Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.3450

The symplectic geometry of cotangent bundles from a categorical viewpoint

Kenji Fukaya·Paul Seidel·Ivan Smith
View original source ↗Read source document ↗Export metadata ↓

Abstract

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.

Explore related subjects

math.SGmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kenji Fukaya, Paul Seidel, Ivan Smith. 2007-09-26. The symplectic geometry of cotangent bundles from a categorical viewpoint. https://arxiv.org/abs/0705.3450

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

KEEP EXPLORING

Related papers

arXiv2026

Localization of Lagrangian Floer homology for $\mathbb{Z}_p$-actions

For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a $p$-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free $\mathbb{Z}_p$-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.

math.SG↗
arXiv2026

Exotic symplectomorphisms of strong fillings

We prove a criterion for detecting exotic symplectomorphisms of strong fillings of closed contact manifolds. The criterion concerns symplectomorphisms induced by free contact circle actions on the boundary, and it is expressed purely in topological terms. As an application, we show that one-point blow-ups of an infinite family of Brieskorn varieties admit exotic compactly supported symplectomorphisms.

math.SG↗
arXiv2026

Conformally symplectic structures and the Lefschetz condition

This short note provides a symplectic analogue of Vaisman's theorem in complex geometry. Namely, for any compact symplectic manifold satisfying the hard Lefschetz condition in degree 1, every locally conformally symplectic structure is in fact globally conformally symplectic, whenever there is a mutually compatible almost complex structure.

math.SG↗