Lagrangian intersection Floer theory: anomaly and obstruction. Part I

AMS/IP Studies in Advanced Mathematics, v.46,1 (2009)
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    • "More recent developments have led to non-displaceability results about some non-monotone Lagrangians . For example, Fukaya, Oh, Ohta and Ono [17] found a continuous family of non-displaceable Lagrangian torî T a ⊂ CP 1 ×CP 1 by means of Floer cohomology with bulk deformations, whose general theory was developed in [15] [16]. For other recent methods of proving non-displaceability, see e.g. "
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    ABSTRACT: We introduce a version of Floer cohomology of a non-monotone Lagrangian submanifold which only uses least area holomorphic disks with boundary on it. We use this theory to prove that a continuous family of Lagrangian tori in $\mathbb{C}P^2$, whose Floer cohomology in the usual sense vanishes, is Hamiltonian non-displaceable from the monotone Clifford torus.
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    • "When the intersection L 1 ∩ L 2 is clean, P • L1∩L2 is a constant sheaf and Joyce's Floer cohomology is simply the singular cohomology H * (L 1 ∩ L 2 ). Then there exists a spectral sequence E 2 = H * (L 1 ∩ L 2 ) ⇒ HF * ω θ (L 1 , L 2 ) converging to HF * ω θ (L 1 , L 2 ) whose E 2 page is H * (L 1 ∩L 2 ) with certain coefficient in the Novikov field (see Theorem 6.1.4 of FOOO [28] "
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    ABSTRACT: We study the intersection theory of complex Lagrangian subvarieties inside holomorphic symplectic manifolds. In particular, we study their behaviour under Mukai flops and give a rigorous proof of the Pl\"ucker type formula for Legendre dual subvarieties written down by the second author before. Then we apply the formula to study projective dual varieties in projective spaces.
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    • "For example, if X = Y − × Y is the product of a compact symplectic manifold Y with its dual Y − then the diagonal in X is Hamiltonian non-displaceable by the Arnold conjecture. More generally if X admits an anti-symplectic involution then in many cases the fixed point set of the involution has non-trivial Floer cohomology [39]. Despite this, it has been far from clear how to actually construct Floer-non-trivial (and so Hamiltonian non-displaceable) Lagrangians in symplectic manifolds that are not equipped with anti-symplectic involutions or how to produce generators of the Fukaya category. "
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    ABSTRACT: We prove that small blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with trivial centers create Floer-non-trivial Lagrangian tori. We give examples of explicit mmp runnings and descriptions of Floer non-trivial tori in the case of toric manifolds, polygon spaces, and moduli spaces of flat bundles on punctured two-spheres (moduli of parabolic bundles). These results are part of a conjectural description of generators for the Fukaya category of a compact symplectic manifold with an orbifold running of the minimal model program.
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