C⁰-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

Abstract: A natural question bridging the celebrated Gromov–Eliashberg theorem and the C⁰-flux conjecture is whether the identity component of the group of symplectic diffeomorphisms is C⁰-closed in Symp(M,ω). Beyond surfaces and the cases in which the Torelli subgroup of Symp(M,ω) coincides with the identity component, little is known. In joint work with Cheuk Yu Mak and Wewei Wu, we show that, for all but a few positive rational surfaces, the group of Hamiltonian diffeomorphisms is the C⁰-connected component of the identity in Symp(M,ω), thereby giving a positive answer in this setting. Here, “positive rational surface” essentially means a k-point blow-up of CP² whose symplectic form evaluates positively on the first Chern class.

algebraic geometrydifferential geometrysymplectic geometry

Audience: researchers in the topic

Geometria em Lisboa (IST)

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