Open access
Author
Date
2023Type
- Journal Article
ETH Bibliography
yes
Altmetrics
Abstract
The main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for n = 2, the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to C⁰ symplectic topology. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000665175Publication status
publishedExternal links
Journal / series
Mathematische AnnalenPublisher
SpringerOrganisational unit
02500 - Forschungsinstitut für Mathematik / Institute for Mathematical Research
More
Show all metadata
ETH Bibliography
yes
Altmetrics