- Prove that the canonical almost complex structure, , on a complex manifold is independent of the holomorphic coordinates chosen.
- Show that the following two conditions for an almost complex structure to be a complex structure are equivalent:
- Consider an almost complex structure on given by where , , and is a smooth function of and .
- Check , so this does define an almost complex structure
- Show that condition in exercise 2 is equivalent to
- Deduce that
- The metric associated to the standard inner product on is almost Hermitian, i.e. for all . Show that this is equivalent to being isotropic for the complexification of , i.e. for all .
- Express as a matrix relative to
- Show that, for a Hermitian form, .
- Prove the following lemma:
If with then , i.e.
- Show that projective manifolds are Kähler. The idea is to show that (complex) projective space is Kähler, which requires us to define the Fubini-Study Metric structure on .