Residues in intersection homology and L_p-cohomology

Abstract

Suppose $M^{n+1}$ is a complex manifold and K is a hypersurface with isolated singularities. Let $\omega$ be a holomorphic form on $M\setminus K$ with the first order pole on K. The Leray residue of such form gives an element in the n-th homology of K which is the Alexander dual to $[\omega ]\in H^{n+1}(M\setminus K)$. It always lifts to the intersection homology if 0 does not belong to the spectra of a singular points. We assume that singularities are described by the quasihomogeneous equations in certain coordinate systems. Suppose that oscillation indicators of the singular points are greater then -1. Then we find a metric on $K\setminus\Sigma$ in which the residue form is square integrable (and even L_p-integrable for p>2). Applying the isomorphism of L_p-cohomology and intersection homology we obtain a particular lift of the residue class in homology to intersection homology.