A theory of characteristic currents associated with a singular connection

TL;DR

This paper develops a comprehensive theory of characteristic currents for singular connections, unifying classical topics and providing new formulas and proofs in differential geometry and algebraic geometry.

Contribution

It introduces a general construction of characteristic currents for singular connections, extending classical theories like Chern-Weil and Poincaré-Lelong to singular settings.

Findings

Provides a new proof of the Riemann-Roch Theorem for algebraic curves.

Generalizes the Poincaré-Lelong Formula to smooth contexts.

Derives universal formulas for Thom classes as equivariant characteristic forms.

Abstract

This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps $\alpha : E \rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ \phi \ = \ \text{\rm Res}_{\phi}\Sigma_{\alpha} + dT. $$ Here $\phi$ is a standard charactersitic form, $\text{Res}_{\phi}$ is an associated smooth ``residue'' form computed canonically in terms of curvature, $\Sigma_{\alpha}$ is a rectifiable current depending only on the singular structure of $\alpha$, and $T$ is a canonical, functorial transgression form with coefficients in $\loc$. The theory encompasses such classical topics as: Poincar\'e-Lelong Theory, Bott-Chern Theory, Chern-Weil Theory, and formulas of Hopf. Applications include:\ \ a new proof of the Riemann-Roch Theorem for vector bundles over algebraic curves, a $C^{\infty}$-generalization of the Poincar\'e-Lelong Formula, universal formulas for the Thom class as an equivariant characteristic form (i.e., canonical formulas for a de Rham representative of the Thom class of a bundle with connection), and a Differentiable Riemann-Roch-Grothendieck Theorem at the level of forms and currents. A variety of formulas relating geometry and characteristic classes are deduced as direct consequences of the theory.