Local mirror symmetry via SYZ

Abstract

In this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations.

Family Floer mirror space for local SYZ singularities

“…The left vertical arrow π ∨ 0 is the family Floer mirror fibration in [66]. The top right corner Y agrees with many previous results [4,5,8,9,19,39], etc. Moreover, the upper horizontal map g essentially follows the principle of Gross-Hacking-Keel [43,44], and the right vertical arrow f generalizes a model of § 8].…”

“…Going one step further, the key contribution in the present paper is to construct the non-archimedean fibration f (with singularities) over B, using only symplectic data. Our construction of f nicely combines the ideas in the celebrated papers [43,53], and it also agrees with the very recent results [5,39]. Therefore, the f -fibers over ∆ = B \ B 0 are almost the only reasonable candidates for the 'dual singular fibers'.…”

“…It has been long expected that X m,n is mirror to X n,m by the works of Gross and Siebert (see also [5,39]). With this notation, we can restate Theorem 1.3: X 1,n−1 is SYZ mirror to X n−1,1 .…”

“…With this notation, we can restate Theorem 1.3: X 1,n−1 is SYZ mirror to X n−1,1 . But, unlike [5,39], we focus mainly on SYZ instead of HMS; besides, our result is mainly non-archimedean analytic, with the symplectic geometry only providing the theoretical basis [66]. Hopefully, we would also achieve HMS in the future by studying Abouzaid's family Floer functor [2,3] in the setting of Theorem 1.3 or 1.10.…”

“…By Proposition 4.4, we have the following two affinoid tropical charts on (π) −1 (B ± ): (c.f. ( 31)) (39) τ + :…”