Cohomologies of rational differential forms of degree n in the space \({\mathbb{C}}\) n (Q1108341)

The aim of this paper is to compute explicitly \(H^ n({\mathbb{C}}^ n\setminus V)\) where \(V{\mathbb{C}}^ n\) is an algebraic hypersurface in \({\mathbb{C}}^ n.\) Let \(Q=0\) be the equation of V and \(Q_ q\) the principal part of Q (i.e. the homogeneous part of maximal degree (\(=q)\) of Q). The principal part of Q is called nondegenerate if \(d_ z(Q_ q)\) vanishes only in \(z=0\in {\mathbb{C}}^ n.\)- There are two results: Theorem 1. Suppose V is smooth and the principal part of Q is nondegenerate. Then: \(H^ n({\mathbb{C}}^ n\setminus V)\cong {\mathbb{C}}[z]/\dot {\mathcal I}_ q\) and has dimension \((q-1)^ n\). Moreover if \(\{e_{\beta}\}\) is a monomial base in \({\mathbb{C}}[z]/\dot {\mathcal I}_ q\) then the images of the rational differential forms \(\{e_{\beta}dz_ 1\wedge...\wedge dz_ n/Q\}\) give a base in \(H^ n({\mathbb{C}}^ n\setminus V)\). Here \(\dot {\mathcal I}_ q\) is the jacobian ideal of \(Q_ q.\) Theorem 2. Suppose \(V{\mathbb{C}}^ 2\) is an irreducible curve and the principal part of Q is nondegenerate. Then: \(H^ 2({\mathbb{C}}^ 2\setminus V)\cong {\mathbb{C}}[z]/K_ Q\) where \(K_ Q=\{P\in {\mathbb{C}}[z]| PQ^ 2\) belongs to the jacobian ideal of Q\}. The proof use among other things the Leray exact sequence and the mixed Hodge structure of \(H^{n-1}(V)\).