Residue currents and cycles of complexes of vector bundles

TL;DR

This paper generalizes the Poincaré-Lelong formula by expressing the cycle of a complex of vector bundles using differential forms and residue currents, extending previous results to more general complexes.

Contribution

It introduces a new factorization of the cycle of complexes of vector bundles using residue currents, broadening the classical Poincaré-Lelong formula to more general settings.

Findings

Provides a new factorization formula for cycles of complexes

Extends classical results to complexes beyond locally free resolutions

Connects residue currents with geometric cycles

Abstract

We give a factorization of the cycle of a bounded complex of vector bundles in terms of certain associated differential forms and residue currents. This is a generalization of previous results in the case when the complex is a locally free resolution of the structure sheaf of an analytic space and it can be seen as a generalization of the classical Poincar\'e-Lelong formula.