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Lagrangian intersection Floer theory: anomaly and obstruction. Part I

AMS/IP Studies in Advanced Mathematics, v.46,1 (2009)
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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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    • "In mirror symmetry, the enumeration of holomorphic discs is of great importance . Holomorphic discs are building blocks of the Fukaya category [12] [29] [15]. "
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    ABSTRACT: We define the counting of holomorphic cylinders in log Calabi-Yau surfaces. We prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the wall-crossing formula for the focus-focus singularity. Our tools include Berkovich spaces, tropical geometry, Gromov-Witten theory and the GAGA theorem for non-archimedean analytic stacks.
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    • "In particular, homological mirror symmetry (HMS) is a relationship between symplectic and algebraic geometry. 4 Currently it is " widely believed " ([15] section 1.4) that to each Calabi-Yau manifold, such a mirror dual exists and is also Calabi-Yau. "
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    ABSTRACT: Here we carefully construct an equivalence between the derived category of coherent sheaves on an elliptic curve and a version of the Fukaya category on its mirror. This is the most accessible case of homological mirror symmetry. We also provide introductory background on the general Calabi-Yau case of The Homological Mirror Symmetry Conjecture.
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