Differential operators and immersions of a Riemann surface into a Grassmannian

Abstract

We consider equivariant holomorphic immersions of a universal cover X of a compact Riemann surface X into a Grassmannian G(n,C²ⁿ) satisfying a nondegeneracy condition. The equivariance condition says that there is a homomorphism ρ of the Galois group to GL(2n,C) that takes the natural action of the Galois group on X to the action of the Galois group on G(n,C²ⁿ) defined using ρ. We prove that the space of such embeddings are in bijective correspondence with the space of all holomorphic differential operators of order two on a rank n vector bundle over X with the property that the symbol of the operator is an isomorphism.