3 Projective surfaces and associated bundles

A natural case to consider is \(n=2\), where \(\mathcal{F}\) is just the square of the \(L^2\)-norm of \(A_{[g]}\) taken with respect to \([g]\). We will henceforth consider the surface case only.

There are several natural geometric spaces fibering over an oriented projective surface which we will discuss next. Before doing so, we recall a result of Cartan [9], which canonically associates a principal bundle together with a “connection” to every projective manifold. The reader interested in a description of Cartan’s construction using modern language may also consult [26]. For additional background on Cartan geometries the reader may also consult [8].

3.1 Cartan’s normal projective connection

Let \(\Sigma\) be an oriented surface and let \(\mathrm{G}\simeq \mathbb{R}_2\rtimes \mathrm{GL}^+(2,\mathbb{R})\) denote the two-dimensional orientation preserving affine group which we think of as the subgroup of \(\mathrm{SL}(3,\mathbb{R})\) consisting of matrices of the form \[b\rtimes a=\begin{pmatrix} \det a^{-1} & b \\ 0 & a\end{pmatrix},\] for \(b \in \mathbb{R}_2\) and \(a \in \mathrm{GL}^+(2,\mathbb{R})\). We denote by \(\upsilon : F^+\to \Sigma\) the principal right \(\mathrm{GL}^+(2,\mathbb{R})\)-bundle of coframes that are orientation preserving with respect to the chosen orientation on \(\Sigma\) and the standard orientation on \(\mathbb{R}^2\). We define a right \(\mathrm{G}\)-action on \(F^+\times \mathbb{R}_2\) by the rule \[\tag{3.1} (u,\xi)\cdot (b\rtimes a)=\left(\det a^{-1} a^{-1} \circ u,\xi a \det a + b\det a\right),\] for all \(b\rtimes a \in \mathrm{G}\). Here \(\xi : F^+ \times \mathbb{R}_2 \to \mathbb{R}_2\) denotes the projection onto the latter factor. This action turns \(\pi : F^+\times \mathbb{R}_2 \to \Sigma\) into a principal right \(\mathrm{G}\)-bundle over \(\Sigma\), where \(\pi : F^+\times \mathbb{R}_2\to \Sigma\) denotes the natural basepoint projection. Suppose \(\nabla\) is a torsion-free connection on \(T\Sigma\) with connection \(1\)-form \(\eta=(\eta^i_j)\) on \(F^+\) so that we have the structure equations² \[\begin{aligned} \mathrm{d}\omega^i&=-\eta^i_j\wedge\omega^j,\\ \mathrm{d}\eta^i_j&=-\eta^i_k\wedge\eta^k_j+(\delta^i_{[k}S_{l]j}-S_{[kl]}\delta^i_j)\omega^k\wedge\omega^l, \end{aligned}\] where \(S=(S_{ij})\) represents the projective Schouten tensor \(\mathrm{Schout}(\nabla)\) of \(\nabla\) and \(\omega^i\) the components of the tautological \(\mathbb{R}^n\)-valued \(1\)-form \(\omega\) on \(F\). Recall that the Schouten tensor is defined as \[\tag{3.2} \mathrm{Schout}(\nabla)=\mathrm{Ric}^{+}(\nabla)-\frac{1}{3}\mathrm{Ric}^{-}(\nabla),\] where \(\mathrm{Ric}^{\pm}(\nabla)\) denote the symmetric and anti-symmetric part of the Ricci curvature of \(\nabla\). On \(P=F^+\times \mathbb{R}_2\) we define the \(\mathfrak{sl}(3,\mathbb{R})\)-valued \(1\)-form \[\tag{3.3} \theta=\begin{pmatrix} -\frac{1}{3}\operatorname{tr}\eta-\xi \omega & \mathrm{d}\xi-\xi\eta-(S\omega)^t-\xi\omega\xi\\ \omega & \eta-\frac{1}{3}\mathrm{I}\operatorname{tr}\eta+\omega\xi \end{pmatrix}.\] The reader may check that the pair \((\pi : P\to \Sigma,\theta)\) defines a Cartan geometry of type \((\mathrm{SL}(3,\mathbb{R}),\mathrm{G})\), that is, \(\pi : P\to \Sigma\) is a principal right \(\mathrm{G}\)-bundle and \(\theta\) is an \(\mathfrak{sl}(3,\mathbb{R})\)-valued \(1\)-form on \(P\) satisfying the following properties:

• \(\theta_u : T_uP\to \mathfrak{sl}(3,\mathbb{R})\) is an isomorphism for every \(u \in P\);

• \((R_{g})^*\theta=g^{-1}\theta g\) for every \(g \in \mathrm{G}\);

• \(\theta(X_v)=v\) for every fundamental vector field \(X_v\) generated by an element \(v\) in the Lie algebra of \(\mathrm{G}\).

