Immersed Lagrangian floer theory

Abstract

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 ofThe 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, . . ..

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 H 2 (M ; R) = R 2 .…”

“…As η is defined using only coordinates on the real slice R 3 \ {0} and annihilates the radial direction, this is a closed form on C 3 \ iR 3 . We shall choose such that < 1/8R and apply a translation x → x + (0, 0, 3 2 ). It is easy to show that η is now well defined on M , so that in the Lefschetz fibration M in → D R , η is a closed, S 1 -equivariant 2-form supported in the region lying over {|x 3 − 3 2 | < 1 4 } and the sphere A has some nonzero area with respect to η.…”

“…We can do likewise for the path γ 1 to obtain the matching cycle V t 1 . By the same argument, for any t > 0, we can take a straight line 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 obtain the critical point, then we would have found a Lagrangian in a homology class of positive symplectic area.…”

Homological mirror symmetry for the quintic 3–fold

“…Let P ƒ N D P .V˝C ƒ N / be the projective space over ƒ N and Y q be the hypersurface defined by q y Let be the abelian subgroup of PSL nC2 .C/ defined in (2). Each E q;k admits .nC2/ n -linearizations, so that one obtains .nC2/ nC1 objects of…”

“…Strictly speaking, the work of Fukaya, Oh, Ohta and Ono [6] that we rely on in this section gives not a full-fledged A 1 -category but an A 1 -algebra for a Lagrangian submanifold and an A 1 -bimodule for a pair of Lagrangian submanifolds. While there is apparently no essential difficulty in generalizing their work to construct an A 1 -category (for transversally intersecting sequence of Lagrangian submanifolds, one can regard it as a single immersed Lagrangian submanifold and use the work of Akaho and Joyce [2]), we do not attempt to settle this foundational issue in this paper. Sections 4 and 5 are at the heart of this paper.…”