Let @math be a Laurent polynomial in @math variables, and let @math be a generic smooth fiber of @math . In RSTZ Ruddat-Sibilla-Treumann-Zaslow give a combinatorial recipe for a skeleton for @math . In this paper, we show that for a suitable exact symplectic structure on @math , the RSTZ-skeleton can be realized as the Liouville Lagrangian skeleton.
@cite_3: We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective DM toric stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using Mikhalkin-Viro patchworking, we compute the skeleton of the hypersurface. The result matches the [FLTZ] skeleton and is naturally realized as a Legendrian in the cosphere bundle of a torus. By [GPS1, GPS2, GPS3], we trade wrapped Fukaya categories for microlocal sheaf theory. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.
In Gammage-Shende @cite_3 , as one ingredient in proving HMS for the toric boundary of a toric variety, they constructed the Liouville skeleton for the same hypersurface as considered here. However, their results [Theorem 3.4.2] GS depends on the following hypothesis that, there exists some tropicalization function @math and some identification @math , such that the tropical amoeba polytope @math contains @math as an interior point, and @math restricts to each face @math of @math has a minimum in the interior of @math . This hypothesis is true in two-dimension, and can be verified in certain examples, e.g. mirror to weighted projective spaces. But in general, one does not know if it is always true, it would be interesting to find a proof or construct a counter-example. We thank Gammage and Shende for this clarification. Our approach here does not rely on this hypothesis, which is equivalent to " @math is adapted to the tropical amoeba polytope @math " for some choice of @math . We avoid this by considering a more flexible choice of Kahler potentials on than @math , and our approach works for any choice of @math compatible with @math .