Applications of singular connections in symplectic and almost complex geometry

TL;DR

This paper applies the theory of singular connections to derive a Lelong-Poincaré formula for vector bundles on almost complex manifolds and proves a convergence theorem for divisors related to symplectic submanifolds.

Contribution

It introduces new applications of singular connection theory to almost complex and symplectic geometry, extending previous results to broader contexts.

Findings

Established a Lelong-Poincaré formula for vector bundles on almost complex manifolds.

Proved a convergence theorem for divisors associated with symplectic submanifolds.

Extended previous hypersurface results to more general cases.

Abstract

In this paper, we give two direct applications of the theory of singular connections developped by Harvey-Lawson [10]. The first one is a version of Lelong-Poincar\'e formula for vector bundle over an almost complex manifold. The second is a convergence theorem for divisors associated to symplectic submanifolds constructed by Auroux in [2]. The case of hypersurfaces was done by Donaldsson in [4].