Journal of Symplectic Geometry

Volume 18 (2020)

Number 2

Stability conditions and Lagrangian cobordisms

Pages: 463 – 536



Felix Hensel (Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland)


In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category $\mathcal{DFuk} (M)$ of a symplectic manifold $(M, \omega)$ induces a stability condition on the derived Fukaya category of Lagrangian cobordisms $\mathcal{DFuk} (\mathbb{C} \times M)$. In addition, using stability conditions, we provide general conditions under which the homomorphism $\Theta : \Omega_{Lag} (M) \to K_0 (\mathcal{DFuk} (M))$, introduced by Biran and Cornea [6, 7], is an isomorphism. This yields a better understanding of how stability conditions affect and it allows us to elucidate Haug’s result, that the Lagrangian cobordism group of $T^2$ is isomorphic to $K_0 (\mathcal{DFuk} (T^2))$ [23].

The author was partially supported by the Swiss National Science Foundation (grant number 200021-156000).

Received 10 August 2018

Accepted 30 April 2019

Published 8 June 2020