3 Reductions of \(\beta\)-integrable \(2\)-Segre structures
We will henceforth consider the \(\beta\)-integrable case and assume \(\rho : X_\mathcal{S}\to M\) to be equipped with its canonical integrable almost complex structure \(\mathfrak{J}\) with respect to which \((X_\mathcal{S},\mathfrak{J})\) is a quasiholomorphic fibre bundle. By construction the sections of the bundle \(\rho : X_\mathcal{S}\to M\) correspond to reductions of the principal \(H\)-bundle \(\pi : F_\mathcal{S}\to M\) with structure group \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\). Note that an \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction of a \(\beta\)-integrable \((2,n)\)-Segre structure \(\pi : F_\mathcal{S}\to M\) equips \(M\) with an almost complex structure preserving the Segre cones \(\mathcal{S}_p\) and for which the \(\beta\)-planes are totally real. In this section we will show that the torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reductions of \(F_\mathcal{S}\) are in one-to-one correspondence with the sections \(\sigma : M \to X_\mathcal{S}\) having holomorphic image \(\sigma(M)\subset X_\mathcal{S}\). This will done using exterior differential systems (eds). The notation and terminology for eds are chosen to be consistent with [5].
A basis for the Lie algebra of \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\) is given by \[a\otimes \mathrm{I}_n, \quad \mathrm{I}_2\otimes e^i_k.\] Suppose \(R \to M\) is a torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure with adapted connection \(\theta\). Write \[\theta=a\alpha\otimes \mathrm{I}_n+\mathrm{I}_2\otimes \beta,\] for some \(1\)-form \(\alpha\) and some \(\mathfrak{gl}(n,\mathbb{R})\)-valued \(1\)-form \(\beta\) on \(R\). Let \(\zeta\) denote the pullback of the canonical \(\mathbb{C}^n\)-valued \(1\)-form to \(R\), then \(\zeta\) satisfies \[\tag{3.1} \mathrm{d}\zeta=-\left(\mathrm{i}\alpha\, \mathrm{I}_n+\beta\right) \wedge \zeta,\] as was already observed in [8].
We will need the following Lemma whose proof is straightforward and thus omitted:
Let \((X,J)\) be a complex \((n+1)\)-manifold, \((\mu^1,\ldots,\mu^n,\kappa) \in \mathcal{A}^1(X,\mathbb{C})\) a basis for the \((1,\!0)\)-forms of \(J\) and \(f : \Sigma \to X\) a \(2n\)-submanifold with \[\tag{3.2} f^*\left(\mathrm{i}\mu^1\wedge\bar{\mu}^1\wedge\cdots\wedge\mathrm{i}\mu^n\wedge\bar{\mu}^n\right)\neq 0.\] Then \((f,\Sigma)\) is a complex submanifold if and only if \[f^*\left(\kappa\wedge\mu^1\wedge\cdots\wedge\mu^n\right)=0.\] Moreover through every point \(p \in X\) passes such a complex submanifold.
On \(F_\mathcal{S}\) define the exterior differential system \[\mathcal{I}=\langle \xi\wedge\zeta^1\wedge\cdots\wedge\zeta^n\rangle\] together with the independence condition \[Z=\mathrm{i}\zeta^1\wedge\bar{\zeta}^1\wedge\cdots\wedge\mathrm{i}\zeta^n\wedge\bar{\zeta}^n.\] The eds \((\mathcal{I},Z)\) is of interest because of the following:
Let \(\sigma : M \to X_\mathcal{S}\) be an \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction of \(F_\mathcal{S}\) and \(\tilde{\sigma} : U \to F_\mathcal{S}\) a local coframing covering \(\sigma\). Then \(\tilde{\sigma}\) is an integral manifold of \((\mathcal{I},Z)\) if and only if \(\sigma\vert_U : U \to X_\mathcal{S}\) is a complex submanifold.
