Computing Hodge-theoretic invariants of singularities

Abstract

Let Y = V(F) ⊂ ℙⁿ⁺¹ be a smooth hypersurface of degree d. By the exact sequence

$$0 \to H^{n + 1} \left( {\mathbb{P}^{n + 1} \backslash Y,\mathbb{C}} \right) \to H^n \left( {Y,\mathbb{C}} \right), \to H^{n + 2} \left( {\mathbb{P}^{n + 1} ,\mathbb{C}} \right) \to 0,$$

the primitive cohomology of 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 ℙⁿ⁺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.