scholarly journals Convergence of polarizations, toric degenerations, and Newton–Okounkov bodies

2021 ◽  
Vol 29 (5) ◽  
pp. 1183-1231
Author(s):  
Mark Hamilton ◽  
Megumi Harada ◽  
Kiumars Kaveh
Keyword(s):  
Toric Degenerations

scholarly journals Hilbert schemes and toric degenerations for low degree Fano threefolds

2016 ◽  
Vol 2016 (717) ◽  
Author(s):  
Jan Arthur Christophersen ◽  
Nathan Ilten
Keyword(s):  
Hilbert Schemes ◽  
Fixed Degree ◽  
Fano Threefolds ◽  
Low Degree ◽  
Toric Degenerations

AbstractFor fixed degree

scholarly journals Newton–Okounkov Bodies of Flag Varieties and Combinatorial Mutations

2020 ◽  
Author(s):  
Naoki Fujita ◽  
Akihiro Higashitani
Keyword(s):  
Mirror Symmetry ◽  
Projective Variety ◽  
Fundamental Problem ◽  
Flag Varieties ◽  
Systematic Method ◽  
Fano Manifolds ◽  
Projective Varieties ◽  
Combinatorial Properties ◽  
Toric Degenerations ◽  
Higher Rank

Abstract A Newton–Okounkov body is a convex body constructed from a projective variety with a globally generated line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of projective varieties. Its combinatorial properties heavily depend on the choice of a valuation, and it is a fundamental problem to relate Newton–Okounkov bodies associated with different kinds of valuations. In this paper, we address this problem for flag varieties using the framework of combinatorial mutations, which was introduced in the context of mirror symmetry for Fano manifolds. By applying iterated combinatorial mutations, we connect specific Newton–Okounkov bodies of flag varieties including string polytopes, Nakashima–Zelevinsky polytopes, and Feigin–Fourier–Littelmann–Vinberg polytopes.

scholarly journals Okounkov bodies and toric degenerations

2012 ◽  
Vol 356 (3) ◽  
pp. 1183-1202 ◽  
Author(s):  
Dave Anderson
Keyword(s):  
Toric Degenerations

scholarly journals Mirror symmetry and toric degenerations of partial flag manifolds

2000 ◽  
Vol 184 (1) ◽  
pp. 1-39 ◽  
Author(s):  
Victor V. Batyrev ◽  
Ionuţ Ciocan-Fontanine ◽  
Bumsig Kim ◽  
Duco Straten
Keyword(s):  
Mirror Symmetry ◽  
Flag Manifolds ◽  
Toric Degenerations

scholarly journals Integrable systems, toric degenerations and Okounkov bodies

2015 ◽  
Vol 202 (3) ◽  
pp. 927-985 ◽  
Author(s):  
Megumi Harada ◽  
Kiumars Kaveh
Keyword(s):  
Integrable Systems ◽  
Toric Degenerations

scholarly journals Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

2014 ◽  
Vol 12 (3) ◽  
pp. 473-509 ◽  
Author(s):  
Mark D. Hamilton ◽  
Hiroshi Konno
Keyword(s):  
Flag Manifolds ◽  
Toric Degenerations

scholarly journals Toric degenerations of cluster varieties and cluster duality

2020 ◽  
Vol 156 (10) ◽  
pp. 2149-2206
Author(s):  
Lara Bossinger ◽  
Bosco Frías-Medina ◽  
Timothy Magee ◽  
Alfredo Nájera Chávez
Keyword(s):  
Toric Variety ◽  
Mirror Symmetry ◽  
Duke Math ◽  
Cluster Algebras ◽  
Canonical Bases ◽  
Toric Degeneration ◽  
The Family ◽  
Toric Degenerations ◽  
Cluster Varieties ◽  
Natural Way

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.

scholarly journals On toric degenerations of flag varieties

2016 ◽  
pp. 187-232 ◽  
Author(s):  
Xin Fang ◽  
Ghislain Fourier ◽  
Peter Littelmann
Keyword(s):  
Flag Varieties ◽  
Toric Degenerations

scholarly journals Wall-Crossing for Newton–Okounkov Bodies and the Tropical Grassmannian

2020 ◽  
Author(s):  
Laura Escobar ◽  
Megumi Harada
Keyword(s):  
Projective Variety ◽  
Tropical Geometry ◽  
Technical Condition ◽  
Complex Projective Variety ◽  
Wall Crossing ◽  
Toric Degenerations ◽  
Complex Projective

Abstract Tropical geometry and the theory of Newton–Okounkov bodies are two methods that produce toric degenerations of an irreducible complex projective variety. Kaveh and Manon showed that the two are related. We give geometric maps between the Newton–Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of $X$. Under a technical condition, we produce a natural “algebraic wall-crossing” map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian $Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one $T$-varieties; Ilten also explains the connection to the “combinatorial mutations” studied by Akhtar–Coates–Galkin–Kasprzyk.

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