3 Kähler metrisability

Proof. Let \(g\) be a (pseudo-)Kähler metric on \((M,J)\) and write \(g=g_{i\bar\jmath}\,\mathrm{d}z^i\circ \mathrm{d}\overline{z^\jmath}\) for local holomorphic coordinates \(z=(z^1,z^2) : U \to \mathbb{C}^2\) on \(M\). Denoting by \(\nabla\) the Levi-Civita connection of \(g\), on \(U\) the identity \(\nabla g=0\) is equivalent to \[\frac{\partial g_{k\bar\jmath}}{\partial z^i}=g_{s\bar\jmath}\Gamma^s_{ik}\qquad \text{and}\qquad \frac{\partial g_{k\bar\jmath}}{\partial \overline{z^{\imath}}}=g_{k\bar s}\overline{\Gamma^s_{ij}},\] where \(\Gamma^i_{jk}\) denote the complex Christoffel symbols of \(\nabla\). Abbreviate \(G=\det g_{i\bar\jmath}\), then we obtain \[\frac{\partial G}{\partial z^i}=G\,\Gamma^s_{si}.\] Hence, the partial derivative of \(h_{k\bar\jmath}=g_{k\bar\jmath}\,G^{-2/3}\) with respect to \(z^i\) becomes \[\begin{aligned} \frac{\partial h_{k\bar\jmath}}{\partial z^i}&=g_{l\bar\jmath}\,\Gamma^l_{ik}\,G^{-2/3}-\frac{2}{3}g_{k\bar\jmath}\,\Gamma^s_{si}\,G^{-2/3}=h_{l\bar\jmath}\left(\Gamma^l_{ik}-\frac{2}{3}\Gamma^s_{si}\delta^l_k\right)\\ &=h_{l\bar\jmath}\left(\Gamma^l_{ik}-\frac{1}{3}\Gamma^s_{si}\delta^l_k-\frac{1}{3}\Gamma^s_{sk}\delta^l_i\right)-\frac{1}{3}h_{l\bar\jmath}\left(\Gamma^s_{si}\delta^l_k-\Gamma^s_{sk}\delta^l_i\right). \end{aligned}\] Note that the last two summands in the last equation are antisymmetric in \(i,k\), so that we may write \[-\frac{1}{3}h_{l\bar\jmath}\left(\Gamma^s_{si}\delta^l_k-\Gamma^s_{sk}\delta^l_i\right)=\overline{h_j}\varepsilon_{ik}\] for unique complex-valued functions \(h_i\) on \(U\). We thus get \[\tag{3.2} \frac{\partial h_{k\bar\jmath}}{\partial z^i}=h_{s\bar\jmath}\Pi^s_{ik}+\overline{h_j}\varepsilon_{ik}.\] In entirely analogous fashion we obtain \[\tag{3.3} \frac{\partial h_{k\bar\jmath}}{\partial \overline{z^{\imath}}}=h_{k\bar s}\overline{\Pi^s_{ij}}+h_k\overline{\varepsilon_{ij}}.\] Recall from Theorem 2.2 that the coordinate system \(z : U \to \mathbb{C}^2\) induces a unique section \(\sigma_z : U \to B\) of Cartan’s bundle such that \[\tag{3.4} \left(\sigma_z\right)^*\theta^0_0=0, \qquad \left(\sigma_z\right)^*\theta^i_0=\mathrm{d}z^i, \qquad \left(\sigma_z\right)^*\theta^i_j=\Pi^i_{jk}\mathrm{d}z^k.\] Consequently, using (3.2), (3.3), (3.4) we see that (3.1) is necessary.

