Problem 0.1. (a) Show that a morphism f : X Y is of finite type if and only if it is locally of finite type and quasi-compact. (b) Conclude from this that f is of finite type if and only if for every
PROBLEM SET 1
DUE TUESDAY, JANUARY 18
0 Which of the following have you seen before: smooth manifolds, dierential forms, analytic
functions, residues (of meromorphic functions), (p, q )-forms, line bu
PROBLEM SET 5
DUE FEBRUARY 15, 2011
1. (Runges approximation theorem in one variable). The point of this problem is to prove the
following result: Let K be a connected, simply connected, compact, subs
NOTES FOR FEBRUARY 3
CHRIS FRASER
We are headed towards proving:
If U a polydisc, W U a closed smooth C-submanifold of dimension d then the topological
k
cohomology Htop (W, C), with coecients in C, i
PROBLEM SET 6
DUE FEBRUARY 24, 2011 NOTE UNUSUAL DATE
1. Let L be a trivial complex line bundle on X , some real manifold. Let
be a connection on
L. If we choose an isomorphism between L and the produ
Notes for February 10
Scribe: Georey Scott
Last class, we proved Cartans Lemma.
Cartans Lemma: Given two polyboxes K, L that share an edge, open sets U and V around
K and L respectively, and
H : U V G
NOTES FOR FEBRUARY 8
ADAM KAYE
Lemma (Cartan). Let K, L be closed polyboxes in Cn with a common side, U, V
open neighborhoods of K, L respectively. Given H : U V GLr (C) holomorphic in
each coordinate
Notes for February 15
Scribe: Justin Campbell
Our goal for this class is to put together all of the results from the previous lectures. At this point, not
much is left.
Let X Cn be an open n-dimension
NOTES FOR FEBRUARY 17
BROOKE ULLERY
Three ways to think about vector bundles:
(1) as ber bundles
(2) in terms of gluing data
(3) as locally free sheaves
1. Vector Bundles as Fiber Bundles
First, well
NOTES FOR 24 FEBRUARY 2011
KEVIN CARDE
1. Terminological Clarifiation
Lets start by clearing up and making precise some abuses of terminology.
Suppose E, F X are two smooth vector bundles over X . Con
NOTES FOR FEBRUARY 22
XIN ZHOU
1. Defining Connections
Motivation for connections: Wed like to compare vectors in dierent bers of a vector bundle.
At rst, well talk about how to do this innitesimally,
PROBLEM SET 7
DUE MARCH 15, 2011
1. This problem takes place on R2 , with its standard inner product.
(1) Given a 1-form = a(x, y )dx + b(x, y )dy , what is ?
(2) With the notation above, what is d ?
PROBLEM SET 4
DUE FEBRUARY 8, 2011
1. Let O be the sheaf of holomorphic functions on Cn . Let H be a hypersurface in Cn , dened
by F = 0 for some analytic function F . Let OH be the sheaf where OH (U
Notes for February 1
Scribe: Yi Su
February 4, 2011
Recap from last time: Given that K is a closed polydisc, U is an open polydisc, K U and
a (p, q )-from on U with = 0, then there is a smaller polyd
NOTES FOR JANUARY 13
TENGREN ZHANG
1. Definitions and Examples of Sheaves and Presheaves
Let X be a topological space. A presheaf E gives a set E (U ) for every open set U X such that
for every inclus
NOTES FOR JANUARY 20
E. HUNTER BROOKS
On Tuesday, we saw the following result: if we have an exact sequence
0 E C0 C1 . . .
of sheaves on a space X , and we know that
H q (C k ) = 0
for all k and for
PROBLEM SET 2
DUE JANUARY 25, 2011
1. (A complex analysis lemma) Let s1 < s2 be real numbers, let A be the annulus cfw_z : s1 <
|z | < s2 in C. Let f be an analytic function on A. For every integer n
THE MISSING PARTS OF CARTANS THEOREM
Let D Cn be a rectangular box (product of rectangles in each coordinate). Let E be a sheaf
of O-modules on D. We dened E to have depth k if there is an exact seque
PROBLEM SET 3
DUE FEBRUARY 1, 2011
1. Compute:
(1) |z |2 , on C.
(2) Re(z )/z .
(3) xy 1 (xx + y y )1 , on C C .
2. Let g be the function on C given by g (z ) = 1 if |z | 1 and g (z ) = 0 if |z | > 1.
NOTES FOR JANUARY 18
PEDRO ACOSTA
Recall that, given a short exact sequence of sheaves on X
0 A B C 0,
we have the long exact sequence
0 A(X ) B (X ) C (X ).
Sheaf cohomology will extend this to an in
NOTES FOR 25 JANUARY 2011
KEVIN CARDE
1. Sheaf Cohomology Examples
Basic point: Sheaf cohomology depends on the space and the sheaf. In basic topology, for X a
space, A an abelian group, we have group
NOTES FOR 1-27
EMILY CLADER
We are headed toward an analogue of the Poincar Lemma for the operator. Our eventual goal
e
is to prove that the sequence of sheaves
0 Hol C 0,1 0,2 0,n 0
is exact, where H
THE MISSING PARTS OF CARTANS THEOREM
Let R Cn be a polybox (product of rectangles in each coordinate). Let E be a sheaf of
O-modules on R. We dened E to have co-depth k if there is an exact sequence:
THE MISSING PARTS OF CARTANS THEOREM
Let R Cn be a polybox (product of rectangles in each coordinate). Let E be a sheaf of
O-modules on R. We dened E to have co-depth k if there is an exact sequence:
PROBLEM SET 8
DUE MARCH 22, 2011
Working with Hermitian forms in coordinates
1. On C2 , let the coordinates be z1 = x1 + iy1 and z2 = x2 + iy2 . Consider the Hermitian form
pdz1 dz1 + (q + ir)dz1 dz2
NOTES FOR MARCH 10
ADAM KAYE
This material coincides with chapter 5 in Voisins book. Last time we looked at
X a smooth manifold with positive denite symmetric bilinear form on T X and
dened
: k nk (a
PROBLEM SET 11
DUE APRIL 12, 2011
1. (Riemann-Roch for curves) Let X be a compact complex curve and let p be a point of X .
Let Cp be the skyscraper sheaf at p, meaning that Cp (U ) = C if p U and 0 i
NOTES FOR APRIL 5TH
SCRIBE: XIN ZHOU
k
1. The map H k (X, Z) HDR (X, C)
Given a geometric cocycle, how does it decompose under Hodge decomposition?
The bridge between the two is deRham cohomology thro
NOTES FOR APRIL 12
E. HUNTER BROOKS
1. Statement of the Kodaira embedding theorem
Last time, we considered a complex manifold X , a holomorphic line bundle L on X , and a
positive denite Hermitian for