PMATH 955: The AtiyahSinger Index Theorem
Assignment 4; due Thursday, 13 March 2014
[1] Let (M, g) be a compact oriented Riemannian manifold, and let S be a Cliord bundle over M with Dirac
operator D. Let (D) be the spectrum of D. Given > 0, dene a bounde
PM 955: Complex and Khler Manifolds
a
Assignment 01; due Monday, 24 January 2011
[1]
Let f be a smooth map f : M N between manifolds. The pullback map f takes dierential forms on
N back to dierential forms on M . Show that the pullback commutes with the e
PMATH 955: The AtiyahSinger Index Theorem
Assignment 2; due Thursday, 06 February 2014
[1] Let E be a Kr -vector bundle over M . Let
be a connection on E, and denote again by
the
induced connection on End(E) and by d the induced exterior covariant derivat
PMATH 955: The AtiyahSinger Index Theorem
Assignment 1; due Thursday, 23 January 2014
[1] Let K = R or C and let r 0 be an integer. Let M be a manifold with an open cover U = cfw_U , A.
Suppose that for any , A such that U U = , we are given smooth functi
PM 955: Complex and Khler Manifolds
a
Assignment 02 (REVISED); due Friday, 04 February 2011
[1]
Let M be a complex manifold of complex dimension n. A function f : M C is called holomorphic if
f 1 : (U ) C is holomorphic for every holomorphic chart (U, ) o
PMATH 955: The AtiyahSinger Index Theorem
Assignment 3; due Thursday, 27 February 2014
[1] Let (M n , g) be a compact oriented Riemannian manifold, and let : k (M ) nk (M ) be the associated
Hodge star operator.
[a] Let 1 (M ) and k (M ). Show that
= (1)
PM 955: Complex and Khler Manifolds
a
Assignment 03; due Friday, 18 February 2011
[1]
Let M be a connected complex manifold of complex dimension m > 1 and let g be a Khler metric on
a
M . Show that g is the only Khler metric in its conformal class. That i
PMATH 955: The AtiyahSinger Index Theorem
Assignment 5; due Tuesday, 1 April 2014
[1] Recall the denitions of the Sobolev spaces of C-valued functions W k (T n ) on the torus: for a smooth
function f : T n C, we dene the Sobolev k-form |f |k of f by
|f (p
PM 955: Complex and Khler Manifolds
a
Assignment 05; due Friday, 01 April 2011
[1]
Let M be a compact Khler manifold.
a
(a) Prove that the total volume of M depends only on the cohomology class of its Khler form .
a
(b) Show that there exists no global Kh
PM 955: Complex and Khler Manifolds
a
Assignment 04; due Monday, 14 March 2011
[1]
Let (M 2m , J, g, ) be a complex Hermitian manifold. We can extend the action of J to k-forms as follows:
J : k (T M ) k (T M ),
2m
(Jek ) (ek ),
k=1
where e1 , . . . , e2m
PMATH 955: The AtiyahSinger Index Theorem
Assignment 6; due Monday, 14 April 2014
[1] Let A be a ltered algebra. This means that for all k Z, we have subspaces Ak such that Ak Ak+1 ,
with A = kZ Ak and Ak Al Ak+l . Let G be a graded algebra. This means th