Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2112.12368

Construction of a linear K-system in Hamiltonian Floer theory

Kenji Fukaya·Yong-Geun Oh·Hiroshi Ohta·Kaoru Ono
View original source ↗Read source document ↗Export metadata ↓

Abstract

The notion of linear K-system is introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer's equation in the book [FOOO14]. The purpose of the present article is to provide a geometric realization of the linear K-system associated with solutions to Floer's equation in the Morse-Bott setting. Immediate consequences (when combined with the abstract theory from [FOOO14]) are construction of Floer cohomology for periodic Hamiltonian system on general compact symplectic manifold without any restriction, and construction of an isomorphism over the Novikov ring between the Floer cohomology and the singular cohomology of the underlying symplectic manifold. The present article utilizes various analytical results on pseudo-holomorphic curves established in our earlier papers and books. However the paper itself is geometric in nature, and does not presume the readers' prior knowledge much on Kuranishi structure and its construction but assumes only the elementary part of thereof, and results from [FOOO11] and [FOOO,Chapter 8] on its construction, and the standard knowledge on Hamiltonian Floer theory. We explain the general procedure of the construction of a linear K-system by explaining in detail the inductive steps of ensuring the compatibility conditions for the system of Kuranishi structures leading to a linear K-system for the case of Hamiltonian Floer theory. A certain minor errors are corrected in this version. (to appear in the special volume dedicated to Claude Viterbo of Journal of Fixed Point Theory and Applications.)

Explore related subjects

math.SGmath.DGmath.DS

Keep this discovery

BibTeXRIS

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono. 2021-12-23. Construction of a linear K-system in Hamiltonian Floer theory. https://arxiv.org/abs/2112.12368

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

KEEP EXPLORING

Related papers

arXiv2026

Extended Future Tube Conjecture for Unipotent Subgroups

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

math.SG↗
arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $\Lambda$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact K\"ahler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient K\"ahler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗
arXiv2026

Classification of Legendrian doubles and suspensions

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

math.SG↗