On Kähler metrisability of two-dimensional complex projective structures

1 Introduction

Recall that an equivalence class of affine torsion-free connections on the tangent bundle of a smooth manifold \(N\) is called a (real) projective structure [11, 38, 39]. Two connections \(\nabla\) and \(\nabla^{\prime}\) are projectively equivalent if they share the same unparametrised geodesics. This condition is equivalent to \(\nabla\) and \(\nabla^{\prime}\) inducing the same parallel transport on the projectivised tangent bundle \(\mathbb{P}TN\).

It is a natural task to (locally) characterise the projective structures arising via the Levi-Civita connection of a (pseudo-)Riemannian metric. R. Liouville [25] made the crucial observation that the Riemannian metrics on a surface whose Levi-Civita connection belongs to a given projective class precisely correspond to nondegenerate solutions of a certain projectively invariant finite-type linear system of partial differential equations. In [3] Bryant, Eastwood and Dunajski used Liouville’s observation to solve the two-dimensional version of the Riemannian metrisability problem. In another direction it was shown in [29] that on a surface locally every affine torsion-free connection is projectively equivalent to a conformal connection (see also [28]). Local existence of a connection with skew-symmetric Ricci tensor in a given projective class was investigated in [36] (see also [23] for a connection to Veronese webs). Liouville’s result generalises to higher dimensions [30] and the corresponding finite-type differential system was prolonged to closed form in [14, 30]. Several necessary conditions for Riemann metrisability of a projective structure in dimensions larger than two were given in [33]. See also [7, 16] for the role of Einstein metrics in projective geometry.

Now let \(M\) be a complex manifold of complex dimension \(d>1\) with integrable almost complex structure map \(J\). Two affine torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) which preserve \(J\) are called complex projectively equivalent if they share the same generalised geodesics (for the notion of a curved complex projective structure on Riemann surfaces see [5]). A generalised geodesic is a smoothly immersed curve \(\gamma \subset M\) with the property that the \(2\)-plane spanned by \(\dot{\gamma}\) and \(J \dot\gamma\) is parallel along \(\gamma\). Complex projective geometry was introduced by Otsuki and Tashiro [35, 37]. Background on the history of complex projective geometry and its recently discovered connection to Hamiltonian \(2\)-forms (see [1] and references therein) may be found in [26].

In the complex setting it is natural to study the Kähler metrisability problem, i.e. try to (locally) characterise the complex projective structures which arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Similar to the real case, the Kähler metrics whose Levi-Civita connection belongs to a given complex projective class precisely correspond to nondegenerate solutions of a certain complex projectively invariant finite-type linear system of partial differential equations [12, 26, 31].

In this note we prolong the relevant differential system to closed form in the surface case. In doing so we obtain necessary conditions for Kähler metrisability of a complex projective structure \([\nabla]\) on a complex surface and show in particular that the generic complex projective structure is not Kähler metrisable. Furthermore we show that the space of Kähler metrics compatible with a given complex projective structure is algebraically constrained by the complex projective Weyl curvature of \([\nabla]\). We also show that the (pseudo-)Kähler metrics defined on some domain in \(\mathbb{CP}^2\) which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms on \(\mathbb{C}^3\) whose rank is at least two. A result whose real counterpart is a well-known classical fact. This note concerns itself with the complex 2-dimensional case, but there are obvious higher dimensional generalisations that can be treated with the same techniques.

The reader should be aware that the results presented here can also be obtained by using the elegant and powerful theory of Bernstein–Gelfand–Gelfand (BGG) sequences developed by Čap, Slovák and Souček [10] (see also the article of Calderbank and Diemer [6]). In particular, the prolongation computed here is an example of a prolongation connection of a first BGG equation in parabolic geometry and may be derived using the techniques developed in [18].

This note aims at providing an intermediate analysis between the abstract BGG machinery and pure local coordinate computations. This is achieved by carrying out the computations on the parabolic Cartan geometry of a complex projective surface.

