On Holomorphic Tubular Neighborhoods of Compact Riemann Surfaces

Let C be a smooth compact Riemann surface holomorphically embedded in a non-singular complex surface M with the unitary flat line bundle \(N_{C/M}\) $N_{C/M}$ . We give a sufficient condition for the existence on a holomorphic tubular neighborhood of C in M. Our sufficient condition is described by an arithmetical condition of \(N_{C/M}\) $N_{C/M}$ in \(\textrm{Pic}^0(C)\) ${\text{Pic}}^0(C)$ which can be regarded as an analogue of the Brjuno condition for irrational numbers which appears in the theory of 1-variable complex dynamics.