COMPLEX GEOMETRY Course notes

(5) C and C ∗ = C − {0}.
(6) The half-plane model H = {z / Im(z) > 0}.
(7) The Poincaré disk model ∆ = D = {z / |z| < 1} and ∆ ∗ = ∆ − {0}. The models H and D are related
by the map
z ↦→ z − a
z − a
The exponential function exp : C −→ C ∗ is a universal covering.
Definition 2.1.2. The manifold C/Γ (and more generally its nontrivial holomorphic images) is called an
elliptic curve. The manifold CP 1 is called a rational curve.
Note that CP 1 has positive curvature, C has zero curvature (in other words, C is said to be flat), and H and
D have negative curvature. Note that Γ is a lattice {n + α / α ∈ H}. The parameter space of elliptic
curves is the quotient H/SL(2, Z). Note that C/Γ has genus 1.
Example 2.1.2.
(1) Let f(z 0 , . . . , z n ) be a homogeneous polynomial. Then
C = V (f) = {[z 0 : z 1 : z 2 ] / f(z 0 , z 1 , z 2 ) = 0} ⊆ CP
is called an algebraic plane curve over C. This cuve C is smooth (or non-singular) if it is a
submanifold (only need to check df(p) ≠ 0 for every p ∈ C to have C smooth).
(2) x d + y d + z d gives the Fermat curve of degree d in CP 2 . It is smooth since df ≠ (0, 0, 0) on C,
df = (x d−1 dx, y n−1 dy, z n−1 dz).
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