Chern currents of coherent sheaves

TL;DR

This paper constructs explicit Chern currents for coherent sheaves using Hermitian metrics and connections, linking them to fundamental cycles and residue currents, thus extending classical Chern class theory to singular settings.

Contribution

It introduces a method to explicitly construct Chern currents for coherent sheaves from resolutions and Hermitian data, relating them to residue currents and fundamental cycles.

Findings

The constructed Chern current represents the sheaf's Chern class.

For pure dimension sheaves with (1,0)-connections, the current's first component matches the fundamental cycle.

The approach generalizes the Poincaré-Lelong formula to coherent sheaves.

Abstract

Given a finite locally free resolution of a coherent analytic sheaf $\mathcal F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $\mathcal F$, that represents the Chern class of $\mathcal F$ and has support on the support of $\mathcal F$. If the connections are $(1,0)$-connections and $\mathcal F$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $\mathcal F$. The proof of this goes through a generalized Poincar\'e-Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.