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Let $X,Y$ be complex manifolds and let $f: X \rightarrow Y$ be a continuous map. When exactly do we say that $f$ is "holomorphic"? I am interested in the basic definition (possibly using charts), not an equivalent characterization.

Thanks.

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1 Answer 1

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If $p$ is a point of $X$, then we say $f$ is holomorphic at $p$ if there is a chart $(U, \phi)$ on $X$ with $p \in U$, and a chart $(V, \psi)$ on $Y$ with $\phi(p) \in V$, such that $f(U) \subseteq V$ and the induced map

$$ \phi(U) \stackrel{\phi^{-1}}\longrightarrow U \stackrel{f}{\to} V \stackrel{\psi}{\to}\mathbb{C}^n $$

is holomorphic in the familiar sense (using $\phi(U) \subseteq \mathbb{C}^n$).

We say $f$ is holomorphic if it is holomorphic at all points.

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Do we say that if $f$ satisfies the property you described using charts for a point $p$, then it is "locally holomorphic"? –  Manos Sep 18 '12 at 18:04

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