445-2 Dierential Geometry
Northwestern University
Solution of Homework 1
1) Let p, q be two positive integers and consider the C-action on (Cp \cfw_0)
given by
(Cq \cfw_0)
t (z1 , . . . , zp , w1 . . . , wq ) = (et z1 , . . . , et zp , eit w1 , . . . , ei
445-2 Dierential Geometry
Northwestern University
Solution of Homework 2
1) Let Xp,q be the complex manifold dieomorphic to S 2p1 S 2q1
which you constructed in Homework 1, problem 1. For which values of p, q
is Xp,q Khler?
a
Solution. When p = q = 1 we g
445-2 Dierential Geometry
Northwestern University
Solution of Homework 3
1) Let (X n , g ) be a Khler manifold of complex dimension n > 1, and
a
let f be a smooth positive nonconstant real function on X . Prove that the
conformally rescaled Hermitian metr
445-2 Differential Geometry Northwestern University Solution of Homework 4 1) Let X be a compact complex manifold. Prove that there cannot exist two Khler metrics , on X (no assumption on the cohomology classes) a with Ric() > 0 and Ric( ) 0. Solution. Co
445-2 Differential Geometry Northwestern University Solution of Homework 5 1) Let (X, ) be a Khler manifold of complex dimension n a that it is Khler-Einstein if and only if a Ric() = R , n 2. Prove
where R = g ij Rij is the scalar curvature. Solution. If
445-2 Differential Geometry Northwestern University Solution of Homework 6 1) Let (X, ) be a compact Khler manifold of complex dimension n > 1, a and let u : X R be a smooth positive function that satisfies u -Au,
where A 1 is a constant, and is the Lapla
445-2 Differential Geometry Northwestern University Solution of Homework 7 1) Let (X, ) be a compact Khler manifold of complex dimension n > 1, a F : X R be a smooth function with X eF n = X n and let : X R be a smooth function with + -1 > 0 solving the c
445-2 Differential Geometry Northwestern University Solution of Homework 8 1) Let (X, ) be a compact Khler manifold of complex dimension n a with Ric() - for some > 0. Let be another Khler metric on X ~ a ~ -A for some A > 0. with scalar curvature R ~n Ap