HOMEWORK ASSIGNMENTS
Homework Assignment I (for the undergraduates in 18.117.) Due Friday April 12
- Problem 1 (Page 12 in the course notes) Prove that for polydisks the
Dolbeault complex is exact.
- Problem 2 (Page 16 in the course notes) Prove that for convex open subsets
of $C^n$ the holomorphic De Rham complex is exact.
- Problem 3 (Also on page 16.) Let U be a polydisk and ω a closed
(p,q) form on U. Show that there exists a (p-1,q-1) form μ
that satisfies the Dolbeault equation on line 15. Extra credit: Describe
the set of all solutions of this equation.
- Problem 4 (Page 35 in the course notes) Prove that the Fubini-Study form
on complex projective n-space is Kaehler-Einstein.
- Problem 5 (Lecture 15 in the course notes.) Let X be a complex
n-dimensional manifold. Prove a Dolbeault version of the De Rham theorem
for the sheaf of holomorphic p-forms on X.
- Problem 6 (Pages 52 and 53 in the course notes.) Prove that the Fredholm
alternative holds for smoothing operators