Tropical Lagrangian hypersurfaces are unobstructed

Hicks, Jeffrey Stephen

doi:10.1112/topo.12165

Cited by 17 publications

(23 citation statements)

References 25 publications

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“…The goal of this paper is to extend the constructions of [19] to Lagrangian fibrations X → Q which are almost toric (and so may admit some fibers with singularities). In doing so, we shed some light on questions laid out in [26, section 6.3] regarding the homological mirror symmetry interpretation of monotone tropical Lagrangian tori in toric del-Pezzos.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…In doing so, we shed some light on questions laid out in [26, section 6.3] regarding the homological mirror symmetry interpretation of monotone tropical Lagrangian tori in toric del-Pezzos. This paper first provides an alternate description of the tropical Lagrangian submanifolds from [19] using the combinatorics of dimers (classically, an embedded bipartite graph G ⊂ T 2 ). To a dimer we construct an exact Lagrangian in (C * ) n whose valuation projection lies near a tropical curve (Definition 3.1.4, Corollary 3.1.10).…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…This was achieved by using tropical geometry to construct Lagrangian sections of val : (C * ) n → R n , and to show that these were mirror to line bundles onX . In [19], it was shown that the tropical-Lagrangian and tropical-complex correspondences are compatible with this mirror functor, in the sense that when a tropical hypersurface V is approximated byv al(D) for a divisor D, the Lagrangian L(V ) is mirror to the sheaf O D . This extends the relation between homological and SYZ mirror symmetry to sheaves beyond line bundles and skyscrapers of points.…”

Section: Homological Mirror Symmetry and Tropical Geometrymentioning

confidence: 99%

“…In the complex setting, this degeneration was studied by [24,27], where a correspondence between the valuations of complex curves (called the amoeba) and tropical curves was established. More recently, tropical-Lagrangian correspondences have been constructed in the parallel works of [19,26,28,30]. These papers show that for a given tropical curve V ⊂ Q there exists a Lagrangian submanifold L(V ) ⊂ X with val(X ) approximating V .…”

Section: Homological Mirror Symmetry and Tropical Geometrymentioning

confidence: 99%

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Tropical Lagrangians in toric del-Pezzo surfaces

Hicks

2021

Sel. Math. New Ser.

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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 dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $$L_{T^2}$$ L T 2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $$(\mathbb {CP}^2{\setminus } E, W), {\check{X}}_{9111}$$ ( CP 2 \ E , W ) , X ˇ 9111 . We find a symplectomorphism of $$\mathbb {CP}^2{\setminus } E$$ CP 2 \ E interchanging $$L_{T^2}$$ L T 2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on $${\check{X}}_{9111}$$ X ˇ 9111 .

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Section: Statement Of Main Resultsmentioning

confidence: 99%

Section: Statement Of Main Resultsmentioning

confidence: 99%

Section: Homological Mirror Symmetry and Tropical Geometrymentioning

confidence: 99%

Section: Homological Mirror Symmetry and Tropical Geometrymentioning

confidence: 99%

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Tropical Lagrangians in toric del-Pezzo surfaces

Hicks

2021

Sel. Math. New Ser.

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“…The idea of construction is motivated by Strominger, Yau and Zaslow's (SYZ) conjecture [56] and the construction of cycles in [47]. Parallel results without connection to enumerative geometry have very recently been achieved independent from us in the situation where the symplectic manifold is noncompact [38], [37], [25,26], a toric variety [41], [27] or a torus bundle over torus [54].…”

Section: Introductionmentioning

confidence: 99%

Tropically constructed Lagrangians in mirror quintic threefolds

Mak

Ruddat²

2020

Forum of Mathematics, Sigma

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We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve. As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

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