Immersed Lagrangian Floer Theory

Abstract

Let (M, ω) be a compact symplectic 2n-manifold, and L a compact embedded Lagrangian submanifold in M . Fukaya, Oh, Ohta and Ono [9] construct Lagrangian Floer cohomology for such M, L, yielding groups HF * (L, b; Λnov) for one Lagrangian or HF * `(L 1 , b 1 ), (L 2 , b 2 ); Λnov ´for two, where b, b 1 , b 2 are choices of bounding cochains, and exist if and only if L, L 1 , L 2 have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, ω), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend [9] to immersed Lagrangians L in M with immersion ι : L → M , with transverse self-intersections. In the embedded case, Floer cohomology HF * (L, b; Λnov) is a modified, 'quantized' version of singular homology H n− * (L; Λnov) over the Novikov ring Λnov. In our immersed case, HF * (L, b; Λnov) turns out to be a quantized version of H n− * (L; Λnov)⊕ L (p − ,p + )∈R Λnov •(p − , p + ), where R = ˘(p − , p + ) : p − , p + ∈ L, p − = p + , ι(p − ) = ι(p + ) ¯is a set of two extra generators for each selfintersection point of L, and (p − , p + ) has degree η (p − ,p + ) ∈ Z, an index depending on how L intersects itself at ι(p − ) = ι(p + ).The theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring ΛCY. The proofs involve associating a gapped filtered A∞ algebra over Λ 0 nov or Λ 0 CY to ι : L → M , which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A N,0 algebras for N = 0, 1, 2, . . ..

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

“…Section 2.5 explains Lagrangian Floer cohomology, obstructions to HF * , and derived Fukaya categories D b F (M ) for embedded Lagrangians in Calabi-Yau m-folds, and §2.6 considers the extension to immersed Lagrangians. Some references are McDuff and Salamon [57] for symplectic geometry, the author [47] and Harvey and Lawson [30] for Calabi-Yau m-folds and special Lagrangians, Mantegazza [55], Smoczyk [78] and Neves [61] for (Lagrangian) MCF, Fukaya [21,22], Fukaya, Oh, Ohta and Ono [23] and Seidel [73] for Lagrangian Floer cohomology and Fukaya categories for embedded Lagrangians, and Akaho and the author [2] for the extension to immersed Lagrangians.…”

“…(ii) The derived Fukaya category D b F (M ) must be enlarged to include immersed Lagrangians as in [2] in dimension m 2, and certain classes of singular Lagrangians in dimension m 3, for the programme to work.…”

Distinguishing between exotic symplectic structures

“…This sphere is precisely the matching cycle associated to that line. We can do the same with the part of M ∩ R x 3 , y 1 , y 2 living over the interval [2,3] to find another Lagrangian sphere B and we shall take A and B to define our standard basis of…”

“…By the same argument, for any t > 0 we can take a straightline path given by the interval [1,3], which goes over the central critical point at 2, and say that this too will define a matching cycle: in the central nonsmooth fibre we shall either get, by S 1 -symmetry, the critical point or some circle. However, if we obtained the critical point, then we would have found a Lagrangian in a homology class of positive symplectic area.…”

“…In fact, topologically it looks like S 3 with some S 1 in it collapsed to a point. Presumably, pseudoholomorphic curve theory with respect to this cycle is very badly behaved, so that a Floer theory along the lines of [3] cannot be made to work here, although see [14] for some analysis of holomorphic discs on certain similar special Lagrangian cones.…”