Notes for 1-27

Let us first examine this sequence in the 1-dimensional case. When the underlying manifold is C, the claim is that 0→ Hol→ C∞ ∂ −→ Ω → 0 is exact. Clearly, Hol → C∞ is injective, and we observed last class that Hol is precisely the kernel of the map ∂ : C∞ → Ω0,1, so the only difficulty in verifying exactness is in showing that ∂ : C∞ → Ω0,1 is surjective. That is, given a smooth function g : C→ C, we would like to produce a smooth function f : C→ C with ∂f = g dz, or in other words, ∂ ∂z f = g.