Let $\pi: E\rightarrow B$ be a $G$-holomorphic principal bundle where $E,B$ are complex manifolds and $G$ is connected complex Lie group. If $f$ is a holomorphic function on $E$ such that $f$ is constant on each fiber $G$. How to show that $f$ induces a holomorphic function on $B$?
2026-03-27 17:59:33.1774634373
Constant function on fibers induces holomorphic function on base
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At least set-theoretically $f$ surely induces a function $g : B \to \mathbb C$. We just need to show that $g$ is holomorphic. The question is local so we can assume that $E$ is the trivial bundle $G \times B$, but then $s : B \to B \times G, b \mapsto (b,e_G)$ is holomorphic and $g = f \circ s$.