Moreover, writing \(\theta=(\theta^i_j)_{i,j=0,1,2}\), the Cartan geometry \((\pi : P\to \Sigma,\theta)\) also satisfies:

• for every non-zero \(x \in \mathbb{R}^2\), the integral curves of the vector field \(X_{x}\) defined by the equations \[\theta(X_x)=\begin{pmatrix} 0 & 0 \\ x & 0 \end{pmatrix}\] project to \(\Sigma\) to become geodesics of \(\mathfrak{p}\) and conversely all geodesics of \(\mathfrak{p}\) arise in this way;

• the \(\pi\)-pullback of an orientation compatible volume form on \(\Sigma\) is a positive multiple of \(\theta^1_0\wedge\theta^2_0\);

• the curvature \(2\)-form \(\Theta=\mathrm{d}\theta+\theta\wedge\theta\) is \[\tag{3.4} \Theta=\mathrm{d}\theta+\theta\wedge\theta=\begin{pmatrix} 0 & L_1\theta^1_0\wedge\theta^2_0 & L_2 \theta^1_0\wedge\theta^2_0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix},\] for unique curvature functions \(L_i : P\to \mathbb{R}\).

A projective structure \(\mathfrak{p}\) on \(\Sigma\) is called flat if every point \(p \in \Sigma\) has a neighbourhood \(U_p\) which is diffeomorphic to a subset of \(\mathbb{RP}^2\) in such a way that the geodesics of \(\mathfrak{p}\) contained in \(U_p\) are mapped onto (segments) of projective lines \(\mathbb{RP}^1\subset \mathbb{RP}^2\). Furthermore, a torsion-free connection \(\nabla\) on \(T\Sigma\) is called projectively flat if \(\mathfrak{p}(\nabla)\) is flat. Using Cartan’s connection, one can show that a projective structure \(\mathfrak{p}\) is flat if and only if the functions \(L_1\) and \(L_2\) vanish identically. Another consequence of Cartan’s result is that there exists a unique \(1\)-form \(\lambda\in\Omega^1(\Sigma,\Lambda^2(T^*\Sigma))\) so that \[\pi^*\lambda=(L_1\theta^1_0+L_2\theta^2_0)\otimes \theta^1_0\wedge\theta^2_0.\] The \(1\)-form \(\lambda\) was first discovered by R. Liouville [32], hence we call \(\lambda\) the Liouville curvature of \(\mathfrak{p}\). In particular, the Liouville curvature is the complete obstruction to flatness of a two-dimensional projective structure.

3.2 The twistor space

Inspired by Hitchin’s twistorial description of holomorphic projective structures on complex surfaces [19], it was shown in [13, 42] how to construct a “twistor space” for smooth projective structures. For what follows it will be convenient to construct the twistor space in the smooth category by using the Cartan geometry of a projective surface.

Let therefore \((\Sigma,\mathfrak{p})\) be an oriented projective surface with Cartan geometry \((\pi : P\to \Sigma,\theta)\). By construction, the quotient of \(P\) by the normal subgroup \(\mathbb{R}_2\rtimes\mathrm{\{Id\}}\subset \mathrm{G}\) is isomorphic to the bundle \(\upsilon : F^+ \to \Sigma\) of orientation preserving coframes of \(\Sigma\). In particular, the choice of a conformal structure \([g]\) on \(\Sigma\) corresponds to a section of the fibre bundle \(\mathrm{C}(\Sigma)\simeq P/\left(\mathbb{R}_2\rtimes\mathrm{CO}(2)\right) \to \Sigma\). Here \(\mathrm{CO}(2)=\mathbb{R}^+\times\mathrm{SO}(2)\) is the linear orientation preserving conformal group. By construction, the typical fibre of the bundle \(\mathrm{C}(\Sigma) \to \Sigma\) is diffeomorphic to \(\mathrm{GL}^+(2,\mathbb{R})/\mathrm{CO}(2)\simeq\mathrm{SL}(2,\mathbb{R})/\mathrm{SO}(2)\), that is, the open unit disk \(D^2\subset \mathbb{C}\).

