Bounds on spectral norms and barcodes

Kislev, Asaf¹ ; Shelukhin, Egor²

¹ School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
² Department of Mathematics and Statistics, University of Montreal, Montreal, QC, Canada

Geometry & topology, Tome 25 (2021) no. 7, pp. 3257-3350.

Résumé

We investigate the relations between algebraic structures, spectral invariants and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov-type low-energy intersection phenomena, and nondegeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.

Détail

DOI : 10.2140/gt.2021.25.3257

Classification : 57R17, 53D12, 53D40
Keywords: Hamiltonian diffeomorphisms, Lagrangian submanifolds, Floer homology, persistence modules, barcodes, spectral norm, spectral invariants

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