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 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

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

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

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