Embeddings and disjunction of Lagrangian pinwheels via rational blow-ups

TL;DR

This paper investigates the existence and disjunction of Lagrangian pinwheels in symplectic rational manifolds using rational blow-ups, providing classifications for certain symplectic forms and answering a specific question about Lagrangian $L_{3,1}$ pinwheels.

Contribution

It introduces new methods to analyze Lagrangian pinwheels via rational blow-ups and classifies symplectic forms that support disjoint Lagrangian projective planes.

Findings

Certain symplectic forms in the threefold blow-up of $\\C P^2$ support disjoint Lagrangian projective planes.

Disjunction of Lagrangian pinwheels is impossible in del Pezzo surfaces with Euler characteristic 4 to 7.

Identifies symplectic forms on $S^2\times S^2$ supporting a Lagrangian $L_{3,1}$ pinwheel.

Abstract

We use the symplectic rational blow-up to study some Lagrangian pinwheels in symplectic rational manifolds. In particular, we determine which symplectic forms in the threefold blow-up of $\C P^2$ carry Lagrangian projective planes that can be made disjoint by a Hamiltonian isotopy. In addition, we show that such a disjunction is not possible in del Pezzo surfaces with Euler characteristic between $4$ and $7$. Finally, we determine which symplectic forms on $S^2\times S^2$ carry a Lagrangian $L_{3,1}$ pinwheel, answering a question of J. Evans.