Residue currents and fundamental cycles

TL;DR

This paper generalizes the Poincaré-Lelong formula by expressing the fundamental cycle of an analytic space through residue currents and differential forms, extending classical results to a broader class of spaces.

Contribution

It introduces a new factorization of the fundamental cycle using residue currents and differential forms, generalizing previous work to more complex analytic spaces.

Findings

Provides a current-based formula for fundamental cycles

Extends classical Poincaré-Lelong formula

Generalizes Lejeune-Jalabert's results for Cohen-Macaulay spaces

Abstract

We give a factorization of the fundamental cycle of an analytic space in terms of certain differential forms and residue currents associated with a locally free resolution of its structure sheaf. Our result can be seen as a generalization of the classical Poincar\'e-Lelong formula. It is also a current version of a result by Lejeune-Jalabert, who similarly expressed the fundamental class of a Cohen-Macaulay analytic space in terms of differential forms and cohomological residues.