Characteristic Currents on Cohesive Modules

Let $\mathcal{F}$ be a coherent sheaf on a complex variety $X$ that has a locally free resolution $E^{\bullet}$. In [19], the authors constructed a pseudomeromorphic current whose support is contained in $supp(E^{\bullet})$ that represents products of Chern classes of $\mathcal{F}.$ In this paper, we show that their construction works for general de-Rham characteristic classes and then generalize it to represent products (in de-Rham cohomology) of characteristic forms of cohesive modules defined by Block. Finally, we state a corollary to a transgression result in [16] that show that it is sufficient to only use the degree-$0$ and degree-$1$ parts of the superconnection to construct currents that represent characteristic forms of cohesive modules in the Bott-Chern cohomology.