Intersection Theory for Lagrangian Immersions

Akaho, Manabu

doi:10.4310/mrl.2005.v12.n4.a8

Cited by 19 publications

(53 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Immersed Floer theory was first developed by Akaho [2], and Akaho and Joyce [3]. In the context of curves on Riemann surfaces it also appears in work of Seidel [22] and Abouzaid [1].…”

Section: Introductionmentioning

confidence: 99%

Exact, graded, immersed Lagrangians and Floer theory

Alston

Bao

2018

Journal of Symplectic Geometry

View full text Add to dashboard Cite

We develop Lagrangian Floer Theory for exact, graded, immersed Lagrangians with clean self-intersection using Seidel's setup [21]. A positivity assumption on the index of the self intersection points is imposed to rule out certain (but not all) disc bubbles. This allows the Lagrangians to be included in the exact Fukaya category. We also study quasi-isomorphism of Lagrangians under certain exact deformations which are not Hamiltonian.

show abstract

“…Immersed Floer theory was first developed by Akaho [2], and Akaho and Joyce [3]. In the context of curves on Riemann surfaces it also appears in work of Seidel [22] and Abouzaid [1].…”

Section: Introductionmentioning

confidence: 99%

Exact, graded, immersed Lagrangians and Floer theory

Alston

Bao

2018

Journal of Symplectic Geometry

View full text Add to dashboard Cite

show abstract

“…. , deg f n below, and also the shifted complexes C si * (L; Q) [1], QX [1] which are ubiquitous in [9] are replaced below by unshifted complexes C si * (L; Q), QX , simplifying the notation. We undo the shift when we define Lagrangian Floer cohomology in (144).…”

Section: Proposition 46 the Virtual Dimension Of The Kuranishi Spacementioning

confidence: 99%

“…Here in the third step 1) are homotopy inverses, the sixth step is similar, and in the fourth step we use (133). For the fifth step, …”

Section: Homotopies Between a N0 Morphismsmentioning

confidence: 99%

“…Suppose that (M, ω) is a compact symplectic manifold, and that ι t : L → M for t ∈ [0, 1] is a smooth 1-parameter family of compact Lagrangian immersions satisfying the exactness condition of Definition 13.14(ii). Let S ⊆ [0, 1] be the open subset of t ∈ [0, 1] for which ι t : L → M has only transverse double self-intersections. Suppose for simplicity that L is oriented and spin; this induces relative spin structures for ι t : L → M for all t ∈ [0, 1], as in §5.1.…”

Section: Bounding Cochains and Lagrangian Floer Cohomologymentioning

confidence: 99%

“…This was done by the first author [1] under the simplifying assumption that π 2 (M, ι(L)) = {1}, which eliminates the issues of disc bubbling, A ∞ algebras and bounding cochains. We now discuss the much more difficult general case.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Immersed Lagrangian floer theory

Akaho¹,

Joyce²

2010

J. Differential Geom.

Self Cite

188

View full text Add to dashboard Cite

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, . . ..

show abstract

Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

Bottman¹,

Wehrheim

2018

Sel. Math. New Ser.

View full text Add to dashboard Cite

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-0 effect, and predict its algebraic consequences -geometric composition extends to a curved A∞-bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schmäschke provides examples of nontrivial figure eight bubbles. arXiv:1503.03486v3 [math.SG] 21 Feb 20185 Disk bubbling gives rise to µ0 terms in all A∞-relations, so that in particular squares of differentials can be nonzero. This can be thought of as allowing curvature, hence we follow e.g. [Au] and denote this generalized type of A∞-relations by the prefix "curved".

show abstract

Intersection Theory for Lagrangian Immersions

Cited by 19 publications

References 5 publications

Exact, graded, immersed Lagrangians and Floer theory

Exact, graded, immersed Lagrangians and Floer theory

Immersed Lagrangian floer theory

Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

Contact Info

Product

Resources

About