Residue currents of cohesive modules and the generalized Poincar\'{e}-Lelong formula on complex manifolds

TL;DR

This paper develops the theory of residue currents for cohesive modules on complex manifolds, establishing duality and comparison principles, and generalizes the Poincaré-Lelong formula to include coherent sheaves without global resolutions.

Contribution

It introduces residue currents for cohesive modules, proving duality and comparison formulas, and extends the Poincaré-Lelong formula to broader classes of coherent sheaves.

Findings

Residue currents for cohesive modules satisfy duality and comparison formulas.

A generalized Poincaré-Lelong formula is established for coherent sheaves.

The results apply to sheaves without globally defined locally free resolutions.

Abstract

Cohesive module provides a tool to study coherent sheaves on complex manifolds by global analytic methods. In this paper we develop the theory of residue currents for cohesive modules on complex manifolds. In particular we prove that they have the duality principle and satisfy the comparison formula. As an application, we prove a generalized version of the Poincar\'{e}-Lelong formula for cohesive modules, which applies to coherent sheaves without globally defined locally free resolutions.