Can someone give an example or a reference on this?
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$\begingroup$ It helps to have some really basic examples... look up the "j-invariant" and "elliptic curves". Also need some little fact like a holomorphic diffeomorphism between smooth complex manifolds is a biholomorphism. $\endgroup$– user36931Sep 15, 2013 at 16:13
1 Answer
An example: the map $z \in \mathbb{C} \mapsto \bar{z} \in \mathbb{C}$. A reference: Kobayashi, Shoshichi, Transformation groups in differential geometry. Springer-Verlag, Berlin, 1995. viii+182 pp. ISBN: 3-540-58659-8 53C10 (53-02), where you will find a proof that any compact complex manifold has only a finite dimensional Lie group of automorphisms. More generally, the holomorphic maps $X \to Y$ of fixed action on homology between two complex manifolds, if $X$ is compact, form a finite dimensional analytic space. But the nonholomorphic maps form an infinite dimensional manifold.