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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
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How to explicitly see the ramification over infinity
One way to see this is the following. Let $U$ be the complement of the points $t_1,\ldots,t_n$ in $\mathbb{P}^1(\mathbb{C})$. The fundamental group $\pi_1(U)$ is generated by elements $\gamma_1,\ldots …