Nevanlinna theory and Plucker identities

The main results of Nevanlinna theory are obtained by integration of the equations given by the local First and Second Main Theorems.; The First Main Theorem for sections of a holomorphic hermitian bundles was proved by Bott and Chern. A section of a vector bundle can be viewed as a special case of a vector bundle homomorphism, and the vanishing of a section is a special (extreme) case of degeneracy of a vector bundle map. In the first chapter we prove the analogue of the local First Main Theorem for degeneracy loci of bundle homomorphisms.; The Second Main Theorem for projective curves relates higher curvature forms of a curve and its singularities. In the second chapter we generalize the local Second Main Theorem to the case of holomorphic curves in grassmannians. To do this we define the sequence of the higher curvature forms on the curve in the grassmanian. These forms are matrix-valued and satisfy the recursive relations similar to the ones that hold for projective curves. In the last section we prove the analogue of the Plucker identities for projective surfaces.