Riemann surface

Summary

A Riemann surface is a connected space that looks locally like an open piece of the complex plane, with changes of coordinates that are holomorphic. It is a complex manifold of complex dimension one. On a Riemann surface one can speak of holomorphic and meromorphic functions. This makes it the natural home of multivalued functions such as $\sqrt{z}$ and $\log z$. Compact Riemann surfaces are the same objects as smooth projective complex algebraic curves.