I am trying to understand how to get hold of the algebraic de Rham cohomology for a smooth projective hypersurface (say the zero set of some polynomial $f$). I think I need to find a hypercohomology(?) but remain unsure what this means so I am stuck. I have been told it is easier to get hold of the algebraic de Rham cohomology than the de Rham cohomology of the corresponding smooth manifold
I have a couple of ideas of how to proceed: one way may be by adjunction formula. An understandable explanation or link to an explanation of how to use this would be great.
Another thought is that (I think) the Hodge to De Rham spectral sequence degenerates at the second page in our case so I think I can conclude(?) $$H_{dR}(X) = \sum_{i+j=n}H^i(X,\Omega^j)$$ which reduces my problem to finding $H^i(X,\Omega^j)$ which may be easier?