposition 15, and the globally defined Kähler structure (ḡ, I) is isomorphic to an extremal Kähler metric which is not locally of cohomogeneity one, on a minimal ruled surface S = ℙ(O⊕ L) → Σ_{g} with g ≥ 1 and deg L > 0.

Proof. We inspect the possible compact non-integrable almost Kähler 4-manifolds given by (i)–(iv) of Proposition 15.

The case of constant scalar curvature is described by Proposition 15(i) & (ii). Our claim then follows by [46] and [6, Th.2].

Suppose that s is not constant, i.e., that K = Jgrad_{g}s is a non-trivial Killing vector field by Lemma 12. Let x_{0} ∈ M be a zero of K; then, the isotropy subgroup H(x_{0}) of the connected group of isometries of (M, g) is a compact group of dimension at least one; one can therefore take an S^{1} in H(x_{0}). Hodge theory implies that on a compact almost Kähler manifold any isometry which is homotopic to the identity inside the group of diffeomorphisms is a symplectomorphism (see e.g. [38]); hence the chosen isometric S^{1}-action is symplectic with respect to ω. Since x_{0} is a fixed point of the S^{1}-action, we obtain a hamiltonian S^{1}-action on (M, ω) [39]. The manifold is then equivariantly (and orientedly) diffeomorphic to a rational or a ruled complex surface endowed with a holomorphic circle action [10, 2, 33]. Moreover, in this case (M, g, J, ω) is given by Proposition 15 (iii) or (iv).

Consider first the case (iii). Since s is not constant, the selfdual Weyl tensor W^{+} does not vanish [5, Cor.1]. By the Chern-Weil formulae, the signature of M is strictly positive, and therefore M is diffeomorphic to ℂP^{2} [11]. Combining the results of [30] and [45], one sees that on ℂP^{2} the only selfdual conformal structure with non-trivial (conformal) Killing vector field is the standard one. Thus, modulo diffeomorphisms, we may assume that g is conformal to the standard Kähler metric (g_{0}, ω_{0}). Since ω and ω_{0} are both harmonic selfdual 2-forms on (ℂP^{2}, g_{0}), and since b^{+}(ℂP^{2})=1, we conclude that ω = const.ω_{0}, showing that J is integrable, a contradiction.

Suppose (M, g, J, ω) is as in Proposition 15(iv). Now ρ_{0} determines an integrable almost complex structure I compatible with g and with the opposite orientation of (M, J), such that (ḡ = λ^{−2}g, I) is an extremal Kähler metric with Igradḡs̄ = const.K. Denote by \overline{M} the smooth manifold M endowed with the orientation induced by I. Thus, the oriented smooth 4-manifolds M and \overline{M} both admit complex structures. Since M is the underlying smooth manifold of a rational or a ruled complex surface, we conclude as in the proof of Theorem 5 that the complex surface (\overline{M}, I) is a ruled surface of the form ℙ(E), where E → Σ_{g} is a holomorphic rank 2 bundle over a compact Riemann surface Σ_{g} of genus g, and which splits as E = O ⊕ L for a holomorphic line bundle L of degree deg (L) > 0.

We have to prove that g ≥ 1. Indeed, if g = 0, we obtain the Hirzebruch surface F_{k} = ℙ(O ⊕ O(−k)), where k is a positive integer. As we have already observed in the proof of Theorem 5, the extremal Kähler metrics on these surfaces are the Calabi cohomogeneity-one U(2)-metrics, i.e. ḡ is given by the Calabi construction (67)–(68); since g = const.s̄^{2}ḡ, it follows that g is a cohomogeneity-one metric as well and therefore gradḡs̄ = const.grad_{g}s, showing that JK = IK. Since s is not constant, by Lemma 13, J is integrable on the open dense subset where ds ≠ 0, hence everywhere. We thus conclude that g ≥ 1. Note that the above local argument applies to any extremal Kähler ḡ which is locally of cohomogeneity one, so that the last part of the theorem also follows. □

We do not have any examples in case (ii) of the above theorem. Indeed, the only examples we know of extremal Kähler metrics on the minimal ruled surfaces in (ii) are locally cohomogeneity-one Calabi-type metrics [48].

5.3. Almost Kähler 4-manifolds of constant Lagrangian sectional curvature. In this section we deduce another global result from Theorem 6. An almost Kähler 4-manifold (M, g, ω) is said to have (pointwise) constant Lagrangian sectional curvature if the sectional curvature of g, at each point of M, is constant on the set of Lagrangian 2-planes at that point—recall that the latter are the planes X ∧ Y with ω(X, Y) = 0. One can make the same definition for almost Kähler manifolds of any dimension, but for 2n > 4, any almost Kähler 2n-manifold of constant Lagrangian sectional curvature is in fact Kähler with constant holomorphic sectional curvature [23, 24]. Conversely, it is easy to see that any complex space form has constant Lagrangian sectional curvature (see e.g. [25]).

