Rational Integrals of the second kind on a complex projective manifold and its primitive cohomology

TL;DR

This paper investigates the mixed Hodge structure and Hodge filtration of cohomology groups on a complex algebraic manifold and its hyperplane section using rational integrals, providing a detailed algebraic geometric analysis.

Contribution

It offers a new description of the mixed Hodge structure and Hodge filtration in terms of rational integrals of the second kind and the generalized Poincare residue map.

Findings

Explicit description of the mixed Hodge structure on H^p(X-Y,C)

Hodge filtration characterization of primitive cohomology group H^n(Y,C)_0

Application of rational integrals to algebraic geometric invariants

Abstract

Let X be a complex algebraic manifold of dimension n+1 embedded in a sufficiently higher dimensional complex projective space, and Y a generic hyperplane section of X. We describe the mixed Hodge structure on H^p(X-Y,C) and the Hodge filtration of the middle primitive cohomology group H^n(Y,C)_0 of Y in terms of rational integrals on X. (Key words: Primitive cohomology, Rational integral of the 2nd kind, Generalized Poincare residue map, Hodge filtration, Mixed Hodge structure)