4 The flat case
In this section we apply the obtained results to the Grassmannian of oriented \(2\)-planes in \(\mathbb{R}^{n+2}\) which carries a \(2\)-oriented torsion-free \(2\)-Segre together with an adapted connection of vanishing curvature.
Here a \(2\)-Segre structure \(\pi : F_\mathcal{S}\to M\) is called \(2\)-oriented if the structure group \(H(2,n)\) has been reduced to \(H^+(2,n)=\mathrm{GL}^+(2,\mathbb{R})\otimes \mathrm{GL}(n,\mathbb{R})\).
Using Theorem 2.13 we also get: If \(F_\mathcal{S}\to M\) is a \(\beta\)-integrable \(2\)-oriented \(2\)-Segre structure, then \(\rho : X_\mathcal{S}=F_\mathcal{S}/(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})) \to M\) together with its canonical almost complex structure \(\mathfrak{J}\) is a quasiholomorphic fibre bundle with fibre \(\mathrm{GL}^+(2,\mathbb{R})/\mathrm{GL}(1,\mathbb{C})\simeq D^2\), the open unit disk in \(\mathbb{C}\).
4.1 The Grassmannian of oriented 2-planes
The projective linear group \(\mathrm{PL}(n+2,\mathbb{R})=\mathrm{GL}(n+2,\mathbb{R})/Z\) acts transitively from the left on the Grassmannian \(G^+_2(\mathbb{R}^{n+2})\) of oriented \(2\)-planes in \(\mathbb{R}^{n+2}\) and the stabiliser subgroup of any element \(\Pi\in G^+_2(\mathbb{R}^{n+2})\) may be identified with the subgroup \(S\) consisting of elements of the form \[\left[\begin{array}{cc} a & b \\ 0 & c \end{array}\right]\] where \(a \in \mathrm{GL}^+(2,\mathbb{R}), c \in \mathrm{GL}(n,\mathbb{R})\) and \(b \in M_{\mathbb{R}}(2,n)\) is a real \((2\times n)\)-matrix. Let \(\mu : \mathrm{PL}(n+2,\mathbb{R}) \to G^+_2(\mathbb{R}^{n+2})\simeq \mathrm{PL}(n+2,\mathbb{R})/S\) be the quotient projection and write \[\tilde{\theta}=\left(\begin{array}{cc} \alpha & \beta\\ \eta & \gamma\end{array}\right)\] for the Maurer-Cartan form of \(\mathrm{PL}(n+2,\mathbb{R})\). The real matrix-valued \(1\)-forms \(\alpha,\beta,\gamma,\eta\) have sizes according to the block decomposition of the Lie group \(S\) and satisfy \(\text{Tr}(\alpha)+\text{Tr}(\gamma)=0\). Let \(H^i_k\) denote the vector fields dual to the forms \(\eta\) with respect to the coframing \(\tilde{\theta}\). Let \(N\subset \mathrm{PL}(n+2,\mathbb{R})\) be the closed normal subgroup given by \[N=\left\{\left[\begin{array}{cc} \mathrm{I}_2 & b \\ 0 & \mathrm{I}_n \end{array}\right]\;\bigg\vert\; b \in M_{\mathbb{R}}(2,n)\right\}\] whose elements will be denoted by \([b]\). The quotient Lie group \(S/N\) is isomorphic to \(H^+(2,n)\) and thus \(\mathrm{PL}(n+2,\mathbb{R})/N\) is the total space of a right principal \(H^+(2,n)\)-bundle over \(G^+_2(\mathbb{R}^{n+2})\). Consider the smooth map \[\varphi^i_k : \mathrm{PL}(n+2,\mathbb{R}) \to TG^+_2(\mathbb{R}^{n+2}), \quad p \mapsto \mu^{\prime}_p(H^i_k(p))\\ \] The Maurer-Cartan equation \(\mathrm{d}\tilde{\theta}+\tilde{\theta}\wedge\tilde{\theta}=0\) implies that the form \(\eta\) is basic for the quotient projection \(\mathrm{PL}(n+2,\mathbb{R}) \to \mathrm{PL}(n+2,\mathbb{R})/N\). Therefore the maps \(\varphi^i_k\) are invariant under the right action of \(N\) and thus descend to smooth maps \(\mathrm{PL}(n+2,\mathbb{R})/N \to TG^+_2(\mathbb{R}^{n+2})\). The images \(\varphi^i_k(p)\) for a given point \(p \in \mathrm{PL}(n+2,\mathbb{R})\) are linearly independent and thus induce a map \(\varphi\) into the coframe bundle of \(G^+_2(\mathbb{R}^{n+2})\). The maps \(\varphi^i_k\) can be arranged so that the induced map \(\varphi\) from \(\mathrm{PL}(n+2,\mathbb{R})/N\) into the coframe bundle of \(G^+_2(\mathbb{R}^{n+2})\) pulls back the components of the canonical \(\mathbb{C}^n\)-valued \(1\)-form to \(\mathrm{i}(\eta^k_1+\mathrm{i}\eta^k_2)\). It follows again with the Maurer-Cartan equation that \(\varphi\) embeds \(\mathrm{PL}(n+2,\mathbb{R})/N\) as a smooth right principal \(H^+(2,n)\)-subbundle of the coframe bundle of \(G^+_2(\mathbb{R}^{n+2})\). This subbundle will be denoted by \(\pi_0 : F_0 \to G^+_2(\mathbb{R}^{n+2})\) and the projection \(\mathrm{PL}(n+2,\mathbb{R}) \to F_0\) by \(\upsilon\). Write \[\begin{align} \tilde{\omega}&=\alpha^2_1,\\ 2\,\tilde{\xi}&=\left(\alpha^1_2+\alpha^2_1\right)+\mathrm{i}\left(\alpha^2_2-\alpha^1_1\right),\\ \tilde{\phi}&=\gamma-\mathrm{I}_n\alpha^2_2,\\ \zeta^i&=\mathrm{i}(\eta^i_1+\mathrm{i}\eta^i_2).\\ \end{align}\] Then straightforward computations show that the forms \((\tilde{\omega},\tilde{\xi},\tilde{\phi})\) transform under the right action of \(H^+(2,n)\) as the connection forms of an \(\mathcal{S}_0\)-adapted connection do. Moreover we have \[\mathrm{d}\zeta=-\left(\mathrm{i}\left(\tilde{\omega}-\tilde{\xi}\right)\mathrm{I}_n+\tilde{\phi}\right)\wedge\zeta-\mathrm{i}\tilde{\xi}\mathrm{I}_n \wedge \bar{\zeta},\] This implies that there exists an adapted torsion-free connection \((\omega,\xi,\phi)\) on \(F_0\) such that \[\tag{4.1} \upsilon^*(\omega,\xi,\phi)=(\tilde{\omega},\tilde{\xi},\tilde{\phi}), \; \text{mod}\; \eta,\] i.e. (4.1) holds up to linear combinations of the elements of \(\eta\). Furthermore the Maurer-Cartan equation implies that the curvature forms of this connection all vanish. Summarising we have proved:
The Grassmannian of oriented \(2\)-planes in \(\mathbb{R}^{n+2}\) admits a \(2\)-oriented \(2\)-Segre structure \(\mathcal{S}_0\) together with an adapted, torsion-free, flat connection \((\omega,\xi,\phi)\) such that \(\upsilon^*(\omega,\xi,\phi)=(\tilde{\omega},\tilde{\xi},\tilde{\phi}), \; \text{mod}\; \eta,\) holds.
