Random fun fact with two totally different proofs

Here’s a fun fact: let X be a smooth proper surface over a perfect field k of positive characteristic p > 0, if there is no global 2-form on X, then the Hodge-to-de-Rham spectral sequence (for X) degenerates at E_1 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 E_2 page and is contained in the [0,2] \times [0,2] box. By Serre duality,  both E_2^{0,2} and E_2^{2,0} are zero. Therefore by plain formalism of spectral sequence, it degenerates at E_2 page. Recall that the Hodge-to-de-Rham spectral sequence (for X) degenerates at E_1 page is equivalent to the conjugae spectral sequence degenerates at E_2 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 H^1(\mathcal{O}_X) \to H^1(\Omega^1_X) is zero. By Serre duality, we see that H^2(\mathcal{O}_X) = 0 which implies that the Picard variety of X is reduced. Therefore the pullback map H^1(\mathrm{Alb}(X), \mathcal{O}) \to H^1(X,\mathcal{O}_X) is an isomorphism, where \mathrm{Alb}(X) is the Albanese variety of X. 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!”.


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