Theta functions and quiver Grassmannians

Cheung, Man-Wai

doi:10.48550/arxiv.1906.12299

Cited by 2 publications

(2 citation statements)

References 20 publications

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“…The present paper also provides the link to the algorithmic construction of walls [44], and to previous mirror constructions in two-dimensions [37] and for cluster varieties [38], also giving rise to rich combinatorial structures, see e.g. [22][23][24]62,63,74,75]. All these previous constructions generalize known and tested mirror constructions such as [12,13,35,51], and they have also been independently tested, see e.g.…”

Section: The Canonical Wall Structure and The Syz Interpretation Of I...supporting

confidence: 52%

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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Section: The Canonical Wall Structure and The Syz Interpretation Of I...supporting

confidence: 52%

The canonical wall structure and intrinsic mirror symmetry

Gross¹,

Siebert

2022

Invent. math.

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“…The last outgoing wall passes through (2, −2, −1, −1) and has scattering function 1 + z (−2,2,1,1) −2 . See [Rei10], [GP10], and [Che19] for details.…”

Section: Scattering Diagram Examplesmentioning

confidence: 99%

Compactifications of cluster varieties and convexity

Cheung¹,

Magee²,

Chávez³

2019

Preprint

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In [GHKK18], Gross-Hacking-Keel-Kontsevich discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let D be the scattering diagram of a cluster variety V (of either type-A or X ), and let S be a closed subset of (V ∨ ) trop (R)-the ambient space of D. The set S is positive if the theta functions corresponding to the integral points of S and its N-dilations define an N-graded subalgebra of Γ(V, OV ). In particular, a positive set S defines a compactification of V through a Proj construction applied to the corresponding N-graded algebra. In this paper we give a natural convexity notion for subsets of D, called broken line convexity, and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of D, or to check positivity of a given subset.

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Theta functions and quiver Grassmannians

Cited by 2 publications

References 20 publications

The canonical wall structure and intrinsic mirror symmetry

The canonical wall structure and intrinsic mirror symmetry

Compactifications of cluster varieties and convexity

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