Acknowledgments

The author was supported by Forschungsinstitut für Mathematik (FIM) at ETH Zürich and Schweizerischer Nationalfonds SNF via the postdoctoral fellowship PA00P2_142053. The author is grateful to Friedrich-Schiller-Universität Jena for financial support for several trips to Jena where a part of the writing for this article took place and to V. S. Matveev and S. Rosemann for introducing him to the subject of complex projective geometry through many stimulating discussions. The author also would like to thank R. L. Bryant for sharing with him his notes [4] which contain the proofs of the counterparts for real projective surfaces of Theorem 3.4 and Corollary 3.5. Furthermore, the author also would like to thank the referees for several valuable comments.

2 Complex projective surfaces

2.1 Definitions

Let \(M\) be a complex \(2\)-manifold with integrable almost complex structure map \(J\) and \(\nabla\) an affine torsion-free connection on \(TM\). We call \(\nabla\) complex-linear if \(\nabla J=0\). A generalised geodesic for \(\nabla\) is a smoothly immersed curve \(\gamma \subset M\) with the property that the \(2\)-plane spanned by \(\dot{\gamma}\) and \(J \dot\gamma\) is parallel along \(\gamma\), i.e. \(\gamma\) satisfies the reparametrisation invariant condition \[ \nabla_{\dot\gamma}\dot\gamma\wedge \dot\gamma\wedge J\dot\gamma=0.\](2.1) We call two complex linear torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(M\) complex projectively equivalent, if they have the same generalised geodesics. An equivalence class of complex projectively equivalent connections is called a complex projective structure and will be denoted by \([\nabla]\). A complex \(2\)-manifold equipped with a complex projective structure will be called a complex projective surface.

Extending \(\nabla\) to the complexified tangent bundle \(T^{\mathbb{C}}M \to M\), it follows from the complex linearity of \(\nabla\) that for every local holomorphic coordinate system \(z=(z^i) : U \to \mathbb{C}^2\) on \(M\) there exist unique complex-valued functions \(\Gamma^i_{jk}\) on \(U\), so that \[\nabla_{\partial_{z^j}}\partial_{z^k}=\Gamma^i_{jk}\partial_{z^i}.\] We call the functions \(\Gamma^i_{jk}\) the complex Christoffel symbols of \(\nabla\). Tashiro showed [37] that two torsion-free complex linear connections \(\nabla\) and \(\nabla^{\prime}\) on \(M\) are complex projectively equivalent if and only if there exists a \((1,\! 0)\)-form \(\beta \in \Omega^{1,0}(M,\mathbb{R})\) so that \[ \nabla^{\prime}_ZW-\nabla_ZW=\beta(Z)W+\beta(W)Z\](2.2) for all \((1,\! 0)\) vector fields \(Z,W \in \Gamma(T^{1,0}M)\). In analogy to the real case one can use (2.2) to show that \(\nabla\) and \(\nabla^{\prime}\) are complex projectively equivalent if and only if they induce the same parallel transport on the complex projectivised tangent bundle \(\mathbb{P}T^{1,0}M\).

Writing \(\Gamma^i_{jk}\) and \(\hat{\Gamma}^i_{jk}\) for the complex Christoffel symbols of \(\nabla\) and \(\nabla^{\prime}\) with respect to some holomorphic coordinates \(z=(z^i)\) and \(\beta=\beta_i \mathrm{d}z^i\), equation (2.2) translates to \[ \hat{\Gamma}^i_{jk}=\Gamma^i_{jk}+\delta^i_j\beta_k+\delta^i_k\beta_j.\](2.3) Note that formally equation (2.3) is identical to the equation relating two real projectively equivalent connections on a real manifold. In particular, similarly to the real case (see [11, 38]), the functions \[ \Pi^i_{jk}=\Gamma^i_{jk}-\frac{1}{3}\left(\Gamma^l_{lj}\delta^i_k+\Gamma^l_{lk}\delta^i_j\right)\](2.4) are complex projectively invariant in the sense that they only depend on the coordinates \(z\). Moreover locally \([\nabla]\) can be recovered from the functions \(\Pi^i_{jk}\) and two torsion-free complex linear connections are complex projectively equivalent if and only if they give rise to the same functions \(\Pi^i_{jk}\) in every holomorphic coordinate system.

