Topics in real and complex analysis  Riemann surfaces
PMATH 800

Winter 2014
PMATH 800: Assignment 3
Due: Friday, 21 March, 2014
1. Prove that the holomorphic 1form
dz
,
1 + z2
which is dened on C\cfw_i, can be extended to a holomorphic 1form on P1 \cfw_i. Let
p := tan : C P1 \cfw_i
(see problem 3 on Assignment 2) and nd p .
2.
Topics in real and complex analysis  Riemann surfaces
PMATH 800

Winter 2014
PMATH 800: Assignment 4
Due: Friday, 4 April, 2014
1. Let X be a Riemann surface. Consider the skyscraper sheaf Cp supported at p X (see Assignment
2). Prove that H 0 (X, Cp ) = C and H 1 (X, Cp ) = 0.
2. Let X be a Riemann surface and U X be an open subs
Topics in real and complex analysis  Riemann surfaces
PMATH 800

Winter 2014
PMATH 800: Assignment 2
Due: Wednesday, 26 February, 2014
1. Open Mapping Theorem. The Open Mapping Theorem on C states that if D C is a domain and
f : D C is a nonconstant holomorphic function, then f is open. Show that the theorem extends
to Riemann su