Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow
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DOI：
10.4171/emss/8
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发表时间：
2014-01
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通讯作者：
D. Joyce
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作者：
D. Joyce
设M $是Calabi-Yau $m $-fold，并考虑M $中的紧的分次Lagrange $L $。托马斯and Yau math. DG/0104196，math. DG/0104197证明了这样的$L $应该有一个"稳定性"的概念，如果$L $是稳定的，那么拉格朗日平均曲率流$\{L ^t：t\in [0，\infty）\}$应该一直存在，并且$L ^\infty =\lim_{t\to\infty} L ^t $应该是$L $的Hamilton合痕类中唯一的特殊拉格朗日量。本文试图对丘-陶关系进行更新，并对相关问题进行讨论。这是一个民间传说的猜想，存在一个Bridgeland稳定性条件$（Z，\mathcal P）$在导出的福谷范畴$D ^B\mathcal F（M）$$上，使得$D ^B\mathcal F（M）$中的同构类是$（Z，\mathcal P）$-半稳定的，当（可能仅当）它包含一个特殊的拉格朗日量，然后它必须是唯一的。我们猜想，如果$（L，E，B）$是$D ^B\mathcal F（M）$的一个放大版本中的一个对象，其中$L $是$M $中的一个紧的分次拉格朗日量（可能是浸入的，或具有"稳定奇点"），$E\to M $a秩1局部系统，$B $a对于$（L，E）$在Lagrangian Floer上同调中的有界上链，则存在唯一族$\（L ^t，E ^t，b ^t）：t\in [0，\infty）\}$使得$（L^0，E^0，b^0）=（L，E，b）$，且$（L ^t，E ^t，b ^t）\cong（L，E，b）$D ^b\mathcal F（M）$中的$对于所有$t $，以及$\{L ^t：t\in [0，\infty）\}$满足拉格朗日MCF，手术时间为奇异时间T_1，T_2，\dots，在分次拉格朗日积分流中，我们有$\lim_{t\to\infty} L ^t = L_1 +\cdots + L_n $，其中$L_j $是相位$e ^{i\pi\phi_j}$的特殊拉格朗日积分电流，其中$\phi_1>\cdots>\phi_n $，并且$（L_1，\phi_1），\ldots，（L_n，\phi_n）$对应于将$（L，E，b）$分解为$（Z，\mathcal P）$-半稳定对象。我们还详细地讨论了拉格朗日MCF在有限奇异时刻T_1，T_2，...
Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 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 that there exists a Bridgeland stability condition $(Z,\mathcal P)$ on the derived Fukaya category $D^b\mathcal F(M)$ of $M$, 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. We conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\mathcal F(M)$, where $L$ is a compact, graded Lagrangian in $M$ (possibly immersed, or with "stable singularities"), $E\to M$ a rank one local system, and $b$ a bounding cochain for $(L,E)$ in Lagrangian Floer cohomology, 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 $e^{i\pi\phi_j}$ for $\phi_1>\cdots>\phi_n$, and $(L_1,\phi_1),\ldots,(L_n,\phi_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,\mathcal P)$-semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\ldots.$