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Zbl 1153.14036
Gubler, Walter
Tropical varieties for non-archimedean analytic spaces. (English)
[J] Invent. Math. 169, No. 2, 321-376 (2007). ISSN 0020-9910; ISSN 1432-1297/e

This paper proves three main theorems, the third of which provides a key step in the author's proof of the Bogomolov conjecture for totally degenerate abelian varieties over function fields [{\it W. Gubler}, Invent. Math. 169, 377--400 (2007; Zbl 1153.14029)]. \par For the first main result, let $\Bbb K$ be an algebraically closed field, complete with respect to a non-archimedean absolute value $\vert \cdot\vert $; let $v=-\log\Vert \cdot\Vert $ be the associated valuation; and let $\Gamma=v(\Bbb K^\times)$ be the value group. Let $X$ be an irreducible closed analytic subvariety of $\Bbb G_{\text{m}}^n$ over $\Bbb K$ of dimension $d$. In this paper, analysis over $\Bbb K$ refers to Berkovich's theory of analytic spaces. Define a map $\text {val}\colon \Bbb G_{\text{m}}^n(\Bbb K)\to\Bbb R^n$ by $\bold x\mapsto(v(x_1),\dots,v(x_n))$, where $x_1,\dots,x_n$ are fixed coordinates on $\Bbb G_{\text{m}}^n$. The closure of the image of $X(\Bbb K)$ is a tropical variety. The first result of this paper is that this tropical variety is a connected totally concave locally finite union of $d$-dimensional $\Gamma$-rational polytopes. (The terminology from convex geometry is defined in the paper's appendix.) This result generalizes a theorem of {\it M. Einsiedler, M. Kapranov} and {\it D. Lind} [J. Reine Angew. Math. 601, 139--157 (2006; Zbl 1115.14051)], which treats the case where $X$ is an algebraic subvariety of $\Bbb G_{\text{m}}^n$. \par For the second result, let $\Bbb K$ and $v$ be as above, let $\Bbb K^\circ$ denote the valuation ring of $\Bbb K$, and let $\widetilde{\Bbb K}$ denote its residue field. An abelian variety $A$ over $\Bbb K$ is {\it totally degenerate\/} if $A^{\text{an}}\cong\bigl(\Bbb G_{\text{m}}^n\bigr)_{\Bbb K}^{\text{an}}/M$ for a discrete subgroup $M$ of $\Bbb G_{\text{m}}^n(\Bbb K)$ mapped isomorphically by $\text {val}$ to a complete lattice $\Lambda$ in $\Bbb R^n$. Let $A$ be a totally degenerate abelian variety over $\Bbb K$. Then there is a canonical map $\overline{\text {val}}\colon A^{\text{an}}\to\Bbb R^n/\Lambda$, and hence a tropical variety $\overline{\text {val}}(X^{\text{an}})$ associated to a closed analytic subvariety $X$ of $A$. This tropical variety is again a connected totally concave locally finite union of $d$-dimensional $\Gamma$-rational polytopes. The second result of this paper is that, if $f\colon X'\to A$ is a morphism over $\Bbb K$ and if $\Cal X'$ is a model for $X'$ over $\Bbb K^\circ$ satisfying certain conditions, then the special fiber of $\Cal X'$ has a $\widetilde{\Bbb K}$-rational point contained in at least $1+\dim f(X')$ irreducible components. \par For the third result, let $K$ be a field with a discrete valuation and let $\Bbb K$ be the completion of the algebraic closure of the completion of $K$. Let $A$ be an abelian variety over $K$ for which $A_{\Bbb K}^{\text{an}}$ is totally degenerate, and let $X$ be a closed subvariety of $A$ of pure dimension $d$. Let $\overline L_1,\dots,\overline L_d$ be ample line bundles on $A$ endowed with canonical metrics. The author notes that Berkovich theory does not currently have a definition for the Chern forms $c_1(\overline L_i)$, but measures $c_1(\overline L_1\big\vert _X)\wedge\dots\wedge c_1(\overline L_d\big\vert _X)$ on $X_{\Bbb K}^{\text{an}}$ analogous to the top-dimensional forms in differential geometry have been defined by {\it A. Chambert-Loir} [J. Reine Angew. Math. 595, 215--235 (2006; Zbl 1112.14022)]. This paper develops these measures further, and as its third main result shows that $\overline{\text {val}}(c_1(\overline L_1\big\vert _X) \wedge\dots\wedge c_1(\overline L_d\big\vert _X))$ is a strictly positive piecewise Haar measure on the polytopal set $\overline{\text {val}}\bigl(X_{\Bbb K}^{\text{an}}\bigr)$. \par The proof of this result is based on a study of Mumford's model $\Cal A$ of $A$ associated to a $\Gamma$-rational polytopal decomposition of $\Bbb R^n/\Lambda$. In a subsequent paper (cited earlier in this review), the author uses this result to prove a {\it tropical equidistribution result\/} modeled after an equidistribution result of {\it S. Zhang} [Ann. Math. (2) 147, 159--165 (1998; Zbl 0991.11034)]. This then leads (in the author's subsequent paper) to a proof of the Bogomolov conjecture over function fields for abelian varieties that are totally degenerate at at least one place.
[Paul Vojta (Berkeley)]

MSC 2000:: *14P99 Real algebraic and real analytic geometry
32P05 Non-Archimedean complex analysis
32J25 Transcendental methods of algebraic geometry
12J25 Non-Archimedean valued fields
52A41 Convex functions and convex programs (convex geometry)

Keywords: tropical algebraic geometry; Berkovich space; toric variety; convex geometry

Citations: Zbl 1115.14051; Zbl 1112.14022; Zbl 0991.11034; Zbl 1153.14029

Cited in: Zbl 1153.14029

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