This article deals with the Floer cohomology (with Z 2 coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. This description is used to show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. Finally, we develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.We do not have a general formula for the rank of the Floer cohomology. However, motivated by [4] and [8], we associate a potential function W c to each torus fiber L c and then are able to show Theorem 1.2. The Floer cohomology of the pair (R, (L c , L ρ )) is well-defined if and only if W c (ρ) = 0. Furthermore, HF (R, (L c , L ρ )) = 0 if and only if ∇W c (ρ) = 0.This fact explains why we use values in F 2 rather than Z 2 : Z 2 is usually not big enough to find solutions to the equations W c = 0, ∇W c = 0. As mentioned before, it is shown in [4] that the condition ∇W c (ρ) = 0 is equivalent to the non-vanishing of HF (L c , L ρ ). Therefore Theorem 1.2 implies that, when defined, the cohomology HF (R, (L c , L ρ )) = 0 if and only if HF (L c , L ρ ) = 0.When ∇W c (ρ) = 0 but W c (ρ) = 0, the Floer cohomology HF (R, (L c , L ρ )) is not defined. Nevertheless, we can still show that R and L c are non-displaceable. Following an idea of Miguel Abreu and Leonardo Macarini we consider the product of the toric manifold with itself. Then HF (R × R, (L c × L c , L ρ ⊕ L ρ )) is well defined and we obtain Theorem 1.3. Suppose there exists a locally constant sheaf L ρ such that ∇W c (ρ) = 0 and let φ be a Hamiltonian diffeomorphism such that φ(R) and L c intersect transversely. ThenThe fact that F 2 is algebraically closed allows us to obtain one more result.Corollary 1.4. Suppose X P is monotone and L 0 is the unique monotone torus fiber. Then there exists L ρ such that ∇W 0 (ρ) = 0. Therefore the Lagrangians R and L 0 are non-displaceable, i.e. for any Hamiltonian diffeomorphism φ, φ(R) ∩ L c = ∅.
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.
is finite. The space of all holomorphic strips that run from p ∈ L 0 ∩L 1 to q ∈ L 0 ∩L 1 will be denoted M J (L 0 , L 1 : p, q). LetIf the linearization D u∂J of∂ J is surjective for every u ∈ M J (L 0 , L 1 ) then each M J (L 0 , L 1 : p, q) is a smooth manifold (different components may have different dimensions). Let J reg denote the set of all such J. J reg is a set of the second category, and from now on we assume J ∈ J reg . If u ∈ M J (L 0 , L 1 : p, q) then dim(T u M J (L 0 , L 1 : p, q)) = Index(D u∂J ).The index of D u∂J is equal to the spectral flow of∂ J along u, and this in turn is an invariant of the homotopy class of u and is equal to µ(u), the Maslov index of u.
Let X = { X 5 0 + • • • + X 5 4 = 0 } be the Fermat quintic threefold. The set of real solutions L forms a Lagrangian submanifold of X. Multiplying the homogeneous coordinates of X by various fifth roots of unity gives automorphisms of X; the images of L under these automorphisms defines a family of 625 different Lagrangian submanifolds, called real Lagrangians. In this paper we try to calculate the Floer cohomology between all pairs of these Lagrangians. We are able to complete most of the calculations, but there are a few cases we cannot do.The basic idea is to explicitly describe some low energy moduli spaces and then use this knowledge to calculate the differential on the E 2 page of the standard spectral sequence for Floer cohomology. It turns out that this is often enough to calculate the cohomology completely. Several techniques are developed to help describe these low energy moduli spaces, including a formula for the Maslov index, a formula for the obstruction bundle, and a way to relate holomorphic strips and discs to holomorphic spheres. The real nature of the Lagrangians is crucial for the development of these techniques.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.