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Register a new userParallel Computation of tropical varieties, their positive part, and tropical Grassmannians
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Added by
Benjamin Schr\\\"oter
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the exception of $23$ special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr$^+(3,8)$ and compare them to the cluster complex of the classical Grassmannian Gr$(3,8)$.
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In this paper we prove that the cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. On the way to establish these results, we introduce and prove other results of independent interest. This includes a generalization of the results of Adiprasito-Huh-Katz, Hodge theory for combinatorial geometries, to any unimodular quasi-projective fan having the same support as the Bergman fan of a matroid, a tropical analog for Bergman fans of the pioneering work of Feichtner-Yuzvinsky on cohomology of wonderful compactifications (treated in a separate paper, recalled and used here), a combinatorial study of the tropical version of the Steenbrink spectral sequence, a treatment of Kahler forms in tropical geometry and their associated Hodge-Lefschetz structures, a tropical version of the projective bundle formula, and a result in polyhedral geometry on the existence of quasi-projective unimodular triangulations of polyhedral spaces.
The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of smooth tropical varieties. Among the resulting applications, we prove the Hodge isomorphism theorem which asserts that the Chow rings of smooth unimodular tropical fans are isomorphic to the tropical cohomology rings of their corresponding canonical compactifications, and prove a slightly weaker statement for any unimodular fan. We furthermore introduce a notion of shellability for tropical fans and show that shellable tropical fans are smooth and thus enjoy all the nice homological properties of smooth tropical fans. Several other interesting properties for tropical fans are shown to be shellable. Finally, we obtain a generalization, both in the tropical and in the classical setting, of the pioneering work of Feichtner-Yuzvinsky and De Concini-Procesi on the cohomology ring of wonderful compactifications of complements of hyperplane arrangements. The results in this paper form the basis for our subsequent works on Hodge theory for tropical and non-Archimedean varieties.
We consider Gromovs homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseaux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and we prove a weak form of our conjecture for the nonarchimedean amoeba of a variety over the complex Puiseaux field. One of our main tools is Jonssons limit theorem for tropical varieties.
In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the tropical skeleton of $X$ as the subset of the associated Berkovich space $X^{rm an}$ which collects all Shilov boundary points in the fibers of the Kajiwara--Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When $X$ is schon and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm--Katz.
We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in $mathbb{TP}^5$. This follows from a result of Sturmfels and Speyer which classifies these tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian $operatorname{Gr}(3,6)$. Speyer and Williams show that the positive part $operatorname{Gr}^+(3,6)$ of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type $D_4$. We provide a structural bijection between the rays of $operatorname{Gr}^+(3,6)$ and the almost positive roots of type $D_4$ which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type $D_4$ to describe the equivalence of positive tropical planes in $mathbb{TP}^5$, giving a combinatorial model which characterizes the combinatorial types of tropical planes using automorphisms of pseudotriangulations of the octogon.
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