Let \(\rho : X_0 \to G^+_2(\mathbb{R}^{n+2})\) be the \(D^2\)-bundle associated to the \(2\)-oriented torsion-free \(2\)-Segre structure \(\pi_0 : F_0 \to G^+_2(\mathbb{R}^{n+2})\) and \(\mathfrak{J}_0\) its canonical almost complex structure which makes \((X_0,\mathfrak{J}_0)\) into a quasiholomorphic fibre bundle with fibre \(D^2\). Its total space \(X_0\) can be identified with the quotient \(\mathrm{PL}(n+2,\mathbb{R})/\tilde{P}\) where \(\tilde{P}\) is the closed Lie subgroup \[\tilde{P}=\left\{\left[\begin{array}{cc} a & b \\ 0 & c \end{array}\right]\;\bigg\vert\; a \in \mathrm{GL}(1,\mathbb{C}), b \in M_{\mathbb{R}}(2,n), c \in \mathrm{GL}(n,\mathbb{R}) \right\}.\] Write an element \([g] \in \mathrm{PL}(n+2,\mathbb{R})\) as \([g_1,\ldots,g_{n+2}]\) where the elements \(g_k\) are column-vectors well defined up to a common non-zero factor. Consider the smooth map \[\lambda : \mathrm{PL}(n+2,\mathbb{R}) \to \mathbb{CP}^{n+1}\setminus \mathbb{RP}^{n+1},\quad [g_1,g_2,\ldots,g_{n+2}] \mapsto [g_1+\mathrm{i}g_2].\] Clearly \(\lambda\) is a surjective submersion whose fibres are the \(\tilde{P}\)-orbits and thus induces a diffeomorphism \(\varphi : X_0 \to \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\). Therefore \(\rho_0=\rho \circ \varphi^{-1} : \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\to G^+_2(\mathbb{R}^{n+2})\) is a bundle with fibre \(D^2\). Explicitly \(\rho_0\) is given by \([z] \mapsto \mathbb{R}\{\operatorname{Re}(z),\operatorname{Im}(z)\}\) and the \(2\)-plane \(\mathbb{R}\{\operatorname{Re}(z),\operatorname{Im}(z)\}\) is oriented by declaring \(\operatorname{Re}(z),\operatorname{Im}(z)\) to be a positively oriented basis.
There exists a biholomorphic fibre bundle isomorphism \(\varphi : (X_0,\mathfrak{J}_0) \to \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\) covering the identity on \(G^+_2(\mathbb{R}^{n+2})\).
Proof. Using Lemma 2.7 and Proposition 4.1 it sufficient to show that \(\lambda\) pulls-back the \((1,\!0)\)-forms of \(\mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\) to linear combinations of the forms \(\zeta^1,\ldots,\zeta^n, \tilde{\xi}\). This is a computation which causes no difficulties and so we omit it.
4.2 Smooth quadrics without real points
If \(V\) is a real vector space, \(V_{\mathbb{C}}=V\otimes \mathbb{C}\) will denote its complexification and \(\mathbb{P}(V_{\mathbb{C}})=\left(V_{\mathbb{C}} \setminus \{0\}\right)/\mathbb{C}^*\) its complex projectivisation. An element \([z] \in \mathbb{P}(V_{\mathbb{C}})\) for which \(z\) is a simple vector is called a real point.
The aim of this subsection is to show that the smooth quadrics \(Q \subset \mathbb{CP}^{n+1}=\mathbb{P}(\mathbb{R}^{n+2}_\mathbb{C})\) without real points are in one-to-one correspondence with the sections of the bundle \(\rho_0 : \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\to G^+_2(\mathbb{R}^{n+2})\) having holomorphic image. This is done by reducing the problem to the case \(n=1\) which was shown to be true in [17] (see also [7]).
