Symplectic packings in dimension 4 and singular curves

TL;DR

This paper introduces new constructive methods for symplectic embeddings in four dimensions, highlighting the connection between packings and singular curves, and demonstrates maximal packings of the projective plane with up to eight balls.

Contribution

It develops a novel inflation technique at the manifold level, providing constructive proofs for symplectic packing existence results and linking them to singular symplectic curves.

Findings

Constructive proofs for symplectic packing existence

New inflation technique at the manifold level

Maximal packings of P^2 with 6, 7, or 8 balls

Abstract

The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff's inflations on the blow-ups, is revisited through a new inflation technique that lives at the level of the manifold. As an application, we explain constructions of maximal symplectic packings of $P^2$ by 6, 7 or 8 balls.