Tropical varieties for non-archimedean analytic spaces
Tropical varieties for non-archimedean analytic spaces
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非阿基米德分析空间的热带品种
DOI：
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发表时间：
2006
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通讯作者：
Walter Gubler
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作者：
Walter Gubler
从热带代数几何的建设推广，我们关联到每个（不可约d维）封闭解析子簇$\mathbb{G}_{m}^{n}$的热带品种在一个完整的非阿基米德的地方。通过解析几何和形式几何的方法，我们证明了热带簇是d维多面体的全凹局部有限并。对于到全退化阿贝尔簇A的代数态射f：X '→ A，我们给出了f（X'）的维数由X '的严格半稳定模型的奇异性给出的一个界. A的一个闭d维子簇X诱导一个周期性的热带簇。芒福德构造的推广产生了X和A的模型，可以用复曲面簇的理论来处理。对于A上的正则度量化线丛L ∈ L，测度c1（L ∈ L）|X）d是X上的分段Haar测度。使用凸几何的方法，我们给出了一个明确的描述，这些措施的热带几何。在随后的论文中，这是一个关键的一步证明博戈莫洛夫猜想完全退化阿贝尔品种的功能领域。
Generalizing the construction from tropical algebraic geometry, we associate to every (irreducible d-dimensional) closed analytic subvariety of $\mathbb{G}_{m}^{n}$ a tropical variety in ℝn with respect to a complete non-archimedean place. By methods of analytic and formal geometry, we prove that the tropical variety is a totally concave locally finite union of d-dimensional polytopes. For an algebraic morphism f:X’→A to a totally degenerate abelian variety A, we give a bound for the dimension of f(X’) in terms of the singularities of a strictly semistable model of X’. A closed d-dimensional subvariety X of A induces a periodic tropical variety. A generalization of Mumford’s construction yields models of X and A which can be handled with the theory of toric varieties. For a canonically metrized line bundle L̄ on A, the measures c1(L̄|X)∧d are piecewise Haar measures on X. Using methods of convex geometry, we give an explicit description of these measures in terms of tropical geometry. In a subsequent paper, this is a key step in the proof of Bogomolov’s conjecture for totally degenerate abelian varieties over function fields.