Let Y = V(F) ⊂ ℙ
n+1 be a smooth hypersurface of degree d. By the exact sequence
0 ® Hn + 1 ( \mathbbPn + 1 \Y,\mathbbC ) ® Hn ( Y,\mathbbC ), ® Hn + 2 ( \mathbbPn + 1 ,\mathbbC ) ® 0,0 \to H^{n + 1} \left( {\mathbb{P}^{n + 1} \backslash Y,\mathbb{C}} \right) \to H^n \left( {Y,\mathbb{C}} \right), \to H^{n + 2} \left( {\mathbb{P}^{n + 1} ,\mathbb{C}} \right) \to 0,
the primitive cohomology of
... [Show full abstract] Y in degree n is identified with the cohomology of its complement in projective space. On the other hand this group can be described, by
Grothendieck’s algebraic de Rham theorem [5], as the space of rational differential n + 1-forms on ℙ
n+1 with poles only along Y modulo exact forms. According to Griffiths [4], this space is filtered by the order of pole of representatives along X and the resulting filtration on H
n
(Y, ℂ) is its Hodge filtration.