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 open affine subset V = Spec B of Y , f -1 (V ) can be c
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 bundles, vector bundles, cohomology (of topological space
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, subset of C. Let U be an open
set containing K and let f be
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, is the cohomology of the complex :
0 H0 (W ) H1 (W ) H2
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 product line bundle C X , then sections of L
can be identied
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 GLr C
holomorphic, then (possibly after shrinking U, V t
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 then (after possibly shrinking U and V to U and V , st
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-dimensional polydisc and W X a d-dimensional complex submanifold
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 introduce some notation and vocabulary:
Let : E X be a
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 . Consider the sheaves C E, C F ,
the sheaves of smooth sect
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, by taking derivatives of sections. Later,
well talk ab
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 ?
(3) Given a function f on R2 , what is d f ?
2. Let X b
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 ) is the holomorphic
functions on U H . Construct a sho
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 polydisc V , where K V U , and a
(p, q 1) form on V such tha
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 inclusion of open sets V U X , we have a map U : E (U ) E (V
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 q > 0, then the cohomology of E is the cohomology of th
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, let
fn =
1
2i
f (z )
dz
z n+1
where the integral is o
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 sequence:
0 Obk Obk1 Ob0 E 0
where all of the maps are maps
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. Find a
continuous function f : C C such that, o of the
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 innite long exact sequence,
0
A(X )
B (X )
C (X )
H 1 (X
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 groups H k (X, A); the coecients A matter. For reasonable
sp
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 Hol is the sheaf of holomorphic functions, C is the shea
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:
0 Obk Obk1 Ob0 E 0
where all of the maps are maps of O-
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:
0 Obk Obk1 Ob0 E 0
where all of the maps are maps of O-
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 + (q ir)dz2 dz1 + sdz2 dz2 for real numbers p, q , r an
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 map of vector bundles)
d : k k1
d = dd + d d.
If X was
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 if p U . Let D be
any divisor on X .
(1) Show that there
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 through the following sequence of
maps:
k
H k (X, Z) H k (X
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 form , on L. In this setup, we get a connection on L,
=D+D