COMPLEX GEOMETRY Course notes

This shows that C ∼ = hol P 1 . For example, we have the Fermat curve z 2 0 + z 2 1 = z 2 2. If P 1 = {[u : v]}, then
define a map z 0 = (u − v) 2 , z 1 = 2uv and z 2 = (u + v) 2 .
Note that π 1 (S) = 0 where S = P 1 , C, D. Recall that if π 1 (S) = Z then S is not compact.
Example 2.3.4. S 1 = C ∗ , S 2 = D − 1 2D. These two examples are not biholomorphic Riemann surfaces.
Neither D ∗ is biholomorphic to C ∗ . If so, then a biholomorphic function D ∗ −→ C ∗ produces an extension
D −→ S, getting a contradiction.
The Riemann surfaces H and D are biholomorphic via the map
ω ↦→ w − a
w − a
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