On the Flux Conjectures

TL;DR

This paper proves the Flux conjecture for certain classes of symplectic manifolds, showing Hamiltonian diffeomorphisms are closed in the symplectic diffeomorphism group under specific conditions, and extends results to torus actions.

Contribution

It establishes the Flux conjecture for spherically rational manifolds and those with large or vanishing minimal Chern number, and confirms a version for symplectic torus actions.

Findings

Proves Flux conjecture for spherically rational manifolds.

Confirms Flux conjecture for symplectic torus actions.

Shows Hamiltonian diffeomorphisms are C^0-closed in some cases.

Abstract

The ``Flux conjecture'' for symplectic manifolds states that the group of Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic diffeomorphisms. We prove the conjecture for spherically rational manifolds and for those whose minimal Chern number on 2-spheres either vanishes or is large enough. We also confirm a natural version of the Flux conjecture for symplectic torus actions. In some cases we can go further and prove that the group of Hamiltonian diffeomorphisms is C^0-closed in the identity component of the group of all symplectic diffeomorphisms.