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 NC/M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{C/M}$$\end{document}. 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 NC/M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{C/M}$$\end{document} in Pic0(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Pic}<^>0(C)$$\end{document} which can be regarded as an analogue of the Brjuno condition for irrational numbers which appears in the theory of 1-variable complex dynamics.