Floer cohomology of the Chiang Lagrangian

Abstract

We study holomorphic discs with boundary on a Lagrangian submanifold L in a Kähler manifold admitting a Hamiltonian action of a group K which has L as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in CP 3 first noticed by Chiang [13]. We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case it generates the split-closed derived Fukaya category as a triangulated category.Mathematics Subject Classification (2010). 53D12, 53D37, 53D40.The results above imply immediately that: Corollary B. The Chiang Lagrangian is not displaceable from itself or from the Clifford torus via Hamiltonian isotopies. Remark 1.2.5. Note that L ∆ and RP 3 intersect along a pair of circles in their standard positions and it is an interesting open question if they can be displaced from one another. Standard techniques in Floer theory cannot answer this question because HF (L ∆ , RP 3 ) is not well-defined: Floer cohomology can only be defined for Lagrangians with the same m 0 -value and m 0 (RP 3 ) = 0 as RP 3 has minimal Maslov 4.

Floer cohomology of Platonic Lagrangians

“…In this subsection we collect together some of the properties of the quasihomogeneous threefolds X C . Most of the results are contained in [13,Section 4]. We follow the notation of Evans-Lekili.…”

“…This filtration was exploited by Evans-Lekili in the proof of [13,Lemma 3.12]. However, the span of the summands of a given partial index is not determined in general: consider for example (E, F ) = (C 2 , R ⊕ z 1/2 R), which has one obvious splitting by the natural basis e 1 and e 2 of C 2 , but can in fact be split by the basis…”

“…Preliminaries. We begin with the following definition, which differs slightly from that given in [13]: Definition 2.1. If X is a complex manifold carrying an action of a compact Lie group K by holomorphic automorphisms, and L is a totally real submanifold which is an orbit of the K-action, then we say (X, L) is K-homogeneous.…”

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

“…The GC system admits nontorus Lagrangian GC fibers at the lower-dimensional strata of the GC polytope, which makes Floer theory of the system more interesting and challenging. Using non-Abelian symmetry or discrete symmetry, particular fibers of limited cases of Grassmannians have been investigated in [8], [9], and [28].…”

A critical point analysis of Landau--Ginzburg potentials with bulk in Gelfand--Cetlin systems

“…The GC system admits non-torus Lagrangian GC fibers at the lower dimensional strata of the GC polytope, which make Floer theory of the system more interesting and challenging. Using non-abelian symmetry or discrete symmetry, particular fibers of limited cases of Grassmannians have been investigated in [EL15,NU16,EL19].…”