Global residues and intersections on a complex manifold

Abstract:

This paper is the study of a class of forms $\eta$ on a complex manifold V which are smooth on $V - W$ and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form $\varphi$ on W of filtration s (no (p, q) components with $p < s$) is the residue of a form $\eta$ of filtration $s + d$ such that $d\eta$ is smooth on V. This result is used to construct global kernels for $\bar \partial$ which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of ${f^\ast }\eta$ where f is a rather general holomorphic map.