The sheaf $\alpha$ $\bullet$ X

TL;DR

This paper introduces a new coherent sheaf of meromorphic differential forms on complex spaces with universal pull-back properties, integral dependence relations, and local integrability and continuity features.

Contribution

It defines a novel sheaf of meromorphic forms with universal pull-back, explores its relation to Nash transform, and studies its integrability and continuity properties.

Findings

The sheaf has the universal pull-back property for holomorphic maps.

Meromorphic forms in this sheaf satisfy integral dependence equations.

Forms are locally square-integrable and their integrals are locally bounded and continuous.

Abstract

We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf $\omega\_{X}^{\bullet}$ which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $\Omega\_{X}^{\bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $\Omega\_{X}^{\bullet}/torsion$. This sheaf $\alpha\_{X}^{\bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic differential forms are locally square-integrable on any $q-$dimensional cycle in $X$ and that the corresponding functions obtained by integration on an analytic family of $q-$cycles are locally bounded and locally continuous on the complement of closed analytic subset.