Let \(\Pi\subset \mathbb{R}^{n+2}\) be a \(3\)-dimensional linear subspace. Choosing an isomorphism \(\mathbb{R}^3\simeq \Pi\) induces an embedding of the \(2\)-sphere \(S^2\simeq G^+_2(\mathbb{R}^3) \hookrightarrow G^+_2(\mathbb{R}^{n+2})\). Clearly the image of this embedding and its induced smooth structure do not depend on the chosen isomorphism and thus \(\Pi\) determines a smoothly embedded \(2\)-sphere in \(G^+_2(\mathbb{R}^{n+2})\) which will be denoted by \(S_\Pi\). Moreover the isomorphism \(\mathbb{R}^3\simeq \Pi\) induces a holomorphic embedding \(\mathbb{CP}^2\simeq \mathbb{P}(\Pi_{\mathbb{C}})\hookrightarrow \mathbb{CP}^{n+1}\) and thus an embedding \(\mathbb{CP}^2\setminus \mathbb{RP}^2 \hookrightarrow \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\). Again the image of this embedding and its induced complex structure do not depend on the chosen isomorphism and thus \(\Pi\) determines a holomorphically embedded submanifold \(Y_\Pi\subset \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\). Restricting the base point projection \(\rho_0 : \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\to G^+_2(\mathbb{R}^{n+2})\) to \(Y_\Pi\) gives a \(D^2\)-bundle \(\rho_\Pi: Y_\Pi\to S_\Pi\) which is isomorphic to the bundle \(\rho_0^2 : \mathbb{CP}^2 \setminus \mathbb{RP}^2 \to G^+_2(\mathbb{R}^3)\), \([z] \mapsto \mathbb{R}\left\{\operatorname{Re}(z),\operatorname{Im}(z)\right\}\).
Recall that for a smooth algebraic hypersurface \(X \subset \mathbb{P}(V_{\mathbb{C}})\), the Gauss map \(\mathcal{G}_X : X \to G_{n-1}(V_{\mathbb{C}})\) sends a point \(x \in X\) to the tangent hyperplane of \(X\) at \(x\). The dual variety \(X^*\) is now defined to be the image of \(X\) under the Gauss map. Usually \(\mathcal{G}_X\) is assumed to take values in \(\mathbb{P}(V^*_{\mathbb{C}})=\mathbb{P}((V_{\mathbb{C}})^*)\simeq G_{n-1}(V_{\mathbb{C}})\). Note that if \(Q\subset \mathbb{P}(V_{\mathbb{C}})\) a smooth quadric without real points. Then the dual of \(Q\) is a smooth quadric without real points in \(\mathbb{P}(V^*_{\mathbb{C}})\). It follows that the intersection of a smooth quadric without real points with a real \(k\)-plane \(\Pi\) of dimension at least \(2\) gives again a smooth quadric without real points in \(\mathbb{P}(\Pi_{\mathbb{C}})\).
The sections of the disk bundle \(\rho_0 : \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\to G^+_2(\mathbb{R}^{n+2})\) having holomorphic image are in one-to-one correspondence with the smooth quadric \(Q\subset \mathbb{CP}^{n+1}\) without real points.
