Tropical Lagrangians in toric del-Pezzo surfaces

Hicks, Jeffrey Stephen

doi:10.1007/s00029-020-00614-1

Cited by 8 publications

(4 citation statements)

References 36 publications

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“…Example 4.2.2 (Lagrangian pair of pants). In [Mat18], it was shown that the tropical pair of pants centered at the origin was an exact Lagrangian submanifold; a similar proof was given in [Hic21b] which showed that all tropical Lagrangian submanifolds constructed from the data of a dimer are exact.…”

Section: Geometric Lagrangians Versus Lagrangian Branesmentioning

confidence: 78%

“…It is possible for non-regular Maslov index 0 disks to appear with boundary on L ε V , even in simple examples (see example 4.2.11). More generally, [Hic21b] shows that there exists a "wall-crossing" phenomenon which occurs for isotopies between tropical Lagrangian submanifolds, and that the count of these Maslov index 0 holomorphic disks play a crucial role in understanding coordinates on the moduli space of tropical Lagrangian submanifolds.…”

Section: Geometric Lagrangians Versus Lagrangian Branesmentioning

confidence: 99%

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Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Hicks¹

2022

Preprint

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We say that a tropical subvariety V ⊂ R n is B-realizable if it can be lifted to an analytic subset of (Λ * ) n . When V is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift L V ⊂ (C * ) n . We prove that whenever L V has well-defined Floer cohomology, we can find for each point of V a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with L V is non-vanishing. As a consequence, whenever L V is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety V is B-realizable by applying mirror symmetry.As an application, we show that the Lagrangian lift of a genus zero tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-zero tropical curves are B-realizable. We also prove that tropical curves inside tropical abelian surfaces are B-realizable. Contents 1 Introduction 1 2 A guided calculation to the support of the Lagrangian pair of pants 8 3 Geometric realization 20 4 Unobstructed Lagrangian lifts of tropical subvarieties 28 5 Faithfulness: unobstructed lifts as A-realizations 34 6 B-realizability and unobstructedness 40 A Pearly model in symplectic fibrations 50 B Auxiliary results for filtered A ∞ algebras and modules 53

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Section: Geometric Lagrangians Versus Lagrangian Branesmentioning

confidence: 78%

Section: Geometric Lagrangians Versus Lagrangian Branesmentioning

confidence: 99%

Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Hicks¹

2022

Preprint

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“…In recent years, a variation in the form of toric hyperplane arrangements has been studied more prominently, from combinatorial, algebraic, and topological point of views; see [2,3] for examples. Some applications of the theory of toric hyperplane arrangements to toric varieties have been presented in [4,5].…”

Section: Introduction 1hyperplane Arrangementsmentioning

confidence: 99%

Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications

Bergerová

2024

PUMP J. Undergrad. Res.

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A toric hyperplane is the preimage of a point in a circle of a continuous surjective group homomorphism from the n-torus to the circle. A toric hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the combinatorial properties of toric hyperplane arrangements on n-tori which are spanning and in general position. Specifically, we describe the symmetry of f-vectors arising in such arrangements and a few applications of the result to count configurations of hyperplanes.

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“…[4]). Also, all tropical curves of genus ≤ 2 can be lifted to an exact Lagrangian submanifold of (C * ) 2 , as described in [8].…”

Section: Introductionmentioning

confidence: 99%

Affine dimers from characteristic polygons

Holmes¹

2021

Preprint

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Recent work by Forsgård indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torus in general position. We begin the classification of convex lattice polygons arising as characteristic polygons of affine dimers. We present several general constructions of new affine dimers from old, and an algorithm for finding affine dimers with prescribed polygon.With these tools we prove that all lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.

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Tropical Lagrangians in toric del-Pezzo surfaces

Abstract: We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $$(\mathbb {C}^*)^n$$ ( C ∗ ) n whose valuation projection approximates a tropical hypersurface. Each face of the di… Show more

Cited by 8 publications

References 36 publications

Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications

Affine dimers from characteristic polygons

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