Weighted integral formulas on manifolds

Abstract

We present a method of finding weighted Koppelman formulas for $(p,q)$-forms on $n$-dimensional complex manifolds $X$ which admit a vector bundle of rank $n$ over $X \times X$, such that the diagonal of $X \times X$ has a defining section. We apply the method to $\Pn$ and find weighted Koppelman formulas for $(p,q)$-forms with values in a line bundle over $\Pn$. As an application, we look at the cohomology groups of $(p,q)$-forms over $\Pn$ with values in various line bundles, and find explicit solutions to the $\dbar$-equation in some of the trivial groups. We also look at cohomology groups of $(0,q)$-forms over $\Pn \times \Pm$ with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.Comment: 25 page

Explicit representation of membership in polynomial ideals

“…This formula appeared in [18] (Proposition 5.5), and it can be deduced from [19]; however, for the reader's convenience we sketch a direct argument. (1) that is smooth outside the diagonal in P n ζ × P n z .…”

Koppelman formulas on Grassmannians

“…plex manifolds, [8]. The manifolds under consideration in [8] are those satisfying the so called Diagonal Property.…”

“…plex manifolds, [8]. The manifolds under consideration in [8] are those satisfying the so called Diagonal Property. This means that if X has (complex) dimension n, then X Â X should admit a holomorphic vector bundle of rank n with a holomorphic section defining the diagonal D H X Â X , i.e., the section should vanish to first order on D and be non-zero elsewhere.…”

“…This means that if X has (complex) dimension n, then X Â X should admit a holomorphic vector bundle of rank n with a holomorphic section defining the diagonal D H X Â X , i.e., the section should vanish to first order on D and be non-zero elsewhere. The starting point of this paper is the work of the first author in [8]. We will recall only the most essential parts of the general framework in Section 2 and refer to that paper for details.…”

“…We will recall only the most essential parts of the general framework in Section 2 and refer to that paper for details. The statements in Section 2 are, however, somewhat more general than the corresponding ones in [8] since we allow the forms to take values in vector bundles. Section 3 describes some general operations on the weight factors we allow in the formulas.…”