Generalised Lelong-Poincaré formula in complex Bott-Chern cohomology

In this note, we present a topological proof of the generalized Lelong-Poincaré formula. More precisely, when the zero locus of a section has a pure codimension equal to the rank of a holomorphic vector bundle, the top Chern class of the vector bundle corresponds to the cycle class of the schematic zero locus of the section in complex Bott-Chern cohomology.