Here’s a fun fact: let be a smooth proper surface over a perfect field of positive characteristic , if there is no global -form on , then the Hodge-to-de-Rham spectral sequence (for ) degenerates at page. Below we will give two proofs that are totally different which builds on many (very) nontrivial results.
Here’s one proof. Look at the conjugate spectral sequence. It starts at page and is contained in the box. By Serre duality, both and are zero. Therefore by plain formalism of spectral sequence, it degenerates at page. Recall that the Hodge-to-de-Rham spectral sequence (for ) degenerates at page is equivalent to the conjugae spectral sequence degenerates at page, therefore we are done.
Here’s another proof. Again by Serre duality and the configuration of the spectral sequence, it suffices to show that the differential map is zero. By Serre duality, we see that which implies that the Picard variety of is reduced. Therefore the pullback map is an isomorphism, where is the Albanese variety of . Since the differentials on any abelian variety is zero, we get what we want.
Maybe I should propose a vote with the question “which proof is better” and let people choose between “the first one”/”the second one”/”neither because you use a steam-hammer to crack nuts, fool!”.