Modifications of torsion-free coherent analytic sheaves

TL;DR

This paper investigates how torsion-free coherent analytic sheaves transform under proper modifications, demonstrating that they can often be represented as direct images of vector bundles, simplifying their study.

Contribution

It establishes conditions under which torsion-free sheaves are direct images of locally free sheaves, enabling analysis via vector bundles on manifolds.

Findings

Torsion-free sheaves can be realized as direct images of vector bundles.

Application to ideal sheaves and Grauert-Riemenschneider canonical sheaf.

Reduces complex sheaf problems to vector bundle analysis.

Abstract

We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert-Riemenschneider canonical sheaf of holomorphic n-forms.