I'm reading Rick Miranda, Algebraic Curves and Riemann Surfaces. I'm having problems with this exercise about complex charts.
Let $\phi_i: U_i \to V_i$, $i=1,2$, be complex charts on $X$ with $U_1\cap U_2 \ne \emptyset$. Suppose that $\phi_2 \circ \phi_1^{-1}: \phi_1 \left(U_1 \cap U_2 \right) \to \phi_2\left(U_1 \cap U_2 \right)$ is holomorphic. Show that it is bijective with inverse $\phi_1 \circ \phi_2^{-1}: \phi_2 \left(U_1 \cap U_2 \right) \to \phi_1\left(U_1 \cap U_2 \right)$.
It's easy to prove that the map $\phi_2 \circ \phi_1^{-1}$ is bijective with inverse $\phi_1 \circ \phi_2^{-1}$ (over the sets $\phi_i \left( U_1 \cap U_2 \right)$. But I don't know how to prove that this inverse is also holomorphic. Maybe a holomorphism which is also a homeomorphism it's biholomorphism but I'm not sure.
Yes, this follows from the holomorphic inverse function theorem (which should not be too hard to prove if you know how to prove the regular inverse function theorem.)