Projective structures on a Riemann surface, III

In earlier work we developed an algebraic geometric approach to the notion of a projective structure on a compact Riemann surface and obtained various equivalent descriptions. This was motivated by Mathematical Physics, viz. conformal field theory, which also motivated the subsequent generalisation of these descriptions to larger classes of objects. These may be regarded as generalized projective structures, with any two such descriptions being canonically isomorphic. Here we construct a canonical involution on each space of such generalized projective structures and show that, given two such spaces, the canonical isomorphism between them takes one involution to the other. In this way the symmetry of a Greenʹs function, the classical adjoint of a differential operator and duality of projective embeddings are seen from a common perspective.