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Toric degenerations of cluster varieties and cluster duality

Published online by Cambridge University Press:  07 December 2020

Lara Bossinger
Affiliation:
Instituto de Matemáticas UNAM Unidad Oaxaca, León 2, altos, Oaxaca de Juárez, Centro Histórico, 68000 Oaxaca, Mexico lara@im.unam.mx
Bosco Frías-Medina
Affiliation:
Centro de Ciencias Matemáticas UNAM Campus Morelia, Antigua Carretera a Pátzcuaro 8701, Col. Ex Hacienda San José de la Huerta, 58089 Michoacán, Mexico bosco@matmor.unam.mx
Timothy Magee
Affiliation:
Department of Mathematics, Faculty of Natural & Mathematical Sciences, King's College London, Strand, London WC2R 2LS, UK timothy.magee@kcl.ac.uk
Alfredo Nájera Chávez
Affiliation:
CONACYT-Instituto de Matemáticas UNAM Unidad Oaxaca, León 2, altos, Oaxaca de Juárez, Centro Histórico, 68000 Oaxaca, Mexico najera@matem.unam.mx

Abstract

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Type
Research Article
Copyright
© The Author(s) 2020

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Footnotes

This work was partially supported by CONACYT grant CB2016 no. 284621. The first author was partially supported by the Max Planck Institute of Mathematics in the Sciences, Leipzig and the ‘Programa de Becas Posdoctorales en la UNAM 2018’ Instituto de Matemáticas, UNAM. The second and third authors acknowledge the support of Fondo Institucional de Fomento Regional para el Desarrollo Científico, Tecnológico y de Innovación, FORDECYT 265667. The second author was additionally supported by ‘Programa de Becas Posdoctorales 2019’, DGAPA, UNAM during the revisions of this article. The third author was additionally partially supported by EPSRC grant EP/P021913/1 during revisions of this article.

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