In four dimensions, the situation is more interesting. A simple local calculation (cf. e.g. [5]) shows that an almost Kähler 4-manifold has constant Lagrangian sectional curvature if and only if the Ricci tensor is J-invariant, the Weyl tensor is selfdual, and the Kähler form is one of its roots (i.e., M has Hermitian Weyl tensor in the sense of [3]).

The following homogeneous example shows that the integrability for almost Kähler 4-manifolds with constant Lagrangian sectional curvature does not follow locally (nor even for complete metrics). Example 3. Consider the homogeneous Kähler surface M = (SU(2)⋉Sol_{2})/U(1), where Sol_{2} denotes the real two-dimensional solvable subgroup of upper triangular matrices in SL_{2}(ℝ) .

We take the unique left-invariant Kähler structure (g, I) on M, determined by the property that the constant principal Ricci curvatures are equal to (−1, +1), cf. [47]. According to Proposition 15(ii), (M, g) admits an almost Kähler structure J with J-invariant Ricci tensor. Since the scalar curvature of g is zero, g is selfdual (with the orientation opposite to I) [26]; furthermore, J is not integrable [6], and by using the general formulae in [12, Ch.7] one easily checks that (M, g, J) has constant Lagrangian sectional curvature.

In contrast to this example, there are a number of reasons [5, 3] to believe that a compact almost Kähler 4-manifold of constant Lagrangian curvature must be a selfdual Kähler metric. The conjectured integrability of the almost complex structure has been proved under some additional assumptions on curvature [5] or the topology [3] of the manifold, but the general question was left open. As a consequence of Theorem 6 we are now able to give a positive answer.

Corollary 1. A compact, 4-dimensional, almost Kähler manifold has constant Lagrangian sectional curvature if and only if it is a Kähler selfdual surface.

Proof. Suppose (M, g, J, ω) is a compact almost Kähler 4-manifold of constant Lagrangian sectional curvature, but for which J is not integrable. According to [5, Th.2] the scalar curvature s of g is not constant; then, by Theorem 6, the smooth manifold \overline{M} is diffeomorphic to a minimal ruled surface. Since any such surface admits an orientation reversing involution, we conclude that M carries a complex structure as well (which is compatible with the orientation induced by ω). Then, by [3, Cor.2], the almost complex structure J must be integrable, contradicting our assumption. □

References

[1] M. Abreu, Kähler geometry of toric varieties and extremal metrics, Int. J. Math. 9 (1998), 641–651.

[2] K. Ahara and A. Hattori, Four dimensional symplectic S^{1}-manifolds admitting moment map, J. Fac. Sci. Univ. Tokyo, Sect. IA, 38 (1991), 251–298.

[3] V. Apostolov and J. Armstrong, Symplectic 4-manifolds with Hermitian Weyl tensor, Trans. Amer. Math. Soc. 352 (2000), 4501–4513.

[4] V. Apostolov, J. Armstrong, and T. Draghici, Local rigidity of certain classes of almost Kähler 4-manifolds, preprint 1999, available at arXiv: math.DG/9911197.

[5] V. Apostolov and T. Draghici, Almost Kähler 4-manifolds with J-invariant Ricci tensor and special Weyl tensor, Q. J. Math. 51 (2000), 275–294.

[6] V. Apostolov, T. Draghici, and A. Moroianu, A splitting theorem for Kähler manifolds with constant eigenvalues of the Ricci tensor, to appear in Int. J. Math., available at arXiv: math.DG/0007122.

[7] V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs Theorem, Int. J. Math. 8 (1997), 421–439.

[8] V. Apostolov and P. Gauduchon, Self-dual Einstein Hermitian 4-manifolds, preprint 2000, available at arXiv:mathDG/0003162.

[9] J. Armstrong, An Ansatz for Almost-Kähler, Einstein 4-manifolds, preprint 1997, to appear in J. reine angew. Math.

[10] M. Audin, Hamiltoniens périodiques sur les variétés symplectiques de dimension 4, in “Géométrie symplectique et mécanique”, Proceedings 1988 (ed. C. Albert), Springer Lecture Notes in Math. 1416 (1990).

[11] W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1984.

[12] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1987.

[13] R. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001) 623–715.

[14] E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Princeton Univ. Press (1982), 259–290.

[15] E. Calabi, Extremal Kähler metrics II, in: Differential Geometry and Complex Analysis (eds. I. Chavel and H.M. Farkas), Springer-Verlag, 1985.

[16] D. M. J. Calderbank and H. Pedersen, Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, Ann. Inst. Fourier 50 (2000) 921–963.

