A guide to tropicalizations (Q2867777)

Let \(K\) be a non-Archimedean field and \(T\) be a split algebraic torus over \(K\). In tropical geometry one associates to a closed subscheme \(X\) of \(T\) its \textit{tropicalization} \(\mathrm{Trop}(X)\), a weighted rational polyhedral complex in the real vector space \(N_\mathbb{R}\) generated by the cocharacter lattice \(N\) of \(T\). The basic idea of tropical geometry is to relate the combinatorial properties of \(\mathrm{Trop}(X)\) to the algebraic geometric properties of \(X\).NEWLINENEWLINEIt has been observed in [\textit{W. Gubler}, Invent. Math. 169, No. 2, 321--376 (2007; Zbl 1153.14029)] and [\textit{M. Einsiedler, M. Kapranov} and \textit{D. Lind}, J. Reine Angew. Math. 601, 139--157 (2006; Zbl 1115.14051)] that \(\mathrm{Trop}(X)\) naturally arises as a projection of the Berkovich analytic space \(X^{\mathrm{an}}\) associated to \(X\) to \(N_\mathbb{R}\).NEWLINENEWLINEIn his guide Gubler works out many aspects of this theory in their natural generality and provides the reader with accessible foundational knowledge of the non-Archimedean approach to tropical geometry.NEWLINENEWLINEAfter a concise introduction to non-Archimedean analytic geometry he constructs the \textit{tropicalization map} \(\mathrm{trop}:T^{\mathrm{an}}\rightarrow N_\mathbb{R}\) that is given by associating to a seminorm \(| .|_x\) on the coordinate ring \(K[M]\) of \(T\) the element \(m\mapsto -\log| \chi^m|_x\) in \(N_\mathbb{R}=Hom(M,\mathbb{R})\). The \textit{tropical variety} \(\mathrm{Trop}(X)\) associated to \(X\subseteq T\) is then defined as projection of \(X^{\mathrm{an}}\) to \(N_\mathbb{R}\) under the tropicalization map.NEWLINENEWLINEFurthermore Gubler uses the theory of models of algebraic varieties over valuations rings to give an extrinsic geometric definition of the \textit{initial degenerations} associated to \(X\). From the point of view of commutative algebra these initial degenerations can be computed using computer algebra systems and form one of the crucial techniques used when explicitly computing with tropical varieties (see [\textit{D. Maclagan} and \textit{B. Sturmfels}, Introduction to tropical geometry. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1321.14048)]).NEWLINENEWLINEIn a next step Gubler generalizes the theory of toric schemes over discrete valuation rings (see [\textit{G. Kempf} et al., Toroidal embeddings. I. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0271.14017)]) to toric schemes over arbitrary valuation rings of rank one.NEWLINENEWLINEBased on these foundations, he gives a generalization of Tevelev's theory of compactifications of subschemes of tori in toric varieties, so called \textit{tropical compactifications} (see [\textit{J. Tevelev}, Am. J. Math. 129, No. 4, 1087--1104 (2007; Zbl 1154.14039)]), to compactifications in arbitrary toric schemes over valuation rings \(R\) of rank one. This finally leads to a general theory of \textit{tropical multiplicities} encompassing all earlier approaches. The crucial difficulty in this development is that the valuation ring \(R\) may not be Noetherian and, when dealing with the geometry of models over \(R\), one has to leave the familiar realm of Noetherian schemes.NEWLINENEWLINEThe author of this review has profited immensely from reading this survey article and strongly recommends anyone with even only a remote interest in the relationship between tropical and non-Archimedean geometry to read it.NEWLINENEWLINEFor the entire collection see [Zbl 1266.14003].