Global residues and intersections on a complex manifold

Abstract

This paper is the study of a class of forms η \eta on a complex manifold V which are smooth on V − W 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 φ \varphi on W of filtration s (no (p, q) components with p > s p > s ) is the residue of a form η \eta of filtration s + d s + d such that d η 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 ∗ η {f^\ast }\eta where f is a rather general holomorphic map.