[17] D. M. J. Calderbank and H. Pedersen, Selfdual Einstein metrics with torus symmetry, Preprint, University of Edinburgh.

[18] D. M. J. Calderbank and K. P. Tod, Einstein metrics, hypercomplex structures and the Toda field equation, to appear in Diff. Geom Appl. (2001).

[19] J. Carrel, A. Howard and C. Kosniowski, Holomorphic vector fields on compact complex manifolds, Math. Ann. 204 (1973), 73–81.

[20] B.-Y. Chen, Some topological obstructions to Bochner-Kähler metrics and their applications, J. Differential Geom. 13 (1978), 574–588.

[21] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405–433.

[22] T. Draghici, On some 4-dimensional almost Kähler manifolds, Kodai Math. J. 18 (1995), 156–163.

[23] M. Falcitelli and A. Farinola, Six-dimensional almost Kähler manifolds with pointwise constant antiholomorphic sectional curvature, Riv. Mat. Univ. Parma (5) 6 (1997), 143–156.

[24] M. Falcitelli, A. Farinola, and O. Kassabov, Almost Kähler manifolds whose antiholomorphic sectional curvature is pointwise constant, Rend. Mat. Appl. Ser. VII, 18 (1998), 151–166.

[25] G. Ganchev, On Bochner curvature tensors in almost Hermitian manifolds, Pliska Studia Matematica Bulgarica 9 (1987), 33–43.

[26] P. Gauduchon, Surfaces kählériennes dont la courbure vérifie certaines conditions de positivité, in “Géométrie riemannienne en dimension 4”, Sém. A. Besse, 1978/79.

[27] P. Gauduchon and K. P. Tod, Hyperhermitian metrics with symmetry, J. Geom. Phys. 25 (1998) 291–304.

[28] S. I. Goldberg, Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100.

[29] V. Guillemin, Kähler structures on toric varieties, J. Differential Geom. 40 (1994), 285–309.

[30] M. Gursky, Conformal vector fields on four-manifolds with negative scalar curvature, Math. Z. 232 (1999), 265–273.

[31] A. Hwang and G. Maschler, Central Kähler metrics with non-constant central curvature, Preprint 2000.

[32] W. Jelonek, Compact Kähler surfaces with δW^{−} = 0, Preprint 2000.

[33] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds in “Contact and symplectic geometry”, (Cambridge, 1994), 43–47, Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996.

[34] B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemann manifold, Trans. Amer. Math. Soc. 80 (1955), 528–542.

[35] D. Kotschick, Orientations and geometrisations of compact complex surfaces, Bull. London Math. Soc. 29, No. 2 (1997), 145–149.

[36] C. R. LeBrun, Explicit self-dual metrics on ℂP^{2}# ⋯ #ℂP^{2}, J. Differential Geom. 34 (1991), 223–253.

[37] C. R. LeBrun and M. Singer, Existence and deformation theory of scalar flat Kähler metrics on compact complex surfaces, Inv. Math. 112 (1993], 273–313.

[38] A. Lichnerowicz, Théorie des groupes de transformations, Dunod, Paris, 1958.

[39] D. McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), 149–160.

[40] M. Maruyama, On automorphism groups of ruled surfaces, J. Math. Koyoto Univ. 11 (1971), 89–112.

[41] G. Maschler, Central Kähler metrics, Preprint 2000.

[42] M. Matsumoto and S. Tanno, On Kähler spaces with parallel or vanishing Bochner curvature tensor, Tensor N. S. 27 (1973), 291–294.

[43] P. Nurowski and M. Przanowski, A four-dimensional example of Ricci flat metric admitting almost Kähler non-Kähler structure, Classical Quantum Gravity 16 (1999), no. 3, L9–L13.

[44] M. Pontecorvo, On twistor spaces of anti-self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 331 (1992), 653–661.

[45] Y.-S. Poon, Compact self-dual manifolds with positive scalar curvature, J. Differential Geom. 24 (1986), 97–132.

[46] K. Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 36 (1987), 677–684.

[47] H. Shima, On homogeneous Kähler manifolds with non-degenerate canonical Hermitian form of signature (2, 2(n − 1)), Osaka J. Math. 10 (1973), 477–493.

[48] C. Tønnesen-Friedman, Extremal Kähler metrics on minimal ruled surfaces, J. reine angew. Math. 502 (1998), 175–197.

[49] C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), 119–153.

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

E-mail address: apostolo@math.uqam.ca

David M. J. Calderbank, Department of Mathematics and Statistics, University of Edinburgh, King’s Building, Mayfield’s Road, Edinburgh EH9 3JZ, Scotland

E-mail address: davidmjc@maths.ed.ac.uk

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

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