Non-Hamiltonian isotopic Lagrangians on the one-point blow-up of CP^2

TL;DR

This paper demonstrates that Hamiltonian isotopic Lagrangians in complex projective space can induce non-Hamiltonian isotopic Lagrangians in its one-point blow-up, revealing new distinctions in Lagrangian isotopy classes.

Contribution

It shows that Hamiltonian isotopic Lagrangians in CP^2 can become non-Hamiltonian isotopic after blow-up, introducing novel examples of Lagrangians with distinct isotopy classes.

Findings

Two Hamiltonian isotopic Lagrangians induce non-Hamiltonian isotopic Lagrangians after blow-up.

For any integer k>1, there are k Hamiltonian isotopic Lagrangians in CP^2 inducing k non-Hamiltonian isotopic Lagrangians.

The work provides explicit examples of Lagrangians with different isotopy classes in the blow-up.

Abstract

We show that two Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) induce two Lagrangian submanifolds in the one-point blow-up (\widetilde{CP}^2,\widetilde{\omega}_\rho) that are not Hamiltonian isotopic. Furthermore, we show that for any integer k>1 there are k Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) that induce k Lagrangian submanifolds in the one-point blow-up such that no two of them are Hamiltonian isotopic.