1. Prove that the canonical almost complex structure, , on a complex manifold is independent of the holomorphic coordinates chosen.
  2. Show that the following two conditions for an almost complex structure to be a complex structure are equivalent:

  1. Consider an almost complex structure on given by where , , and is a smooth function of and .
    1. Check , so this does define an almost complex structure
    2. Show that condition in exercise 2 is equivalent to

    1. Deduce that
    2. 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 .
    3. Express as a matrix relative to
  1. Show that, for a Hermitian form, .
  2. Prove the following lemma:

If with then , i.e. Starting hint:

  1. 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 .