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.
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 .
@cite_3
2094372703 2734934103 2964067385 2056986231
The amoeba of a holomorphic function @math is, by definition, the image in @math of the zero locus of @math under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function @math with the property that its Monge-Ampere mass is concentrated to the amoeba of @math We obtain results of two kinds; by approximating @math with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of @math ; by computing the Monge-Ampere measure, we find sharp bounds for the area of amoebas in @math . We also consider systems of functions @math and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations. 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. This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of Rd. This leads to natural cell decompositions of the tropical projective space TPmind−1. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007). Let P be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n , k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n , k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n , k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n , k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k > 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.