Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow

Abstract

Let $(M,J,g,\Omega )$ be a Calabi–Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau conjectured that there should be a notion of 'stability' for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in [0,\infty )\}$ with $L^0=L$ should exist for all time, and $L^{\infty }={\lim }_{t\to \infty }{L}^t$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas–Yau conjectures, and discuss related issues.

It is a folklore conjecture, extending [81], that there exists a Bridgeland stability condition $(Z,\mathcal{P})$ on the derived Fukaya category $D^b\mathcal{F}(M)$, such that an isomorphism class in $D^b\mathcal{F}(M)$ is $(Z,\mathcal{P})$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique.

In brief, we conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\mathcal{F}(M)$, then there is a unique family $\{(L^t,E^t,b^t):t\in [0,\infty )\}$ such that $(L^0,E^0,b^0)=(L,E,b)$, and $(L^t,E^t,b^t)\cong (L,E,b)$ in $D^b\mathcal{F}(M)$ for all $t$, and $\{L^t:t\in [0,\infty )\}$ satisfies Lagrangian MCF with surgeries at singular times $T_1,T_2,\dots ,$ and in graded Lagrangian integral currents we have ${\lim }_{t\to \infty }{L}^t=L_1+\cdots +L_n$, where $L_j$ is a special Lagrangian integral current of phase $e^{i\pi {\varphi }_j}$ for ${\varphi }_1>\cdots >{\varphi }_n$, and $(L_1,{\varphi }_1),\dots ,(L_n,{\varphi }_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,P)$-semistable objects.

We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\dots .$