On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

Liat Kessler; Nikolas Wardenski

Home/Archive/arXiv

arXiv · 2505.14000

On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

Liat Kessler·Nikolas Wardenski

View original source ↗Read source document ↗Export metadata ↓

Abstract

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian $S^1$-manifold of dimension six is determined by certain data on the critical levels. We first give counter examples showing that Gonzales' assumptions are not sufficient for a positive answer. Then we prove that it is enough to further assume that the reduced spaces of dimension four are symplectic rational surfaces and the interior fixed surfaces are restricted to at most one level. The additional assumptions allow us to use results proven by $J$-holomorphic methods. Gonzales' answer was applied by Cho in proving that if the underlying symplectic manifold is positive monotone then the space is isomorphic to a Fano manifold with a holomorphic $S^1$-action. We show that our variation is enough for Cho's application.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Liat Kessler, Nikolas Wardenski. 2025-05-20. On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data. https://arxiv.org/abs/2505.14000

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Darboux's Theorem in $p$-adic symplectic geometry

We prove a non Archimedean Darboux's Theorem: any two symplectic forms on a $p$-adic analytic manifold are locally isomorphic. Understanding local problems such as the existence of flows or the normalization of singularities in the theory of integrable systems, is essential to understand the physics behind these systems. Our result tells us that the phase space defined by a $p$-adic manifold is locally standard, allowing us to concentrate on the equations defining the dynamics rather than on the space itself. Our proof uses a non Archimedean version of Moser's Path Method to push one symplectic form onto another one by a flow. A central technical contribution of the paper is the proof that the flow is given by a power series with non zero radius of convergence, which requires geometric analytic estimates and does not follow from algebraic considerations. The main difficulty of the paper is analytic: an ODE will not have a long-time solution even in a neighborhood of a fixed point but we are able to find a condition for a long-time solution to exist, which allows us to tweak the ODE in Moser's Path Method so that it satifies the condition. As a global application, we derive a classification of second-countable $p$-adic analytic symplectic manifolds in terms of $p$-adic volume, which generalizes a classical theorem of J-P. Serre.

math.SG↗

arXiv2026

A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

math.SG↗

arXiv2026

The Legendrian Hopf Link has exactly two Lagrangian fillings

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such Lagrangian fillings: we show these are all. Specifically, we use a type of neck-stretching procedure to construct a pseudoholomorphic conic fibration that makes a given arbitrary exact Lagrangian filling fiber over a real curve, under a global pseudoholomorphic Lefschetz fibration. This then allows for an explicit Hamiltonian isotopy to be constructed from any given Lagrangian filling to one of two known standard fillings.

math.SG↗