Conversely, suppose there exist complex-valued functions \(h_{i\bar\jmath}=\overline{h_{j\bar\imath}}\) and \(h_i\) on \(B\) solving (3.1) for which \[\left(h_{1\bar 1}h_{2\bar 2}-|h_{1\bar 2}|^2\right)\neq 0.\] Setting \(g_{i\bar\jmath}=h_{i\bar\jmath}\left(h_{1\bar 1}h_{2\bar 2}-|h_{1\bar 2}|^2\right)^{-2}\) we get \[\tag{3.5} \mathrm{d}g_{i\bar \jmath}=-g_{i\bar \jmath}\left(\theta^0_0+\bar\theta^0_0\right)+g_{i\bar s}\overline{\theta^s_{j}}+g_{s\bar\jmath}\theta^s_i+g_{i\bar \jmath\bar s}\overline{\theta^s_0}+g_{i\bar \jmath s}\theta^s_0\] with \[g_{i\bar \jmath \bar s}=\frac{(h_{i\bar \jmath}h_{l\bar s}+h_{i\bar s}h_{l\bar \jmath})\varepsilon^{lk}h_k}{(h_{1\bar 1}h_{2\bar 2}-|h_{1\bar 2}|^2)^3}, \quad \text{and} \quad g_{i\bar \jmath k}=\frac{(h_{i\bar \jmath}h_{k\bar s}+h_{k\bar\jmath}h_{i\bar s})\overline{\varepsilon^{su}{h_u}}}{(h_{1\bar 1}h_{2\bar 2}-|h_{1\bar 2}|^2)^3}.\] It follows with (3.5) that there exists a unique \(J\)-Hermitian metric \(g\) on \(M\) such that \(\pi^*g=g_{i\bar\jmath}\,\theta^i_0\circ \overline{\theta^{\jmath}_0}\). Choose local holomorphic coordinates \(z=(z^1,z^2) : U \to \mathbb{C}^2\) on \(M\). By abuse of notation we will write \(g_{i\bar\jmath},g_{i\bar \jmath \bar s},g_{i\bar \jmath k}\) for the pullback of the respective functions on \(B\) by the section \(\sigma_z : U \to B\) associated to \(z\). From (3.5) we obtain \[\frac{\partial g_{i\bar\jmath}}{\partial z^s}=g_{u\bar\jmath}\Pi^u_{is}+g_{i\bar\jmath s}=g_{u\bar\jmath}\left(\Pi^u_{is}+g^{\bar v u}g_{i\bar v s}\right)=g_{u\bar\jmath}\left(\Pi^u_{is}+\delta^u_ib_s+\delta^u_sb_i\right)=g_{u\bar\jmath}\Gamma^u_{is}\] where we write \[b_i=\frac{h_{i\bar s}\overline{\varepsilon^{su}h_u}}{(h_{1\bar 1}h_{2\bar 2}-|h_{1\bar 2}|^2)^{11/3}}\qquad \text{and} \qquad \Gamma^i_{jk}=\Pi^i_{jk}+\delta^i_jb_k+\delta^i_kb_j.\] Likewise we obtain \[\frac{\partial g_{i\bar\jmath}}{\partial \overline{z^{s}}}=g_{i\bar u}\overline{\Gamma^u_{js}}.\] It follows that there exists a complex-linear connection \(\nabla\) on \(U\) defining \([\nabla]\) and whose complex Christoffel symbols are given by \(\Gamma^i_{jk}\). By construction, the connection \(\nabla\) preserves \(g\) and hence must be the Levi-Civita connection of \(g\). Furthermore, \(\nabla\) being complex-linear implies that \(g\) is Kähler. This completes the proof.

Proof. Suppose the complex projective structure \([\nabla]\) has vanishing complex projective Weyl and Liouville curvature. Then the differential system (3.11) may be written as \[\tag{3.12} \mathrm{d}H+\theta H+H\theta^*=0\] with \[H=H^*=\left(\begin{array}{ccc}h & -\overline{h_2} & \overline{h_1}\\ -h_2 & -h_{22} & h_{21}\\ h_1 & h_{12} & -h_{11}\end{array}\right)\] where \({}^*\) denotes the conjugate transpose matrix. Recall that in the flat case \(\theta=g^{-1}\mathrm{d}g\) for some smooth \(g: B \to \mathrm{PSL}(3,\mathbb{C})\), hence the solutions to (3.12) are \[H=g^{-1}C\left(g^{-1}\right)^*\] where \(C=C^*\) is a constant hermitian matrix of rank at least two. The statement now follows immediately with Theorem 3.7.