Stability conditions and Lagrangian cobordisms

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].

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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