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1. Real Tropicalization and Analytification of Semialgebraic Sets

   https://doi.org/10.1093/imrn/rnaa112  

   Jell, Philipp; Scheiderer, Claus; Yu, Josephine (May 2020, International Mathematics Research Notices)

   Abstract Let $$K$$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $$K$$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $$S \subseteq K^n$$ we define a space $$S_r^{{\operatorname{an}}}$$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $$S$$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $$X$$ is an algebraic variety, we show that $$X_r^{{\operatorname{an}}}$$ can be canonically embedded into the real spectrum $$X_r$$ of $$X$$, and we study its relation with the Berkovich analytification of $$X$$. 

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2. Tropical and non-Archimedean Monge–Ampère equations for a class of Calabi–Yau hypersurfaces

   https://doi.org/10.1016/j.aim.2024.109494  

   Hultgren, Jakob; Jonsson, Mattias; Mazzon, Enrica; McCleerey, Nicholas (March 2024, Advances in Mathematics)

   For a large class of maximally degenerate families of Calabi–Yau hypersurfaces of complex projective space, we study non- Archimedean and tropical Monge–Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting. 

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3. Higher connectivity of tropicalizations

   https://doi.org/10.1007/s00208-021-02281-9  

   Maclagan, Diane; Yu, Josephine (October 2021, Mathematische Annalen)

   Abstract We show that the tropicalization of an irreducibled-dimensional variety over a field of characteristic 0 is$$(d-\ell )$$ $(d-\ell )$-connected through codimension one, where$$\ell $$ $\ell$is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety. 

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4. Local tropicalizations of splice type surface singularities

   Cueto, Maria Angelica; Popescu-Pampu, Patrick; Stepanov, Dmitry; Wahl, Jonathan (December 2023, Mathematische Annalen)

   Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham–Brieskorn–Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In this paper, we study these singularities from the tropical viewpoint. We characterize their local tropicalizations as the cones over the appropriately embedded associated splice diagrams. As a corollary, we reprove some of Neumann and Wahl’s earlier results on these singularities by purely tropical methods, and show that splice type surface singularities are Newton non-degenerate complete intersections in the sense of Khovanskii. We also confirm that under suitable coprimality conditions on its weights, the diagram can be uniquely recovered from the local tropicalization. As a corollary of the Newton non-degeneracy property, we obtain an alternative proof of a recent theorem of de Felipe, González Pérez and Mourtada, stating that embedded resolutions of any plane curve singularity can be achieved by a single toric morphism, after re-embedding the ambient smooth surface germ in a higher-dimensional smooth space. The paper ends with an appendix by Jonathan Wahl, proving a criterion of regularity of a sequence in a ring of convergent power series, given the regularity of an associated sequence of initial forms. 

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5. Wall-Crossing for Newton–Okounkov Bodies and the Tropical Grassmannian

   https://doi.org/10.1093/imrn/rnaa230  

   Escobar, Laura; Harada, Megumi (September 2020, International Mathematics Research Notices)

   null (Ed.)

   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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