Relative differential forms and complex polynomials
The authors study the algebraic de Rham complex for polynomials \(f\colon {\mathbb C}^n \to {\mathbb C}\). For \(f\) with isolated singularities on \({\mathbb C}^n\) and with generic fibre cohomologically \((n-2)\)-connected their main result states that \(H^0(\Omega_f^*,d)={\mathbb C}[f]\) and \(H^k(\Omega_f^*,d)=0\) for \(0<k<n-1\). If moreover \(f\) is cohomologically tame or \(M\)-tame the \({\mathbb C}[t]\)-module \(H^{n-1}(\Omega_f^*,d)\) is free and of finite rank equal to the total Milnor number of \(f\). In case \(f\) is semi-weighted homogeneous an explicit basis is produced. The final two sections describe in detail the torsion in \(H^{n-1}(\Omega_f^*,d)\) in the case \(n=2\), when the polynomial \(f\) has non-isolated singularities or non-connected fibres.
scientific article; zbMATH DE number 1547230