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, . . ..

Nearby Special Lagrangians

“…In the virtual method we give X an almost-complex structure which need not be regular, and in the direct method we give X some generic almost-complex structures which we can take to be regular because the two Lagrangians are mutually transverse. Changing almost-complex structures induces an isomorphism of the corresponding cohomology groups, as in Theorem 11.2 (a) of Akaho-Joyce's paper [10], so we conclude that the morphism spaces in the two categories are isomorphism.…”

Nearby Special Lagrangians

“…In the virtual method we give X an almost-complex structure which need not be regular, and in the direct method we give X some generic almost-complex structures which we can take to be regular because the two Lagrangians are mutually transverse. Changing almost-complex structures induces an isomorphism of the corresponding cohomology groups, as in Theorem 11.2 (a) of Akaho-Joyce's paper [10], so we conclude that the morphism spaces in the two categories are isomorphism.…”

A Compactness Theorem for $SO(3)$ Anti-Self-Dual Equation with Translation Symmetry

“…Now we define the notion of holomorphic curves with boundary lying in the immersed Lagrangian. One can see that this notion coincides with that in [AJ10] after ordering the set of marked points W .…”

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

“…Recall that our exact Lagrangian immersions are equipped with choices of potentials making their Legendrian lifts embedded. This is not new material; similar or even more general accounts can be found, for example, in [5], [6] and [30]…”

Nearby Special Lagrangians

“…In the virtual method we give X an almost-complex structure which need not be regular, and in the direct method we give X some generic almost-complex structures which we can take to be regular because the two Lagrangians are mutually transverse. Changing almost-complex structures induces an isomorphism of the corresponding cohomology groups, as in Theorem 11.2 (a) of Akaho-Joyce's paper [10], so we conclude that the morphism spaces in the two categories are isomorphism.…”

A Compactness Theorem for $SO(3)$ Anti-Self-Dual Equation with Translation Symmetry

“…Now we define the notion of holomorphic curves with boundary lying in the immersed Lagrangian. One can see that this notion coincides with that in [AJ10] after ordering the set of marked points W .…”

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

“…Recall that our exact Lagrangian immersions are equipped with choices of potentials making their Legendrian lifts embedded. This is not new material; similar or even more general accounts can be found, for example, in [5], [6] and [30]…”

Nearby Special Lagrangians

“…In the virtual method we give X an almost-complex structure which need not be regular, and in the direct method we give X some generic almost-complex structures which we can take to be regular because the two Lagrangians are mutually transverse. Changing almost-complex structures induces an isomorphism of the corresponding cohomology groups, as in Theorem 11.2 (a) of Akaho-Joyce's paper [10], so we conclude that the morphism spaces in the two categories are isomorphism.…”

A Compactness Theorem for $SO(3)$ Anti-Self-Dual Equation with Translation Symmetry

“…Now we define the notion of holomorphic curves with boundary lying in the immersed Lagrangian. One can see that this notion coincides with that in [AJ10] after ordering the set of marked points W .…”

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

“…Recall that our exact Lagrangian immersions are equipped with choices of potentials making their Legendrian lifts embedded. This is not new material; similar or even more general accounts can be found, for example, in [5], [6] and [30]…”