We write the elements of the group \(\mathbb{R}_2\rtimes \mathrm{CO}(2)\) in the following form \[z\rtimes r\mathrm{e}^{\mathrm{i}\phi}=\begin{pmatrix} r^{-2} & \operatorname{Re}(z) & \operatorname{Im}(z)\\ 0& r\cos\phi & r \sin\phi \\ 0 & -r \sin\phi & r\cos\phi\end{pmatrix}, \quad z \in \mathbb{C},r\mathrm{e}^{\mathrm{i}\phi}\in \mathbb{C}^*.\] Property (iii) of the Cartan geometry \((\pi : P\to \Sigma,\theta)\) implies that the (real – or complex-valued) \(1\)-forms on \(P\) that are semibasic³ for the quotient projection \(\mu : P\to \mathrm{C}(\Sigma)\) are complex linear combinations of the complex-valued \(1\)-forms \[\tag{3.5} \zeta_1=\theta^1_0+\mathrm{i}\theta^2_0, \qquad \zeta_2=\left(\theta^1_1-\theta^2_2\right)+\mathrm{i}\left(\theta^1_2+\theta^2_1\right)\] and their complex conjugates. The equivariance property (ii) of the Cartan geometry gives\[\tag{3.6} \left(R_{z\rtimes r\mathrm{e}^{\mathrm{i}\phi}}\right)^*\begin{pmatrix}\zeta_1 \\ \zeta_2\end{pmatrix}=\begin{pmatrix} \frac{1}{r^3} \mathrm{e}^{\mathrm{i}\phi} & 0\\ \frac{z}{r}\mathrm{e}^{\mathrm{i}\phi} & \mathrm{e}^{2\mathrm{i}\phi} \end{pmatrix}\begin{pmatrix}\zeta_1 \\ \zeta_2\end{pmatrix}.\] It follows that there exists a unique almost complex structure \(\mathfrak{J}\) on \(\mathrm{C}(\Sigma)\) having the property that a complex-valued \(1\)-form on \(P\) is the pullback of a \((1,\! 0)\)-form on \(\mathrm{C}(\Sigma)\) if and only if it is a complex linear combination of \(\zeta_1\) and \(\zeta_2\). Indeed, locally we may use a section \(s\) of the bundle \(\mu : P\to \mathrm{C}(\Sigma)\) to pull down the forms \(\zeta_1,\zeta_2\) onto the domain of definition \(U\subset \mathrm{C}(\Sigma)\) of \(s\). Since \(\zeta_1,\zeta_2\) are semi-basic for the projection \(\mu : P\to \mathrm{C}(\Sigma)\), it follows that the pulled down forms are linearly independent over \(\mathbb{C}\) at each point of \(U\). Hence we obtain a unique almost complex structure \(\mathfrak{J}\) on \(U\) whose \((1,\! 0)\)-forms are \(s^*\zeta_1,s^*\zeta_2\). The equivariance (3.6) implies that \(\mathfrak{J}\) is independent of the choice of the section \(s\) and extends to all of \(\mathrm{C}(\Sigma)\). Using property (vi) of the Cartan geometry the reader may easily verify that \[\mathrm{d}\zeta_1=\mathrm{d}\zeta_2=0, \quad \text{mod}\quad \zeta_1,\zeta_2.\] It follows from the Newlander-Nirenberg theorem that \(\mathfrak{J}\) is integrable, thus giving \(\mathrm{C}(\Sigma)\) the structure of a complex surface which we will denote by \(Z\) and which – abusing language – we call the twistor space of the projective surface \((\Sigma,\mathfrak{p})\).

