Why is a meromorphic section without zeros and poles on a compact Riemann surface necessarily a constant? Thank you very much.
2026-03-27 04:56:40.1774587400
Reading Griffiths Harris: Quick question
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Suppose you have a global holomorphic section $s$ of a line bundle $L$ on a compact, connected Riemann surface $M$. $s$, being nowhere zero, gives a holomorphic trivialization of the line bundle; i.e., $L \cong M\times \Bbb C$. Holomorphic sections of the trivial bundle are holomorphic functions, and the only holomorphic functions on such an $M$ are constant functions (for example, non-constant holomorphic functions are open maps).