Ellipsoidal superpotentials and singular curve counts

TL;DR

This paper introduces new invariants for counting pseudoholomorphic curves with singularities and ellipsoidal ends in symplectic manifolds, revealing their relation to symplectic embedding obstructions and curve classification.

Contribution

It establishes a novel equivalence between counts of singular curves and curves with ellipsoidal ends, providing new geometric insights and classification results in symplectic geometry.

Findings

New invariants for counting singular and punctured pseudoholomorphic curves

Explicit equivalence between singular curve counts and ellipsoidal end counts

Classification of rigid unicuspidal curves in the first Hirzebruch surface

Abstract

Given a closed symplectic manifold, we construct invariants which count (a) closed rational pseudoholomorphic curves with prescribed cusp singularities and (b) punctured rational pseudoholomorphic curves with ellipsoidal negative ends. We prove an explicit equivalence between these two frameworks, which in particular gives a new geometric interpretation of various counts in symplectic field theory. We show that these invariants encode important information about singular symplectic curves and stable symplectic embedding obstructions. We also prove a correspondence theorem between rigid unicuspidal curves and perfect exceptional classes, which we illustrate by classifying rigid unicuspidal (symplectic or algebraic) curves in the first Hirzebruch surface.