Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) \pi _2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.
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Moment maps, nonlinear PDE and stability in mirror symmetry, I: geodesics.
In this paper, the first in a series, we study the deformed HermitianYangMills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold H closely related to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with C1,α regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen's theorem on the existence of C1,α geodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM and special Lagrangians in LandauGinzburg models.
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 Award ID(s):
 1902645
 NSFPAR ID:
 10294726
 Date Published:
 Journal Name:
 Annals of PDE
 Volume:
 7
 Issue:
 no. 1
 ISSN:
 21992576
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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