Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space

Abstract

Let Gr_L⊂ Gr(n, V) be the space of all Lagrangian subspaces C²ⁿ of equipped with the standard symplectic form. Let X^~ be a universal cover of a compact connected Riemann surface X. We consider all immersions f : X^~→ Gr_L satisfying the following two conditions: (1) the map f is equivariant with respect to some homomorphism into Sp(2n,C) of the Galois group of the covering X^~ → X, and (2) the symmetric bilinear form on the pullback, to X, of the tautological vector bundle over Gr_L is fiberwise nondegenerate. Two such maps are called equivalent if they differ by the action of some fixed element of Sp(2n,C). We prove that the equivalence classes of all such maps are bijectively parametrized by pairs of the form (P, (F,∇ ) where P is a projective structure on X and (F,∇ ) is an equivalence class of flat O(n,C)-connection on X. Two flat O(n,C)-bundles are equivalent if the corresponding flat PO(n,C)-bundles are isomorphic.