Newton--Okounkov bodies for cluster varieties

Abstract: Cluster varieties are schemes glued from algebraic tori. Just as tori themselves, they come in dual pairs and it is good to think of them as generalizing tori. Just as compactifications of tori give rise to interesting varieties, (partial) compactifications of cluster varieties include examples such as Grassmannians, partial flag varieties or configurations spaces. A few years ago Gross--Hacking--Keel--Kontsevich developed a mirror symmetry inspired program for cluster varieties. I will explain how their tools can be used to obtain valuations and Newton--Okounkov bodies for their (partial) compactifications. The rich structure of cluster varieties however can be exploited even further in this context which leads us to an intrinsic definition of a Newton--Okounkov body. The theory of cluster varieties interacts beautifully with representation theory and algebraic groups. I will exhibit this connection by comparing GHKK's technology with known mirror symmetry constructions such as those by Givental, Baytev--Ciocan-Fontanini--Kim--van Straten, Rietsch and Marsh--Rietsch.

algebraic geometrydifferential geometryrepresentation theory

Audience: researchers in the topic

Toric Degenerations

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