Proof. Let \(s : \rho^{-1}(U) \to F_\mathcal{S}\) be a local section of the bundle \(\nu : F_\mathcal{S}\to X_\mathcal{S}\) and let \(\mu^i=s^*\zeta^i\) for \(i=1,\ldots,n\) and \(\kappa=s^*\xi\) be a local basis for the \((1,\!0)\)-forms on \(\rho^{-1}(U)\). Then \[\begin{align} \nu^*\mu^i&=(s\circ \nu)^*\zeta^i=(R_t)^*\zeta^i,\\ \nu^*\kappa&=(s\circ \nu)^*\xi=(R_t)^*\xi, \end{align}\] for some smooth function \(t : \pi^{-1}(U) \to S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\). Recall that for \(e^{i\varphi}\cdot b \in S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\) we have \[\begin{align} \big(R_{e^{\mathrm{i}\varphi}\cdot b}\big)^*\xi&=e^{-2\mathrm{i}\varphi}\xi,\\ \big(R_{e^{\mathrm{i}\varphi}\cdot b}\big)^*\zeta&=\left(e^{-\mathrm{i}\varphi} \cdot b^{-1}\right)\zeta. \end{align}\] This yields \[\nu^*\left(\mathrm{i}\mu^1\wedge\bar{\mu}^1\wedge\cdots\wedge \mathrm{i}\mu^n\wedge\bar{\mu}^n\right)=(\det b)^{-2}\,Z\neq 0\] for some smooth map \(b : \pi^{-1}(U) \to \mathrm{GL}(n,\mathbb{R})\) and \[\nu^*\kappa=e^{-2\mathrm{i}\varphi}\xi\] for some smooth function \(\varphi : \pi^{-1}(U) \to \mathbb{R}\). Hence we get \[\left(\sigma\vert_U\right)^*\left(\mathrm{i}\mu^1\wedge\bar{\mu}^1\wedge\cdots\wedge\mathrm{i}\mu^n\wedge\bar{\mu}^n\right)=\left((\det b)^{-2} \circ \tilde{\sigma}\right)\tilde{\sigma}^*Z\] which vanishes nowhere since \(\tilde{\sigma}\) is a \(\pi\)-section. Therefore according to Lemma 3.1, \(\sigma\vert_U : U \to X_\mathcal{S}\) is a complex submanifold if and only if \[\left(\sigma\vert_U\right)^*\left(\kappa\wedge\mu^1\wedge\cdots\wedge\mu^n\right)=\left(\left(\frac{e^{-(n+2)\mathrm{i}\varphi}}{\det b}\right)\circ \tilde{\sigma}\right)\tilde{\sigma}^*\left(\xi\wedge\zeta^1\wedge\cdots\wedge\zeta^n\right)=0.\]
Recall that \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\subset \mathrm{GL}(n,\mathbb{C})\) and we can thus look for reductions \(\sigma : M \to X_\mathcal{S}\) whose associated almost complex structure \(\mathfrak{J}_{\sigma}\) is integrable.
Let \(\sigma : M \to X_\mathcal{S}\) be an \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction of \(\pi : F_\mathcal{S}\to M\). Then the following two statements are equivalent:
The almost complex structure \(\mathfrak{J}_{\sigma}\) is integrable.
Any local coframing \(\tilde{\sigma} : U \to F_\mathcal{S}\) covering \(\sigma\) is an integral manifold of \((\mathcal{I},Z).\)
Proof. Since \(\tilde{\sigma}\) is a \(\pi\)-section we have \(\tilde{\sigma}^*Z \neq 0\). Write \(\chi^i=\tilde{\sigma}^*\zeta^i\). The local coframing \(\tilde{\sigma}\) is adapted to the \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction \(\sigma\) and thus the forms \(\chi^i\) are a local basis of the \((1,\!0)\)-forms of \(\mathfrak{J}_{\sigma}\). By Newlander-Nirenberg \(\mathfrak{J}_{\sigma}\) is integrable if and only if there exist complex-valued \(1\)-forms \(\pi^i_k\) such that \[d\chi^i=\pi^i_k\wedge\chi^k.\] Using the structure equation (2.5) we get \[\tag{3.3} \mathrm{d}\chi^i=\tilde{\sigma}^*d\zeta^i=\tilde{\pi}^i_k\wedge\chi^k-\mathrm{i}\tilde{\sigma}^*\xi\wedge\bar{\chi}^i\] for some complex-valued \(1\)-forms \(\tilde{\pi}^i_k\). Write \[\tilde{\sigma}^*\xi=x_k\chi^k+y_k\bar{\chi}^k\] for some smooth complex-valued functions \(x_k,y_k\) on \(U\). Then (3.3) implies that \(\mathfrak{J}_{\sigma}\) is integrable on \(U\) if and only if the functions \(y_k\) all vanish. This condition is equivalent to \(\tilde{\sigma}\) being an integral manifold of \((\mathcal{I},Z)\).
We are now ready to prove:
Let \(\pi : F_\mathcal{S}\to M\) be a \(\beta\)-integrable \(2\)-Segre structure. Then an \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-reduction \(R \subset F_\mathcal{S}\) is torsion-free if and only if \(\nu(R)\subset X_\mathcal{S}\) is a complex submanifold.
