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

Abstract

We consider the space of all holomorphic immersions of the universal cover of a compact hyperbolic Riemann surface into CP n satisfying the condition that at every point of the image, the order of contact of the image with any hyperplane in CP n is at most n−1. A further restriction that is imposed states as follows: there exists a homomorphism of the Galois group for the universal cover into GL(n+1, C) such that the map from the universal cover to CP n is equivariant for the actions of the Galois group. To each such immersion we associate a i-form on the compact Riemann surface for each i∈[3, n+1], and also associate a projective structure on the Riemann surface. The resulting map from the space of all 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, where the equivalence relation identifies an immersion with any other immersion obtained by composing it with an automorphism of CP n .