A complex projective structure \([\nabla]\) is called holomorphic if the functions \(\Pi^i_{jk}\) are holomorphic in every holomorphic coordinate system. Gunning [17] obtained relations on characteristic classes of complex manifolds carrying holomorphic projective structures. The condition on a manifold to carry a holomorphic projective structure is particularly restrictive in the case of compact complex surfaces. See also the beautiful twistorial interpretation of holomorphic projective surfaces by Hitchin [19] and 8.

2.2 Cartan geometry

A complex projective structure admits a description in terms of a normal Cartan geometry modelled on complex projective space \(\mathbb{CP}^n\), following the work of Ochiai [34]: see  [20] and [40]. The reader unfamiliar with Cartan geometries may consult [9] for a modern introduction. We will restrict to the construction in the complex two-dimensional case.

Let \(\mathrm{PSL}(3,\mathbb{C})\) act on \(\mathbb{CP}^2\) from the left in the obvious way and let \(P\) denote the stabiliser subgroup of the element \([1,0,0]^t \in \mathbb{CP}^2\). We have:

Let \((\pi : B \to M,\theta)\) be the Cartan geometry of a complex projective structure \((J,[\nabla])\) on a simply-connected surface \(M\) whose Cartan connection satisfies \(\mathrm{d}\theta+\theta\wedge\theta=0\). Then there exists a local diffeomorphism \(\Phi : B \to \mathrm{PSL}(3,\mathbb{C})\) pulling back the Maurer-Cartan form of \(\mathrm{PSL}(3,\mathbb{C})\) to \(\theta\) and consequently, a local biholomorphism \(\varphi : M \to \mathbb{CP}^2\) identifying the projective structure on \(M\) with the standard flat structure on \(\mathbb{CP}^2\).

2.3 Bianchi-identities

Theorem 2.1 implies that the curvature form \(\Theta=\mathrm{d}\theta+\theta\wedge\theta\) satisfies \[ \Theta=\mathrm{d}\theta+\theta\wedge\theta=\left(\begin{array}{ccc} 0 & \Theta^0_1 & \Theta^0_2\\ 0 & \Theta^1_1 & \Theta^1_2\\ 0 & \Theta^2_1 & \Theta^2_2\end{array}\right)\](2.6) with \[\Theta^0_i=L_i\theta^1_0\wedge\theta^2_0+K_{il\bar\jmath}\theta^l_0\wedge\overline{\theta^{\jmath}_0}, \quad \Theta^i_k=W^i_{kl\bar\jmath}\theta^l_0\wedge\overline{\theta^\jmath_0}\] for unique complex-valued functions \(L_i,K_{il\bar\jmath}\), and \(W^i_{kl\bar\jmath}\) on \(B\) satisfying \(W^l_{li\bar \jmath}=0\). Note that by construction, with respect to local holomorphic coordinates \(z=(z^i)\), we obtain \[ (\sigma_z)^*W^i_{kl\bar\jmath}=-\frac{\partial \Pi^i_{kl}}{\partial \bar z^j}.\](2.7)

Differentiation of the structure equations (2.6) gives \[0=\mathrm{d}^2\theta^i_0=W^i_{lk\bar\jmath}\theta^l_0\wedge\theta^k_0\wedge\overline{\theta^\jmath_0}, \quad \text{and} \quad 0=\mathrm{d}^2 \theta^0_0=K_{ik\bar\jmath}\theta^i_0\wedge\theta^k_0\wedge\overline{\theta^\jmath_0}\] which yields the algebraic Bianchi-identities \[W^i_{lk\bar\jmath}=W^i_{kl\bar\jmath}, \quad \text{and} \quad K_{ik\bar\jmath}=K_{ki\bar\jmath}.\]