Proof. Let \(\sigma : G^+_2(\mathbb{R}^{n+2})\to \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\) be a \(\rho_0\)-section with holomorphic image \(Q=\operatorname{im}\sigma\). Let \(\Pi\subset \mathbb{R}^{n+2}\) be a \(3\)-dimensional linear subspace and \(\iota_\Pi: S_\Pi\to G^+_2(\mathbb{R}^{n+2})\), \(\tilde{\iota}_\Pi: Y_\Pi\to \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\) the corresponding embedded submanifolds. Then the map \(\sigma \circ \iota_\Pi: S_\Pi\to \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\) is smooth and takes values in \(Y_\Pi\). Consequently the induced map \(\sigma_\Pi: S_\Pi\to Y_\Pi\) is an injective immersion and thus, since \(S_\Pi\) is compact, a smooth embedding. Set \(Q_\Pi=Q\cap Y_\Pi=\sigma_\Pi(S_\Pi),\) then \(Q_{\Pi}\subset Y_\Pi\) is a smoothly embedded submanifold. Now Chow’s theorem implies that \(Q\) is a smooth algebraic hypersurface. Suppose \(P : \mathbb{C}^{n+2} \to \mathbb{C}\) is a homogeneous polynomial defining \(Q\) and let \(P_\Pi: \mathbb{C}^3 \to \mathbb{C}\) denote the homogeneous polynomial obtained by pulling back \(P\) to \(\Pi_{\mathbb{C}}\simeq \mathbb{C}^3\). The map \(P_\Pi\) is a homogeneous polynomial of the same degree as \(P\) which has no real points, since \(P\) has no real points. Under the identification \(Y_\Pi\simeq \mathbb{CP}^2\setminus\mathbb{RP}^2\), \(Q_\Pi\) becomes a smoothly embedded submanifold of \(\mathbb{CP}^2\setminus\mathbb{RP}^2\) defined by the zero locus of the homogeneous polynomial \(P_\Pi\). Since \(Q_\Pi\) is diffeomorphic to the \(2\)-sphere, the genus of \(Q_\Pi\) is \(0\) and thus by the degree-genus formula for smooth plane algebraic curves \(g=(d-1)(d-2)/2,\) the degree of \(P_\Pi\) must be \(1\) or \(2\). However since \(Q_\Pi\) has no real points the degree of \(P_\Pi\) and thus the degree of \(P\) must be \(2\).
Conversely let \(Q\subset \mathbb{CP}^{n+1}\) be a smooth quadric without real points. Let \(\left\{\Pi^{\iota}\right\}_{\iota \in I}\) be a family of \(3\)-dimensional linear subspaces of \(\mathbb{R}^{n+2}\) so that the submanifolds \(S_{\Pi^{\iota}}\) cover \(G^+_2(\mathbb{R}^{n+2})\). Let \(Q_{\Pi^{\iota}}\) denote the intersection of \(Q\) with \(\mathbb{P}(\Pi^{\iota}_{\mathbb{C}})\) which is a smooth quadric without real points. According to [17] each such quadric is the image of a unique section \(\sigma_{\iota} : S_{\Pi^{\iota}} \to X_{\Pi^{\iota}}\). Now for any two \(\Pi^{\iota_1}, \Pi^{\iota_2}\) the spheres \(S_{\Pi^{\iota_1}}\) and \(S_{\Pi^{\iota_2}}\) are either disjoint or intersect in exactly two points. Since for a given \(Q_{\Pi^{\iota}}\) the section \(\sigma_{\iota}\) is unique, it follows that \(Q_{\Pi^{\iota}}\) intersects each \(\rho_{\Pi^{\iota}}\)-fibre in exactly one point. This implies that the sections \(\sigma_{\iota_1}\) and \(\sigma_{\iota_2}\) agree on intersection points and thus the family \(\left\{\sigma_{\iota}\right\}_{\iota \in I}\) gives rise to a unique global section \(\sigma : G^+_2(\mathbb{R}^{n+2})\to \mathbb{CP}^{n+1}\setminus\mathbb{RP}^{n+1}\) with image \(Q\).
The torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reductions \(R \subset F_0\) are in one-to-one correspondence with the smooth quadrics \(Q\subset \mathbb{CP}^{n+1}\) without real points.
Proof. This follows immediately from Theorem 3.4 and Theorem 4.3.
For \(n=2\), the case of conformal \(4\)-manifolds of split-signature, Corollary 4.4 can also be deduced by applying results from [15]. One could also look for \(S^1\cdot \mathrm{GL}(2,\mathbb{R})\)-reductions whose associated almost complex structure is not only integrable, but for which the corresponding conformal structure \([g]\) also contains a Kähler-metric. This, and the related problem in \((4,\! 0)\)-signature has been studied in [10] (see also [15]). Moreover for \(n=2\), Theorem 3.4 has an analogue in \((4,\! 0)\)-signature due to Salamon [19].