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 \[\tag{2.1} \nabla_{\dot\gamma}\dot\gamma\wedge \dot\gamma\wedge J\dot\gamma=0.\] 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 \[\tag{2.2} \nabla^{\prime}_ZW-\nabla_ZW=\beta(Z)W+\beta(W)Z\] 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 \[\tag{2.3} \hat{\Gamma}^i_{jk}=\Gamma^i_{jk}+\delta^i_j\beta_k+\delta^i_k\beta_j.\] 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 \[\tag{2.4} \Pi^i_{jk}=\Gamma^i_{jk}-\frac{1}{3}\left(\Gamma^l_{lj}\delta^i_k+\Gamma^l_{lk}\delta^i_j\right)\] 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 Remark 2.8.

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 2.2 implies that the curvature form \(\Theta=\mathrm{d}\theta+\theta\wedge\theta\) satisfies \[\tag{2.6} \Theta=\mathrm{d}\theta+\theta\wedge\theta=\left(\begin{array}{ccc} 0 & \Theta^0_1 & \Theta^0_2\\ 0 & \Theta^1_1 & \Theta^1_2\\ 0 & \Theta^2_1 & \Theta^2_2\end{array}\right)\] 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 \[\tag{2.7} (\sigma_z)^*W^i_{kl\bar\jmath}=-\frac{\partial \Pi^i_{kl}}{\partial \bar z^j}.\]

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 \[\tag{2.8} \mathrm{d}W^i_{kl\bar\jmath}=\left(W^i_{kl\bar\jmath s}+\delta^i_kK_{sl\bar\jmath}+\delta^i_lK_{sk\bar\jmath}-3\delta^i_sK_{kl\bar\jmath}\right)\theta^s_0+W^i_{kl\bar\jmath\bar s}\overline{\theta^s_0}+\varphi^i_{kl\bar\jmath}\] where \[\tag{2.9} \varphi^i_{kl\bar\jmath}=-W^i_{kl\bar\jmath}\left(\theta^0_0+\overline{\theta^0_0}\right)+W^i_{ls\bar\jmath}\theta^s_k+W^i_{ks\bar\jmath}\theta^s_l-W^s_{kl\bar\jmath}\theta^i_s+W^i_{kl\bar s}\overline{\theta^s_j}.\]

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 \[\tag{2.10} \mathrm{d}L_i=-4L_i\theta^0_0+L_j\theta^j_i+L_{ij}\theta^j_0+L_{i\bar\jmath}\overline{\theta^{\jmath}_0}\] 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 \[\tag{2.11} (L_1\theta^1_0+L_2\theta^2_0)\otimes \left(\theta^1_0\otimes \theta^2_0\right)\] 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 \[\tag{2.12} \mathrm{d}K_{ik\bar\jmath}=\left(K_{ik\bar\jmath s}+\frac{1}{4}\left(\varepsilon_{sk}L_{i\bar\jmath}+\varepsilon_{si}L_{k\bar\jmath}\right)\right)\theta^s_0+K_{ik\bar\jmath\bar s}\overline{\theta^s_0}+\varphi_{ik\bar\jmath}\] 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 2.2 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 2.2. 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 \[\tag{2.13} (\varphi^*\theta)_{(z,p)}=\left(\omega_P\right)_p+\mathrm{Ad}(p^{-1})\circ \phi_z\] 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 \[\tag{2.14} \left(\begin{array}{cc} \det a^{-1} & b \\ 0 & a \end{array}\right)\] 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: