Compactifications of cluster varieties and convexity

Cheung, Man-Wai; Magee, T.R.; Chávez, Alfredo Nájera

doi:10.48550/arxiv.1912.13052

Cited by 2 publications

(7 citation statements)

References 11 publications

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“…The sad fact is that the if we are taking the convex hull of the primitive generators of the walls, the resulting polytope would no longer be positive. This is because the polytope is no longer broken line convex as indicated in [8]. Since we only care about the incoming walls for broken line convexity [8], one can see that the top left polytope in Figure 29 in the next section is broken line convex.…”

Section: Type B 2 and Gmentioning

confidence: 97%

“…This is because the polytope is no longer broken line convex as indicated in [8]. Since we only care about the incoming walls for broken line convexity [8], one can see that the top left polytope in Figure 29 in the next section is broken line convex. The mutation sequence of the G 2 scattering diagrams are similar to those for type A 2 and B 2 .…”

Section: Type B 2 and Gmentioning

confidence: 97%

“…Positive polytopes with respect to the affine structures can be similarly defined to give a graded algebra. Using [25] or the argument in [8], the polytopes are broken line convex. In this case, since all the walls are outgoing, the positive polytopes are simply convex with respect to the affine structures.…”

Section: Canonical Scattering Diagramsmentioning

confidence: 99%

“…For type B 2 , we can again take the primitive generators of the walls and then consider the polytope as the convex hull of those vertices. By using [8], this polytope is a positive polytope. The multiplication of the theta functions tells us [7] that the corresponding space is the del Pezzo surfaces of degree 6.…”

Section: Type B 2 and Gmentioning

confidence: 99%

“…Combinatorially, the construction can be described by 'convex' polytopes, called the positive polytopes [17]. The article [8] showed that the positive polytopes satisfy a convexity condition called 'broken line convexity'. As the seed mutates, the polytope mutates correspondingly.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

Cheung,

Vianna

2020

Preprint

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In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17].In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over R 2 (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].

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Section: Type B 2 and Gmentioning

confidence: 97%

Section: Type B 2 and Gmentioning

confidence: 97%

Section: Canonical Scattering Diagramsmentioning

confidence: 99%

Section: Type B 2 and Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

Cheung,

Vianna

2020

Preprint

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The canonical wall structure and intrinsic mirror symmetry

Gross¹,

Siebert

2022

Invent. math.

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As announced in Gross and Siebert (in Algebraic geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol 97, no 2. AMS, Providence, pp 199–230, 2018) in 2016, we construct and prove consistency of the canonical wall structure. This construction starts with a log Calabi–Yau pair (X, D) and produces a wall structure, as defined in Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022). Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich et al. (Punctured Gromov–Witten invariants, 2020. arXiv:2009.07720v2 [math.AG]). There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that, using the setup of Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022), the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]). While the setup of this paper is narrower than that of Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]), it gives a more detailed description of the mirror.

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Compactifications of cluster varieties and convexity

Cited by 2 publications

References 11 publications

Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

The canonical wall structure and intrinsic mirror symmetry

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