Local tropicalization (Q2867780)

Tropicalization is a process that associates to a closed subvariety \(Y\) of a split algebraic torus \(T\) over a non-Archimedean base field \(k\) a rational polyhedral complex \(\mathrm{Trop}(Y)\) in the real vector space \(N_\mathbb{R}\) spanned by the cocharacter lattice \(N\) of \(T\). Given a non-Archimedean extension \(K\) of \(k\) whose norm is non-trivial, one can identify \(\mathrm{Trop}(Y)\) with the closure of the image of \(Y(K)\) under the coordinate-wise valuation map \(T(K)\rightarrow N_\mathbb{R}\).NEWLINENEWLINEIn its most natural form this definition is given using the language of non-Archimedean analytic spaces in the sense of Berkovich (see [\textit{W. Gubler}, Invent. Math. 169, No. 2, 377--400 (2007; Zbl 1153.14029)] and [\textit{W. Gubler}, Contemp. Math. 589, 125--189 (2013; Zbl 1318.14061)]) or, alternatively, valuation spaces (see [\textit{M. Einsiedler, M. Kapranov} and \textit{D. Lind}, J. Reine Angew. Math. 601, 139--157 (2006; Zbl 1115.14051)]). These approaches naturally generalize to subvarieties of toric varieties with big torus \(T\) (see [\textit{T. Kajiwara}, Contemp. Math. 460, 197--207 (2008; Zbl 1202.14047)] and [\textit{S. Payne}, Math. Res. Lett. 16, No. 2--3, 543--556 (2009; Zbl 1193.14077]).NEWLINENEWLINEUsing the theory of valuation spaces and an in-depth study of the geometry of toric monoids Popescu-Pampu and Stepanov propose in this article a general framework incorporating all of the above approaches to tropicalization and many more. Most notably, they can also deal with subvarieties of formal completions of affine toric varieties along an orbit closure. This allows them to study the \textit{local tropicalization} of a closed subvariety of a toroidal embedding in the sense of [\textit{G. Kempf, F. Knudsen, D. Mumford} and \textit{B. Saint-Donat}, Toroidal embeddings. I. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0271.14017)].NEWLINENEWLINETheir main result is a generalization of the classical Theorem of \textit{R. Bieri} and \textit{J. R. J. Groves} [J. Reine Angew. Math. 347, 168--195 (1984; Zbl 0526.13003)], which describes the tropicalization of an algebraic variety as a rational polyhedral complex, to local tropicalizations of subvarieties of toroidal embeddings. Its proof involves the development of a theory of tropical bases in the formal completions \(k[[\Gamma]]\) of the monoid algebras \(k[\Gamma]\) defined by a toric monoid \(\Gamma\).NEWLINENEWLINEPopescu-Pampu and Stepanov's theory forms the foundation of a global theory of tropicalization for toroidal embeddings, which has been developed by the reviewer using the language of logarithmic geometry (see [\textit{M. Ulirsch}, Math. Z. 280, No. 1--2, 195--210 (2015; Zbl 1327.14267)] and [\textit{M. Ulirsch}, ``Functorial tropicalization of logarithmic schemes: The case of constant coefficients'', Preprint, \url{arXiv:1310.6269}]).NEWLINENEWLINEFor the entire collection see [Zbl 1266.14003].