Tropical cycle classes for non-archimedean spaces and weight decomposition of De Rham cohomology sheaves
Tropical cycle classes for non-archimedean spaces and weight decomposition of De Rham cohomology sheaves
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非阿基米德空间的热带循环类和 De Rham 上同调滑轮的权重分解
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发表时间：
2020
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通讯作者：
Yifeng Liu
中科院分区：
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作者：
Yifeng Liu
本文有三个主要目标。首先，我们定义了非阿基米德域上光滑品种的热带循环类图，这些图用用Chambert-Loir和Ducros引入的实形式定义的Dolbeault上同调来表示。其次，对于特征为零的非阿基米德域上的光滑解析空间，我们构造了de Rham上同轴的泛函分解，称为权分解，推广了Berkovich的构造，解决了自己提出的一个问题。第三，揭示了热带理论与代数de Rham理论之间的联系。作为一个应用，我们证明了在代数de Rham上同调中平凡的代数环作为Dolbeault上同调的电流也是平凡的。
This article has three major goals. First, we define tropical cycle class maps for smooth varieties over non-Archimedean fields, valued in the Dolbeault cohomology defined in terms of real forms introduced by Chambert-Loir and Ducros. Second, we construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over certain non-Archimedean fields of characteristic zero, which generalizes a construction of Berkovich and solves a question raised by himself. Third, we reveal a connection between the tropical theory and the algebraic de Rham theory. As an application, we show that algebraic cycles that are trivial in the algebraic de Rham cohomology are trivial as currents for Dolbeault cohomology as well.