Show that, for a Hermitian form, .
Let us start by just using the definition of in terms of our Riemannian metric
Here we've used
. Now using bilinearity of
Now we recall that
, so we obtain
Prove that any projective manifold admits a Kähler metric.
The solution to this exercise boils down to the construction of the Fubini-Study metric on . Indeed, suppose we are provided with a Kähler metric on the projective space and let be a complex (projective) submanifold and let us denote with the inclusion . We claim that, if is a Kähler form on , then is a Kähler form on : in fact, is a real -form, is positive defined and , because the pullback commutes with the exterior differential and is closed.
Given this, let us show the construction of the Fubini-Study metric on : consider local sections of the tautological bundle , , which can be thought as a function . Then, by definition of line bundle, there is a holomorphic function such that . Therefore, and this fact proves that the quatity does not depend on the section we choose. We thus can define a global real -form by setting
is a local section of the tautological bundle.
Using the identities
, one can easily check that
is closed. In order to see that it is the form associated to a riemannian metric, we write it in local coordinates, choosing
. A straightforward computation shows that, at
and the corresponding Hermitian metric is a real scalar multiple of the identity, which is positive definite. We claim that it is enough to check the positive definiteness of the form only at one point: indeed
acts trasitively on
. This concludes our proof.
Show that 4k dimensional spheres do not admit almost complex structures.
The basic topological feature of the tangent bundle is that it is stably trivial. This is because the sphere sits inside the Euclidean space, and the normal bundle is a 1D trivial bundle. Thus .
This means the Chern classes of the complexified tangent bundle are trivial.
Now suppose we have an almost complex structure, then the complexified bundle splits into (1,0) and (0,1) parts which are complex vector bundles. Most of the Chern classes vanish for purely degree reasons. But the top Chern class of the (1,0) bundle is the Euler characteristic times the fundamental class, by Gauss Bonnet. Now since the top Chern class is we observe the conjugate bundle has the same Chern class. Then an application of Whitney sum formula gives a contradiction of the value of the top Chern class of the complexified tangent bundle.