"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 . 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. "
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.
"In particular, homological mirror symmetry (HMS) is a relationship between symplectic and algebraic geometry. 4 Currently it is " widely believed " ( section 1.4) that to each Calabi-Yau manifold, such a mirror dual exists and is also Calabi-Yau. "
[Show abstract][Hide abstract] 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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