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For a smooth hypersurface $X\subset\mathbb{P}^n_k$, where $k$ is an algebraic closed field of charactersitc $p>0$. How to compute its algebraic de Rham cohomology explicitly? or equivalently its Hodge number?

PS：In characteristic $0$ case, one has Lefschetz hyperplane theorem,Hirzebruch-Riemann-Roch and Hodge theory.

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    $\begingroup$ The same formulas as in characteristic $0$ hold in characteristic $p> 0$. See the exposition by Deligne in SGA 7 II, Expose XI. $\endgroup$
    – naf
    Feb 22, 2012 at 14:33

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