If \((M,\omega)\) is a symplectic manifold, then \(\widetilde{M}\) is the symplectic manifold \(\mathbb{C}\times M\) endowed with the symplectic form \(\widetilde\omega=\omega_{\mathrm{std}}\oplus\omega\) and the projection \(\pi:\widetilde{M}\to\mathbb{C}\) to the complex plane. \(X_{|S}=X\cap\pi^{-1}(S)\) denotes the restriction for subsets \(X\subset\widetilde{M}\) and \(S\subset\mathbb{C}\). The Fukaya category \(\mathcal{F}uk(M)\) of a symplectic manifold \((M,\omega)\) is an \(A_\infty\)-category over some ground field \(K\). \(\mathcal{F}uk(\mathcal{C})\) is the Fukaya category, where \(\mathcal{C}\) is an admissible class of Lagrangians in \(M\). The objects of \(\mathcal{F}uk(\mathcal{C})\) are Lagrangian branes. These are triples \((L,\theta,P)\), where \(L\subset M\) is a Lagrangian in \(M\) of a certain class \(\mathcal{C}\), \(\theta\) is a \(\mathbb{Z}\)-grading on \(L\), and \(P\) is a Pin-structure on \(L\). If \(\mathcal{D}\) is a triangulated category, then the Grothendieck group \(K_0(\mathcal{D})\), or \(K\)-group, of \(\mathcal{D}\) is the free abelian group generated by the objects of \(\mathcal{D}\) modulo the relations \(X-Y+Z=0\) whenever there is an exact triangle \(X\to Y\to Z\to XZ\) in \(\mathcal{D}\). The notion of the derived Fukaya category is a central object of symplectic manifolds. The derived Fukaya category \(D\mathcal{F}uk(M)\) of a symplectic manifold \((M,\omega)\) is a triangulated category which encodes information on the Lagrangian submanifolds of \(M\). In [Geom. Funct. Anal. 24, No. 6, 1731–1830 (2014; Zbl 1306.55003)], P. Biran and O. Cornea developed a theory of Lagrangian cobordism. Two families \((L_i)_{1\le i\le k_-}\) and \((L_j')_{1\le j\le k_+}\) of closed Lagrangian submanifolds of \(M\) are said to be Lagrangian cobordant, \((L_i)\simeq (L_j')\) if there exists a smooth compact cobordism \((V;\coprod_iL_i,\coprod_jL_j')\) and a Lagrangian embedding \(V\subset([0,1]\times\mathbb{R})\times M\) so that for some \(\varepsilon>0\) the following holds: \[ V|_{[0,\varepsilon)\times\mathbb{R}}=\coprod_i([0,\varepsilon)\times\{i\})\times L_i,\\ V|_{(1-\varepsilon,1]\times\mathbb{R}}=\coprod_j((1-\varepsilon,1]\times\{j\})\times L_j'. \] The manifold \(V\) is called a Lagrangian cobordism from the Lagrangian family \((L_j')\) to the family \((L_i)\). This cobordism is denoted by \(V:(L_j')\rightsquigarrow(L_i)\) or \((V;(L_i),(L_j'))\). The Lagrangian cobordism group \(\Omega_{\mathrm{Lag}}(M)\) of \(M\) is defined as the quotient of the free abelian group \(\langle\mathcal{L}\rangle\) modulo the subgroup \(R_{\mathcal{L}}\), where \(\mathcal{L}\) is the set of all Lagrangian branes \((L,\theta,P)\). A stability condition on a triangulated category \(\mathcal{D}\) is a pair \((Z,\mathcal{P})\) consisting of an additive group homomorphism \(Z: K_0(\mathcal{D})\to\mathbb{C}\) and a collection of full additive subcategories \(\mathcal{P}(\varphi)\subset\mathcal{D}\) for each \(\varphi\in\mathbb{R}\), satisfying certain properties. In the above paper, the authors showed that there is a natural surjective group homomorphism \(\Theta:\Omega_{\mathrm{Lag}}(M)\to K_0(D\mathcal{F}uk(M))\) from the Lagrangian cobordism group to the Grothendieck group of the derived Fukaya category of \(M\). It is an interesting question to understand when \(\theta\) is an isomorphism or, more generally, to understand the kernel of \(\Theta\). In [Sel. Math., New Ser. 21, No. 3, 1021–1069 (2015; Zbl 1408.53105)], L. Haug used homological mirror symmetry to show that \(\Theta\) is an isomorphism in the case of the torus \(T^2\).
In this paper, the authors study the interplay between Lagrangian cobordisms and stability conditions. They show that any stability condition on the derived Fukaya category \(D\mathcal{F}uk(M)\) of a symplectic manifold \((M,\omega)\) induces a stability condition on the derived Fukaya category of Lagrangian cobordisms \(D\mathcal{F}uk(\mathbb{C}\times M)\). In addition, using stability conditions, the authors provide general conditions under which the homomorphism \(\Theta:\Omega_{\mathrm{Lag}}(M)\to K_0(D\mathcal{F}uk(M)\) is an isomorphism. This yields a better understanding of how stability conditions affect \(\Theta\). The main result of the paper states that if \((Z^M,\mathcal{P}^M)\) is a stability condition on \(D\mathcal{F}uk(M)\) and \(\kappa\in 2\cdot\mathbb{Z}_+\), then \((Z,\mathcal{P}_\kappa)\) is a stability condition on \(D\mathcal{F}uk(\mathbb{C}\times M)\). Additionally, if \((Z^M,\mathcal{P}^M)\) is locally-finite, then \((Z,\mathcal{P}_\kappa)\) is locally finite as well.