The Nevanlinna's First Main Theorem for Holomorphic Hermitian Line Bundles

In 1925, Nevanlinna's first and second fundamental theorems were established by introducing characteristic function. Later, the first main theorem for complex analytic maps of several variables was researched by S. S. Chern. Then, H.J.W. Ziegler extended the classical Nevanlinna's theory of meromorphic functions to vector-valued meromorphic functions on a complex plane to finite dimensional spaces. This chapter presents the Nevanlinna's first main theorem for holomorphic Hermitian line bundles in order to lay a foundation of value distribution theory in infinite dimensional spaces. It also describes the Green-Residue theorem that yields the first main theorem for Hermitian line bundles.