Proof. Let \(\nu(R)=\sigma(M)\) for some \(\rho\)-section \(\sigma : M \to X_\mathcal{S}\) which has holomorphic image, then by Lemma 3.2 and Proposition 3.3, the almost complex structure \(\mathfrak{J}_{\sigma}\) is integrable. This is equivalent to \(\xi\) satisfying \(\xi=x_k \zeta^k\) for some smooth complex-valued functions \(x_k\) on \(R\). Pulling back the structure equation (2.5) to \(R\subset F_\mathcal{S}\) gives \[\tag{3.4} \mathrm{d}\zeta=-\big(\mathrm{i}\left(\omega-x_k \zeta^k\right)\mathrm{I}_n+\phi\big)\wedge\zeta-\mathrm{i}\,x_k \zeta^k\,\mathrm{I}_n \wedge \bar{\zeta}.\] Define \[\begin{align} \alpha&=\omega-\operatorname{Im}(x_k)\operatorname{Im}(\zeta^k),\\ \beta^j_l&=\phi^j_l-\operatorname{Re}(\mathrm{i}\bar{x}_l\zeta^j)-\delta^j_l\operatorname{Im}(x_k)\operatorname{Re}(\zeta^k),\\ \end{align}\] then the \(1\)-form \(\theta=a\alpha\otimes\mathrm{I}_n+\mathrm{I}_2\otimes\beta\) is a linear connection on \(R\) which satisfies \[\tag{3.5} \mathrm{d}\zeta=-\left(\mathrm{i}\alpha\mathrm{I}_n+\beta\right)\wedge\zeta,\] thus \(R\) is torsion-free. Conversely suppose the reduction \(\sigma : M \to X_\mathcal{S}\) is torsion-free, so that on \(R=(\nu^{-1}\circ \sigma)(M)\) there exists a linear connection \(\theta=a\alpha\otimes\mathrm{I}_n+\mathrm{I}_2\otimes\beta\) satisfying (3.5). Pulling back \((\omega,\xi,\phi)\) to \(R\) gives \[\tag{3.6} \begin{align} \omega&=\alpha+a_k\left(\zeta^k+\bar{\zeta}^k\right)+\mathrm{i}\tilde{a}_k\left(\zeta^k-\bar{\zeta}^k\right)\\ \xi&=x_k\zeta^k+\tilde{x}_{k}\bar{\zeta}^k\\ \phi^i_k&=\beta^i_k+f^i_{kl}\left(\zeta^k+\bar{\zeta}^k\right)+\mathrm{i}\bar{f}^i_{kl}\left(\zeta^k-\bar{\zeta}^k\right) \end{align}\] for some smooth complex-valued functions \(a_k,\tilde{a}_k,x_k,\tilde{x}_k,f^i_{kl},\tilde{f}^i_{kl}\) on \(R\). Subtracting (3.4) from (3.5) and using (3.6) gives in components \[\tag{3.7} 0=\cdots+\mathrm{i}\left(x_k\zeta^k+\tilde{x}_k\bar{\zeta}^k\right)\wedge\bar{\zeta}^i\] where the unwritten summands are not of the form \(\bar{\zeta}^k\wedge\bar{\zeta}^i\). It follows that (3.7) can hold for every \(i=1,\ldots,n\) if and only if \(\tilde{x}_k=0\).
Locally every \(\beta\)-integrable \(2\)-Segre structure \(\pi : F_\mathcal{S}\to M\) can be reduced to a torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure.
Proof. For a given point \(p \in M\), choose \(q \in X_\mathcal{S}\) with \(\rho(q)=p\) and a coordinate neighbourhood \(U_p\). Let \(\mu^i\), \(i=1,\ldots,n\) and \(\kappa\) be a basis for the \((1,\! 0)\)-forms on \(\rho^{-1}(U_p)\) as constructed in Lemma 3.2. Using Lemma 3.1 there exists a complex \(2n\)-submanifold \(f : \Sigma \to \rho^{-1}(U_p)\) passing through \(q\) for which \[f^*\left(\mathrm{i}\mu^1\wedge\bar{\mu}^1\wedge\cdots\wedge\mathrm{i}\mu^n\wedge\bar{\mu}^n\right)\neq 0.\] Since the \(\pi : F_\mathcal{S}\to M\) pullback of a volume form on \(M\) is a nowhere vanishing multiple of \(Z=\mathrm{i}\mu^1\wedge\bar{\mu}^1\wedge\cdots\wedge\mathrm{i}\mu^n\wedge\bar{\mu}^n\), the \(\rho\) pullback of a volume form on \(U_p\) is a nowhere vanishing multiple of \(Z\) and hence \(\rho \circ f : \Sigma \to U_p\) is a local diffeomorphism. Composing \(f\) with the locally available inverse of this local diffeomorphism one gets a local \(\rho\)-section which is defined in a neighbourhood of \(p\) and which is a complex submanifold. Applying Theorem 3.4 it follows that \(\pi : F_\mathcal{S}\to M\) locally has an underlying torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure.