Are the chart maps of a complex manifold necessarily biholomorphic? I know that the transition maps of a complex manifold are biholomorphic, but are the chart maps themselves also biholomorphic? I know that it is the case for real smooth manifolds (here the chart maps are diffeomorphisms). Thanks for your help!
By definition, if $g:U\subset\mathbb{C}\to g(U)\subset X$ is a chart, then the function $g^{-1}:g(U)\to U$ is holomorphic, because $g^{-1}\circ g:U\to U$ is holomorphic for being the identity function on $U$. Likewise, $g:U\to g(U)$ is holomorphic because $g^{-1}\circ g\circ i:U\to U$ is holomorphic, where $i$ is the identity function on $U$, which is also a chart of $\mathbb{C}$. It is a bit redundant because the definition of what functions are holomorphic on $X$ is made such that the charts are iso morphisms between their domain and their range in the corresponding category.
By definition, if $g:U\subset\mathbb{C}\to g(U)\subset X$ is a chart, then the function $g^{-1}:g(U)\to U$ is holomorphic, because $g^{-1}\circ g:U\to U$ is holomorphic for being the identity function on $U$. Likewise, $g:U\to g(U)$ is holomorphic because $g^{-1}\circ g\circ i:U\to U$ is holomorphic, where $i$ is the identity function on $U$, which is also a chart of $\mathbb{C}$. It is a bit redundant because the definition of what functions are holomorphic on $X$ is made such that the charts are iso morphisms between their domain and their range in the corresponding category.