3.3 An indefinite Kähler-Einstein 3-fold

From (3.6) it follows that the holomorphic cotangent bundle \(T^*_{\mathbb{C}}Z^{1,0}\to Z\) is the bundle associated to \(\mu : P\to Z\) via the complex two-dimensional representation \(\rho : \mathbb{R}_2\rtimes \mathrm{CO}(2) \to \mathrm{GL}(2,\mathbb{C})\) defined by the rule \[\tag{3.7} \rho(z \rtimes r\mathrm{e}^{\mathrm{i}\phi})(w_1\;w_2)=(w_1\;w_2)\begin{pmatrix} \frac{1}{r^3} \mathrm{e}^{\mathrm{i}\phi} & 0\\ \frac{z}{r}\mathrm{e}^{\mathrm{i}\phi} & \mathrm{e}^{2\mathrm{i}\phi} \end{pmatrix}\] for all \((w_1\;w_2) \in \mathbb{C}_2\). In particular, the form \(\zeta_1\) is well defined on \(Z\) up to complex-scale and hence may be thought of as a section of the projective holomorphic cotangent bundle \(\mathbb{P}(T^*_{\mathbb{C}}Z^{1,0}) \to Z\). Abusing notation, we write \(\zeta_1(Z) \subset \mathbb{P}(T^*_{\mathbb{C}}Z^{1,0})\) to denote the image of \(Z\) under this section. We now have:

We will henceforth identify \(Y\simeq P/\mathrm{CO}(2)\) and think of \(\tau\) as the projection map onto \(Y\). Denoting the integrable almost complex structure on \(Y\) by \(J\), the first part of the following proposition is therefore clear:

3.4 The canonical flat case

In this subsection we identify the spaces \[Y=P/\mathrm{CO}(2) \quad\text{and} \quad Z=P/\left(\mathbb{R}_2\rtimes \mathrm{CO}(2)\right)\] in the case where \((\Sigma,\mathfrak{p})\) is the canonical flat projective structure on the projective \(2\)-sphere. Recall that in this case \(P=\mathrm{SL}(3,\mathbb{R})\). The group \(\mathrm{SL}(3,\mathbb{R})\) also acts naturally on \(\mathbb{C}_3\) by complexification, that is, by the rule \[g \cdot (\xi+\mathrm{i}\chi)=\xi g^{-1}+\mathrm{i}\chi g^{-1}\] for all \(g \in \mathrm{SL}(3,\mathbb{R})\). Clearly, this action descends to define a left action on \(\mathbb{CP}_2\). However, this action is not transitive, but has two orbits. The first orbit is \(\mathbb{RP}_2\subset \mathbb{CP}_2\), where we think of \(\mathbb{RP}_2\) as those points \([\xi+\mathrm{i}\chi] \in \mathbb{CP}_2\) which satisfy \(\xi\wedge\chi=0\), that is, \(\xi\) and \(\chi\) are linearly dependent over \(\mathbb{R}\). Assume therefore \([\varepsilon]\) is an element in the complement \(\mathbb{CP}_2\setminus\mathbb{RP}_2\) of \(\mathbb{RP}^2\) in \(\mathbb{CP}^2\). Since \(\mathrm{SL}(3,\mathbb{R})\) acts transitively on unimodular triples of vectors in \(\mathbb{R}_3\), we can assume without losing generality that \(\varepsilon=(0\;-\mathrm{i}\; 1)\). For \(g\in\mathrm{SL}(3,\mathbb{R})\) we write \(g=(g_0\;g_1\;g_2)\) with \(g_i \in \mathbb{R}^3\). We will next determine the stabiliser subgroup of \([\varepsilon]\). A simple computation gives \[g\cdot \varepsilon=g_0\wedge\left(g_1+\mathrm{i}g_2\right).\] An elementary calculation shows that \([g\cdot \varepsilon]=[\varepsilon]\) implies that we must have \[\begin{pmatrix} c_1 \\ c_2\end{pmatrix}=\begin{pmatrix}g^2_1 & -g^1_1 \\ g^2_2 & -g^1_2\end{pmatrix}\begin{pmatrix}g^1_0 \\ g^2_0 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \end{pmatrix}.\] Since \[\det g=g^0_2\,c_1-g^0_1\,c_2+g^0_0\,\det\begin{pmatrix}g^2_1 & -g^1_1 \\ g^2_2 & -g^1_2\end{pmatrix}=1,\] it follows that \(g^1_0=g^2_0=0\). Therefore, the stabiliser subgroup of \([\varepsilon]\) is a subgroup of \(\mathbb{R}_2\rtimes \mathrm{GL}(2,\mathbb{R})\). Writing \(a=(a^i_j)\), we obtain \[\left(b\rtimes a\right) \cdot \varepsilon=\det a^{-1}\begin{pmatrix}0 & -a^2_1-\mathrm{i}a^2_2 & a^1_1+\mathrm{i}a^1_2 \end{pmatrix},\] from which it follows that \([(b\rtimes a)\cdot \varepsilon]=[\varepsilon]\) if and only if \(a^1_1=a^2_2\) and \(a^1_2+a^2_1=0\), that is, \(a \in \mathrm{CO}(2)\). Concluding, we have shown \[\mathrm{SL}(3,\mathbb{R})/\left(\mathbb{R}_2\rtimes\mathrm{CO}(2)\right)\simeq \mathbb{CP}_2\setminus \mathbb{RP}_2\] and the projection map is \[\mu:\mathrm{SL}(3,\mathbb{R}) \to \mathbb{CP}_2\setminus \mathbb{RP}_2, \quad \begin{pmatrix} g_0 & g_1 & g_2 \end{pmatrix} \mapsto [g_0\wedge (g_1+\mathrm{i}g_2)],\] where we use \(\mathbb{R}_3\simeq \Lambda^2(\mathbb{R}^3)\).

