Theorems in algebraic geometry which have been proved only by using cohomology There are many theorems in algebraic geometry which were proved using cohomology.
I would like to know examples of such theorems which have been proved only by using cohomology.
In other words, those theorems which have not been so far proven without using cohomology.
I am asking a big list of such examples.
 A: Here are two results usually proved by using cohomology and for which I am not aware of any other proof.  
1) A noetherian scheme $X$ is affine if and only if its reduction $X_{red}$ is affine
The proof uses Serre's characterization of noetherian affine schemes as those noetherian schemes for which  $H^i(X,\mathcal F)=0$  for all   coherent sheaves $\mathcal F$ on $X$ and  all $i\gt 0$ .  
2) A compact complex manifold  $M$ is projective algebraic  if and only if it is a Hodge manifold
A Hodge manifold is a compact complex manifold admitting of a Kähler metric whose associated fundamental form $\phi$ has an integral  De Rham cohomology class: $[\phi]\in H^2_{DR}(M,\mathbb Z)\subset H^2_{DR}(M,\mathbb C)$.
The heart of the proof is that on aHodge manifold there exists a suitable positive line bundle $L$ and for such a positive line bundle we have $H^q(M,\Omega_M^n\otimes L)=0$ for $q\gt 0$ ("Kodaira's vanishing theorem") .
This had been conjectured by Hodge.
Kodaira's proof of that conjecture certainly played a large role in  his being awarded a Fields medal in 1954. 
