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

2014/01/31 by Dominic Joyce · 2 citations
Mathematics · #math.DG #math.SG

paper · pdf · doi:10.4171/emss/8

published as EMS Surveys in Mathematical Sciences 2 (2015), 1-62 · 63 pages. (v2) new section 4 added, discussing exact Lagrangians, and the Kahler-Einstein case. To appear in EMS Surveys in Mathematical Sciences

arxiv created 2014/09/03 · arxiv updated 2015/06/05

Abstract

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∈[0,∞)\ with L⁰=L should exist for all time, and L^∞=lim_t→∞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→ 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∈[0,∞)\ such that (L⁰,E⁰,b⁰)=(L,E,b), and (L^t,E^t,b^t)≅(L,E,b) in D^b\mathcal F(M) for all t, and \L^t:t∈[0,∞)\ satisfies Lagrangian MCF with surgeries at singular times T₁,T₂,…, and in graded Lagrangian integral currents we have lim_t→∞L^t=L₁+⋯+L_n, where L_j is a special Lagrangian integral current of phase e^iπϕ_j for ϕ₁>⋯>ϕ_n, and (L₁,ϕ₁),…,(L_n,ϕ_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 T₁,T₂,….

Cited by

• Proper modules over Ginzburg dg algebras and compact Fukaya categories of plumbings
• On short time existence of Lagrangian mean curvature flow

Related

• On short time existence of Lagrangian mean curvature flow
• Special Lagrangian sections and stability conditions on threefolds
• Special Lagrangians, stable bundles and mean curvature flow
• Nakai-Moishezon criteria and the toric Thomas-Yau conjecture
• Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in Cm