Limit holomorphic sections and Donaldson's construction of symplectic submanifolds

TL;DR

This paper reformulates Donaldson's construction of symplectic submanifolds as a compactness result, linking approximately holomorphic sections to their limit holomorphic sections and their transversality properties.

Contribution

It provides a new perspective on Donaldson's construction by framing it as a compactness theorem relating approximate and actual holomorphic sections.

Findings

Approximately holomorphic sections converge to limit holomorphic sections.

Uniform transversality of approximate sections corresponds to transversality of limits.

Reformulation offers a compactness-based understanding of symplectic submanifold construction.

Abstract

In this note, we reformulate Donaldson's construction as a compactness result. Approximately holomorphic sections accumulate to "limit holomorphic sections" and uniform transversality properties of the approximately holomorphic sections correspond to transversality properties of their limits.