Invariants for a class of equivariant immersions of the universal cover of a compact Riemann surface into a projective space (Q1569013)

Let \(X\) be a compact connected hyperbolic Riemann surface, and \(\pi:\widetilde{X}\to X\) a universal cover of \(X\) with Galois group \(\Gamma\). A map \(\gamma\) from \(\widetilde{X}\) into projective space \(\mathbb{C}\mathbb{P}^n\) is called everywhere locally nondegenerate if for any \(y \in \widetilde{X}\) and any hyperplane \(H\) passing through \(\gamma(y)\) the order of contact of \(\widetilde{X}\) with \(H\) is at most \(n-1\). The author studies the space of pairs \((\rho,\gamma)\), where \(\rho\) is a homomorphism from \(\Gamma\) to \(GL(n+1,\mathbb{C})\) and \(\gamma\) is a \(\Gamma\)-equivariant everywhere locally nondegenerate map (immersion) from \(\widetilde{X}\) to \(\mathbb{C}\mathbb{P}^n\). To each such immersion \(\gamma\) he associates an \(i\)-form on the Riemann surface \(X\) for every \(i \in [3,n+1]\) and a projective structure on \(X\). The resulting map from the space of immersions surjects onto the target space. Moreover, this map gives a bijective correspondence between the target space and the space of all equivalence classes of immersions (two immersions are equivalent if they differ by an automorphism of \(\mathbb{C}\mathbb{P}^n\)).