2.3.1 Complex projective Weyl curvature

The identities \(\mathrm{d}^2\theta^i_k=0\) yield \[\kappa^i_{kl\bar\jmath}\wedge\theta^l_0\wedge\overline{\theta^{\jmath}_0}=0\] with \[\kappa^i_{kl\bar\jmath}=\mathrm{d}W^i_{kl\bar\jmath}+W^i_{kl\bar\jmath}\left(\theta^0_0+\overline{\theta^0_0}\right)+K_{kl\bar\jmath}\theta^i_0-W^i_{ls\bar\jmath}\theta^s_k-W^i_{ks\bar\jmath}\theta^s_l+W^s_{kl\bar\jmath}\theta^i_s-W^i_{kl\bar s}\overline{\theta^s_l}\] which implies that there exist complex-valued functions \(W^i_{kl\bar\jmath\bar s}\) and \(W^i_{kl\bar\jmath s}\) on \(B\) satisfying \[W^i_{kl\bar\jmath\bar s}=W^i_{lk\bar\jmath\bar s}=W^i_{kl\bar s\bar\jmath}, \quad W^k_{kl\bar\jmath\bar s}=W^k_{kl\bar\jmath s}=0, \quad W^i_{kl\bar\jmath s}=W^i_{lk\bar\jmath s}\] such that \[ \mathrm{d}W^i_{kl\bar\jmath}=\left(W^i_{kl\bar\jmath s}+\delta^i_kK_{sl\bar\jmath}+\delta^i_lK_{sk\bar\jmath}-3\delta^i_sK_{kl\bar\jmath}\right)\theta^s_0+W^i_{kl\bar\jmath\bar s}\overline{\theta^s_0}+\varphi^i_{kl\bar\jmath}\](2.8) where \[ \varphi^i_{kl\bar\jmath}=-W^i_{kl\bar\jmath}\left(\theta^0_0+\overline{\theta^0_0}\right)+W^i_{ls\bar\jmath}\theta^s_k+W^i_{ks\bar\jmath}\theta^s_l-W^s_{kl\bar\jmath}\theta^i_s+W^i_{kl\bar s}\overline{\theta^s_j}.\](2.9)

Let \(\mathrm{End}_0(TM,J)\) denote the bundle whose fibre at \(p \in M\) consists of the \(J\)-linear endomorphisms of \(T_pM\) which are complex-traceless. It follows with the structure equations (2.6),(2.8),(2.9) and straightforward computations, that there exists a unique \((1,\!1)\)-form \(W\) on \(M\) with values in \(\mathrm{End}_0(TM,J)\) for which \[W\left(\frac{\partial }{\partial z^l},\frac{\partial }{\partial \overline{z^{\jmath}}}\right)\frac{\partial }{\partial z^k}=(\sigma_z)^*W^i_{kl\bar\jmath}\frac{\partial }{\partial z^i}=-\frac{\partial \Pi^i_{kl}}{\partial \bar z^j}\frac{\partial }{\partial z^i}\] in every local holomorphic coordinate system \(z=(z^i)\) on \(M\). Here, as usual, we extend tensor fields on \(M\) complex multilinearly to the complexified tangent bundle of \(M\). The bundle-valued \(2\)-form \(W\) is called the complex projective Weyl curvature of \([\nabla]\). We obtain:

2.3.2 Complex projective Liouville curvature

From \(\mathrm{d}^2\theta^0_i\wedge\overline{\theta^1_0}\wedge\overline{\theta^2_0}=0\) one sees after a short computation that \[ \mathrm{d}L_i=-4L_i\theta^0_0+L_j\theta^j_i+L_{ij}\theta^j_0+L_{i\bar\jmath}\overline{\theta^{\jmath}_0}\](2.10) for unique complex-valued functions \(L_{i\bar\jmath},L_{ij}\) on \(B\). Using this last equation it is easy to check that the \(\pi\)-semibasic quantity \[ (L_1\theta^1_0+L_2\theta^2_0)\otimes \left(\theta^1_0\otimes \theta^2_0\right)\](2.11) is invariant under the \(P\) right action and thus the \(\pi\)-pullback of a tensor field \(\lambda\) on \(M\) which is called the complex projective Liouville curvature (see the note of R. Liouville [24] for the construction of \(\lambda\) in the real case).

The differential Bianchi-identity (2.8) implies that if the functions \(W^i_{kl\bar\jmath}\) vanish identically, then the functions \(K_{ik\bar\jmath}\) must vanish identically as well. We have thus shown:

2.3.3 Further identities

We also obtain \[0=\mathrm{d}^2\theta^0_i=\kappa_{ik\bar\jmath}\wedge\overline{\theta^{\jmath}_0}\wedge\theta^k_0\] with \[\begin{aligned} \kappa_{ik\bar\jmath}=&-\mathrm{d}K_{ik\bar\jmath}+\frac{1}{2}\varepsilon_{sk}L_{i\bar\jmath}\theta^s_0-K_{ik\bar\jmath}\left(2\theta^0_0+\overline{\theta^0_0}\right)+K_{sk\bar\jmath}\theta^s_i+K_{si\bar\jmath}\theta^s_k-\\ &-W^s_{ik\bar\jmath}\theta^0_s+K_{ik\bar s}\overline{\theta^s_j}. \end{aligned}\] It follows that there are complex-valued functions \(K_{ik\bar\jmath l}\) and \(K_{kl\bar\imath\bar\jmath}\) on \(B\) satisfying \[K_{ik\bar\jmath l}=K_{ki \bar\jmath l}, \quad \text{and} \quad K_{kl\bar \imath \bar\jmath}=K_{lk\bar \imath \bar\jmath}=K_{kl \bar\jmath\bar \imath}\] such that \[ \mathrm{d}K_{ik\bar\jmath}=\left(K_{ik\bar\jmath s}+\frac{1}{4}\left(\varepsilon_{sk}L_{i\bar\jmath}+\varepsilon_{si}L_{k\bar\jmath}\right)\right)\theta^s_0+K_{ik\bar\jmath\bar s}\overline{\theta^s_0}+\varphi_{ik\bar\jmath}\](2.12) where \[\varphi_{ik\bar\jmath}=-K_{ik\bar\jmath}\left(2\theta^0_0+\overline{\theta^0_0}\right)+K_{sk\bar\jmath}\theta^s_{i}+K_{si\bar\jmath}\theta^s_k-W^s_{ik\bar\jmath}\theta^0_s+K_{ik\bar s}\overline{\theta^s_j}.\]

2.4 Complex and generalised geodesics

It is worth explaining how the generalised geodesics of \([\nabla]\) appear in the Cartan geometry \((\pi : B \to M,\theta)\). To this end let \(G\subset P\subset \mathrm{PSL}(3,\mathbb{C})\) denote the quotient group of the group of upper triangular matrices of unit determinant modulo its center. The quotient \(B/G\) is the total space of a fibre bundle over \(M\) whose fibre \(P/G\) is diffeomorphic to \(\mathbb{CP}^1\). In fact, \(B/G\) may be identified with the total space of the the complex projectivised tangent bundle \(\tau : \mathbb{P}(T^{1,0}M) \to M\) of \((M,J)\). Writing \(\theta=(\theta^i_j)_{i,j=0..2}\), Theorem 2.1 implies that the real codimension \(4\)-subbundle of \(TB\) defined by \(\theta^2_0=\theta^2_1=0\) descends to a real rank \(2\) subbundle \(E\subset T\mathbb{P}(T^{1,0}M)\). The integral manifolds of \(E\) can most conveniently be identified in local coordinates. Let \(z=(z^1,z^2) : U \to \mathbb{C}^2\) be a local holomorphic coordinate system on \(M\) and write \(\phi\) for the pullback of \(\theta\) with the unique section \(\sigma_z\) associated to \(z\) in Theorem 2.1. We obtain a local trivialisation of Cartan’s bundle \[\varphi : U \times P \to \pi^{-1}(U)\] so that for \((z,p) \in U\times P\) we have \[ (\varphi^*\theta)_{(z,p)}=\left(\omega_P\right)_p+\mathrm{Ad}(p^{-1})\circ \phi_z\](2.13) where \(\omega_P\) denotes the Maurer-Cartan form of \(P\) and \(\mathrm{Ad}\) the adjoint representation of \(\mathrm{PSL}(3,\mathbb{C})\). Consider the Lie group \(\tilde{P}\subset \mathrm{SL}(3,\mathbb{C})\) whose elements are of the form \[ \left(\begin{array}{cc} \det a^{-1} & b \\ 0 & a \end{array}\right)\](2.14) for \(a\in \mathrm{GL}(2,\mathbb{C})\) and \(b^{t} \in \mathbb{C}^2\). By construction, the elements of \(P\) are equivalence classes of elements in \(\tilde{P}\) where two elements are equivalent if they differ by scalar multiplication with a complex cube root of \(1\). The canonical projection \(\tilde{P} \to P\) will be denoted by \(\nu\). Note that a piece \(N\) of an integral manifold of \(E\) that is contained in \(\tau^{-1}(U)\) is covered by a map \[(z^1,z^2,p) : N \to U \times \tilde{P}\] where \(p : N \to \tilde{P}\) may be taken to be of the form \[p=\left(\begin{array}{ccr} \frac{1}{(a_1)^2+(a_2)^2} & 0 & 0 \\ 0 & a_1 & -a_2 \\ 0 & a_2 & a_1\end{array}\right)\] for smooth complex-valued functions \(a_i : N \to \mathbb{C}\) satisfying \((a_1)^2+(a_2)^2\neq 0\).

We first consider the case where \(N\) is one-dimensional. We fix a local coordinate \(t\) on \(N\). It follows with (2.13) and straightforward calculations that \[\left(\varphi \circ (z^1,z^2,\nu\circ p)\right)^*\theta^2_0=\frac{a_1\dot{z}^2-a_2\dot{z}^1}{\left((a_1)^2+(a_2)^2\right)^2}\mathrm{d}t\] where \(\dot{z}^i\) denote the derivative of \(z^i\) with respect to \(t\). Hence we may take \[a_1=\dot{z}^1 \quad \text{and} \quad a_2=\dot{z}^2.\] Writing \(\beta=\left(\varphi \circ (z^1,z^2,\nu\circ p)\right)^*\theta^2_1\) and using (2.13) again, we compute \[\begin{aligned} \beta=&\left[\dot{z}^1\ddot{z}^2-\dot{z}^2\ddot{z}^1+\left(\dot{z}^1\dot{z}^2(\Pi^2_{21}-\Pi^1_{11})+(\dot{z}^1)^2\Pi^2_{11}-(\dot{z}^2)^2\Pi^1_{12}\right)\dot{z}^1+\right.\\ &\left.+\left(\dot{z}^1\dot{z}^2(\Pi^2_{22}-\Pi^1_{12})+(\dot{z}^1)^2\Pi^2_{12}-(\dot{z}^2)^2\Pi^1_{22}\right)\dot{z}^2\right]\frac{\mathrm{d}t}{(\dot{z}^1)^2+(\dot{z}^2)^2}. \end{aligned}\] Note that since \(\Pi^i_{ik}=0\) for \(k=1,2\), it follows that \(\beta\equiv 0\) is equivalent to \((z^1,z^2)\) satisfying the following ODE system \[\dot{z}^i\left(\ddot{z}^j+\Pi^j_{kl}\dot{z}^k\dot{z}^l\right)=\dot{z}^j\left(\ddot{z}^i+\Pi^i_{kl}\dot{z}^k\dot{z}^l\right), \quad i,j=1,2.\] This last system is easily seen to be equivalent to the system (2.1). Consequently, the one-dimensional integral manifolds of \(E\) are the generalised geodesics of \([\nabla]\).

Note that in the case of two-dimensional integral manifolds the above computations carry over where \(t\) is now a complex parameter, i.e. the two-dimensional integral manifolds are immersed complex curves \(Y \subset M\) for which \(\nabla_{\dot{Y}}\dot{Y}\) is proportional to \(\dot{Y}\) for some (and hence any) \(\nabla \in [\nabla]\). This last condition is equivalent to \(Y\) being a totally geodesic immersed complex curve with respect to \(([\nabla],J)\) (c.f. [32]) . A totally geodesic immersed complex curve \(Y\subset M\) which is maximally extended is called a complex geodesic. Since the complex geodesics are the (maximally extended) two-dimensional integral manifolds of \(E\), they exist only provided that \(E\) is integrable. We will next determine the integrability conditions for \(E\). Recall that \(E\subset T\mathbb{P}(T^{1,0}M)\) is defined by the equations \(\theta^2_1=\theta^2_0=0\) on \(B\). It follows with the structure equations (2.6) that \[\mathrm{d}\theta^2_0=0 \quad \text{mod}\quad \theta^2_0,\theta^2_1\] and \[\mathrm{d}\theta^2_1=W^2_{11\bar\jmath}\theta^1_0\wedge \overline{\theta^{\jmath}_0}\quad \text{mod}\quad \theta^2_0,\theta^2_1.\] Consequently, \(E\) is integrable if and only if \(W^2_{11\bar 1}=W^2_{11\bar 2}=0\). As a consequence of (2.8) and \(W^2_{11\bar 1}=0\) we obtain \[0=\varphi^2_{11\bar 1}=-W^2_{11\bar 1}\left(\theta^0_0+\overline{\theta^0_0}\right)+W^2_{1s\bar 1}\theta^s_1+W^2_{1s\bar 1}\theta^s_1-W^s_{11\bar 1}\theta^2_s+W^2_{11\bar s}\overline{\theta^s_j},\] which is equivalent to \(2W^2_{12\bar 1}=W^1_{11\bar 1}\). Using the symmetries of the complex projective Weyl tensor we compute \[W^1_{11\bar 1}=-W^2_{21\bar 1}=2W^2_{12\bar 1}=2W^2_{21\bar 1},\] thus showing that \(W^1_{11\bar 1}=W^2_{12\bar 1}=0\). From this we obtain \[0=\varphi^1_{11\bar 1}=2W^1_{12\bar 1}\theta^2_1-W^2_{11\bar 1}\theta^1_2+W^1_{11\bar 2}\overline{\theta^2_1}.\] thus implying \(W^1_{12\bar 1}=W^2_{11\bar 1}=W^1_{11\bar 2}=0\). Continuing in this vein allows to conclude that all components of the complex projective Weyl tensor must vanish. We may summarise:

3 Kähler metrisability

In this section we will derive necessary conditions for a complex projective structure \([\nabla]\) on a complex surface \((M,J)\) to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. There exists a complex projectively invariant linear first order differential operator acting on \(J\)-hermitian \((2,\! 0)\) tensor fields on \(M\) with weight \(1/3\), i.e sections of the bundle \(S^2_J(TM)\otimes \left(\Lambda^4(T^*M)\right)^{1/3}\). This differential operator has the property that nondegenerate sections in its kernel are in one-to-one correspondence with (pseudo-)Kähler metrics on \(M\) whose Levi-Civita connection is compatible with \([\nabla]\) (see [12, 26, 31]).

3.1 The differential analysis

We will show that in the surface case, the (pseudo-)Kähler metrics on \((M,J,[\nabla])\) whose Levi-Civita connection is compatible with \([\nabla]\) can equivalently be characterised in terms of a differential system on Cartan’s bundle \((\pi : B \to M,\theta)\).

3.1.1 First prolongation

Differentiating (3.1) yields \[ 0=\mathrm{d}^2 h_{i\bar\jmath}=\varepsilon_{si}\overline{\eta_j}\wedge\theta^s_0+\overline{\varepsilon_{sj}}\eta_i\wedge\overline{\theta^s_0}-(h_{s\bar\jmath}W^{s}_{iv\bar u}+h_{i\bar s}\overline{W^{s}_{ju\bar v}})\overline{\theta^u_0}\wedge\theta^v_0\](3.6) with \[\eta_k=\mathrm{d}h_k+h_k\left(\overline{\theta^0_0}-\theta^0_0\right)-h_j\theta^j_k+\overline{\varepsilon^{ij}}h_{k\bar\jmath}\overline{\theta^0_i}.\] This implies that we can write \[ \eta_i=a_{ij}\theta^j_0\](3.7) for unique complex-valued functions \(a_{ij}\) on \(B\). Equations (3.6) and (3.7) imply \[ \varepsilon_{ki}\overline{a_{jl}}-\overline{\varepsilon_{lj}} a_{ik}=\overline{h_{j\bar s}}W^s_{ik\bar l}-h_{i\bar s}\overline{W^s_{jl\bar k}}\](3.8) Contracting this last equation with \(\overline{\varepsilon^{jl}}\varepsilon^{ik}\) implies that the function \[h=-\frac{1}{2}\overline{\varepsilon^{ij}a_{ij}}\] is real-valued. We get \[a_{jl}=\varepsilon_{jl}h-\frac{1}{2}\overline{\varepsilon^{iu}}h_{s\bar \imath}W^s_{jl\bar u},\] and thus \[\mathrm{d}h_i=h_i\left(\theta^0_0-\overline{\theta^0_0}\,\right)+h_j\theta^j_i+h_{i\bar s}\overline{\varepsilon^{sl}\theta^0_l}+\left(\varepsilon_{ij}h-\frac{1}{2}\overline{\varepsilon^{uv}} h_{s\bar u}W^s_{ij\bar v}\right)\theta^j_0.\] Plugging the formula for \(a_{ij}\) back into (3.8) yields the integrability conditions \[h_{s\bar\jmath}W^s_{ik\bar l}-h_{i\bar s}\overline{W^s_{jl\bar k}}=\frac{1}{2}\overline{\varepsilon_{lj}\varepsilon^{uv}}h_{s\bar u}W^s_{ik\bar v}-\frac{1}{2}\varepsilon_{ki}\varepsilon^{uv}h_{u\bar s}\overline{W^s_{jl\bar v}}.\] This last equation can be simplified so that we obtain:

3.1.2 Second prolongation

We start by computing \[0=\mathrm{d}^2 h_i\wedge\theta^1_0\wedge\theta^2_0=-\left(h_{i\bar\jmath}\overline{\varepsilon^{jk}L_{k}}\right)\theta^1_0\wedge\overline{\theta^1_0}\wedge\theta^2_0\wedge\overline{\theta^2_0}\] which is equivalent to \[\left(\begin{array}{cc} h_{1\bar 1} & h_{1\bar 2} \\ h_{2\bar 1} & h_{2\bar 2}\end{array}\right)\cdot \left(\begin{array}{r} \overline{L_2} \\ -\overline{L_1}\end{array}\right)=0\] which cannot have any solution with \((h_{11}h_{22}-|h_{12}|^2)\neq 0\) unless \(L_1=L_2=0\). This shows:

Assuming henceforth \(L_1=L_2=0\) we also get \[ 0=\mathrm{d}^2h_i=\left(\varepsilon_{ij}\eta+\varphi_{ij}\right)\wedge\theta^j_0\](3.10) with \[\eta=\mathrm{d}h +2h\mathrm{Re}(\theta^0_0)+2\varepsilon^{ij}\mathrm{Re}(h_i\theta^0_j)-\frac{1}{2}\varepsilon^{kl}h_{k\bar \imath}\overline{\varepsilon^{ij}K_{js\bar l}\theta^s_0}\] and \[\varphi_{ij}=\mathrm{d}r_{ij}+r_{ij}\overline{\theta^0_0}-r_{si}\theta^s_j-r_{sj}\theta^s_i-h_lW^l_{ij\bar s}\overline{\theta^s_0}+\frac{1}{2}\overline{\varepsilon^{uv}}\left(h_{i\bar u}\overline{K_{vs\bar\jmath}}+h_{j\bar u}\overline{K_{vs\bar \imath}}\right)\overline{\theta^s_0}\] where \[r_{ij}=-\frac{1}{2}\overline{\varepsilon^{uv}} h_{s\bar u}W^s_{ij\bar v}.\] It follows with Cartan’s lemma that there are functions \(a_{ijk}=a_{ikj}\) such that \[\varepsilon_{ij}\eta+\varphi_{ij}=a_{ijk}\theta^k_0.\] Since \(\varphi_{ij}\) is symmetric in \(i,j\), this implies \[\eta=\frac{1}{2}\varepsilon^{ji}a_{ijs}\theta^s_0.\] Since \(h\) is real-valued, we must have \[\varepsilon^{ji}a_{ijs}=\overline{\varepsilon^{uv}}\varepsilon^{kl}h_{k\bar u}K_{ls\bar v}.\] Concluding, we get \[\mathrm{d}h=-2h\mathrm{Re}(\theta^0_0)+2\varepsilon^{kl}\mathrm{Re}(h_l\theta^0_k)+\frac{1}{2}\overline{\varepsilon^{ij}}\varepsilon^{kl}\mathrm{Re}(h_{k\bar \imath}K_{ls\bar\jmath}\theta^s_0).\] This completes the prolongation procedure.

Using Proposition 3.1 we obtain:

From this we get:

References