TITLE: Cohomology of Hypersurface complement
QUESTION [9 upvotes]: Suppose $Y$ is a smooth hypersurface in projective space $\mathbb{P}^n$, $X = \mathbb{P}^n - Y$ is the hypersurface complement. Is there a general method to compute cohomology of $X$? In particular, for small n, is there any examples or references?

REPLY [3 votes]: Griffiths (On the periods of certain rational integrals. I, II. Ann. of Math. 90 (1969), 460-495 & 90 (1969), 496–541.) gave a procedure to calculate the Hodge numbers of Y. Let $S=\mathbf{C}[x_0,\dots,x_n]$, let $f$ be an equation for $Y$ and $d$ be its degree. Let $J_f$ be the jacobian ideal of $f$, i.e., the ideal generated by the partial derivatives of $f$. Then
$$ h^{i,n-1-i}(Y)=\delta_{i,n-1-i}+\dim (S/J_f)_{id-n-1}.$$
The other Hodge numbers can be obtained from by hyperplane theorems: i.e., if $p+q\neq n-1$ then $h^{p,q}(Y)=1$ if and only if $0\leq p=q\leq n-1$ holds. All other Hodge numbers are zero.
Here, $\delta_{i,j}$ is the Kronecker delta, $S$ has a natural grading, $J_f$ is generated by homogeneous elements and therefore $S/J_f$ is a graded ring. $(S/J_f)_k$ means all elements of degree $k$.
The above formula can be generalized, see e.g., Steenbrink (Intersection form for quasi-homogeneous singularities. Compositio Math. 34 (1977), 211–223) for weighted projective spaces of Dimca (Betti numbers of hypersurfaces and defects of linear systems.
Duke Math. J. 60 (1990), 285–298) for hypersurfaces with isolated weighted homogeneous singularities.