We will show next that \(Y\) is a submanifold of \(F(\mathbb{C}_3)\). Clearly, the action of \(\mathrm{SL}(3,\mathbb{R})\) on the space \(F(\mathbb{C}_3)\) of complete complex flags is not transitive, there is however an open orbit. Let \(F(\mathbb{C}_3)^*\) denote the \(\mathrm{SL}(3,\mathbb{R})\) orbit of the flag \[(\ell,\Pi)=\left(\mathbb{C}\{\varepsilon_1\},\mathbb{C}\{\varepsilon_1,\varepsilon_2\}\right),\] where \[\varepsilon_1=\begin{pmatrix} 0&-\mathrm{i}&1\end{pmatrix}, \quad \varepsilon_2=\begin{pmatrix} 1&0&0\end{pmatrix}.\quad %\eps_3=\begin{pmatrix} 0&1&0\end{pmatrix}.\] We already know that the stabiliser subgroup \(\mathrm{G}_0\) of \((\ell,\Pi)\) must be a subgroup of \(\mathbb{R}_2\rtimes \mathrm{CO}(2)\). For \(b\rtimes a \in \mathbb{R}_2\rtimes \mathrm{CO}(2)\) we write \[b\rtimes a=\begin{pmatrix} \frac{1}{x^2+y^2} & b_1 & b_2 \\ 0 & x & y \\ 0 & -y & x\end{pmatrix},\] with \(x^2+y^2>0\). We compute \[\varepsilon_2 \cdot (b\rtimes a)=\begin{pmatrix} x^2+y^2 & -xb_1-yb_2 & -xb_2+yb_1\end{pmatrix}\] which is easily seen to lie in the complex linear span of \(\varepsilon_1,\varepsilon_2\) if and only if \(b_1=b_2=0\), hence \[\mathrm{SL}(3,\mathbb{R})/\mathrm{CO}(2)\simeq F(\mathbb{C}_3)^*\] and the projection map is \[\tau : \mathrm{SL}(3,\mathbb{R}) \to F(\mathbb{C}_3) \quad \begin{pmatrix} g_0 & g_1 & g_2 \end{pmatrix} \mapsto \left(\mathbb{C}\{\varepsilon_1\},\mathbb{C}\{\varepsilon_1,\varepsilon_2\}\right),\] with \[\varepsilon_1=g_0\wedge (g_1+\mathrm{i}g_2), \qquad \varepsilon_2=g_1\wedge g_2.\] Since \(F(\mathbb{C}_3)\) is real six-dimensional and since \(\dim \mathrm{SL}(3,\mathbb{R})-\dim \mathrm{CO}(2)=6\), it follows that \(F(\mathbb{C}_3)^*\) is open.