COMPLEX GEOMETRY Course notes

Theorem 1.3.3 (Riemann Extension Theorem).
• Part I: If f is a holomorphic function and bounded outside an analytic subset of codimension 2 or
higher, then f extends over the subset as a holomorphic function.
• Part II (also known as the Hartogs Theorem):
(1) Let U = ∆(r) = {(z 1 , z 2 ) / |z 1 | < r and |z 2 | < r} and V = ∆(r ′ ), such that r ′ < r and V ⊂⊂ U,
then every f ∈ O(U − V ) extends to a holomorphic function on U.
(2) If S ⊆ U ⊆ C n has complex codimension greater or equal than 2, where S is an analytic subset
and f ∈ O(U − S), then f extends to a holomorphic function on U.
Proof: We only proof the second part.
(1) Take a slice z 1 = const. Then U − V = {r ′ < |z 2 | < r} on this slice. Set
F (z 1 , z 2 ) = 1 ∫
f(z 1 , w 2 )
dw 2
2πi |w 2|=r w 2 − z 2
Hence F : U −→ C is holomorphic in z 1 since
∂f
= 0 =⇒ ∂F
∂z 1 ∂z 1
and clearly also in z 2 (Cauchy’s Integral Formula). Moreover, F = f on U − V by the Cauchy’s
Integral Formula.
(2) The a 2-dimensional slice and apply (1).
Theorem 1.3.4 (Open Mapping Theorem). If f ∈ O(U) then f is open.
Theorem 1.3.5 (Maximum Principle). |f| has no local maximum unless it is locally constant there.
Theorem 1.3.6 (Analytic continuation). f = 0 in an open subset of U and f ∈ O(U), where U is connected
(or arcwise connected), then f ≡ 0 on U.
Proof: For every path α, the set I = {c / f ◦ α(t) = 0 ∀t < c} is open and closed.
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