Cohomology of real-valued differential forms on Berkovich analytic spaces

Berkovich 解析空间上实值微分形式的上同调

• 批准号: 387554191
• 负责人: Dr. Philipp Jell
• 金额: --
• 依托单位: Lehrstuhl für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/387554191?language=en
• 关键词: Cohomology; real; valued; differential; forms

In algebraic geometry one studies the geometry of the set of solutions of a family of polynomial equations. One method to study integral solutions of such systems of equations is Arakelov theory. It was Arakelov's great insight that to study these solutions, it is very helpful to combine algebraic geometry at the prime numbers, often called finite places, with analytic geometry over the complex numbers. It has always been the hope in Arakelov theory that one can use analytic geometry also at finite places. In particular, one needs a notion of real-valued differential forms at such a finite place. In the 1990s, Berkovich introduced suitable analytic spaces, called Berkovich analytic spaces. In 2012 Chambert-Loir and Ducros introduced smooth real-valued differential forms on Berkovich analytic spaces. Chambert-Loir and Ducros, Gubler and Künnemann as well as Liu showed first results in applying these differential forms in Arakelov theory. My own previous results include a Poincaré lemma for these differential forms, which was crucially used in Liu’s work. Further, in joint work with V. Wanner, we showed that the cohomology with respect to smooth real-valued differential forms of Mumford curves satisfies Poincaré duality and used this to completely calculate that cohomology for Mumford curves. The goal of my research project is to study these smooth real-valued differential forms in a general context and prove results about their cohomology, which are analogous to the results over the complex numbers. In particular, I want to prove that the cohomology of curves satisfies Poincaré duality. Poincaré duality is one of the basic properties of smooth differential forms over the complex numbers. It is both useful in theoretical applications as well as in concrete calculations of the cohomology. Since the definition of smooth-real valued differential forms uses tropical geometry and previous work shows direct relations to invariants in tropical geometry, studying questions in tropical geometry will also be part of the project. In said previous work, which was joint work with K. Shaw and J. Smacka, we further showed that smooth tropical varieties satisfy Poincaré duality. I want to show that more tropical spaces than currently known satisfy Poincaré duality. Also I want to prove that certain tropical spaces, and in particular smooth projective tropical varieties, satisfy symmetry in Hodge numbers.

在代数几何形状中，一个研究多项式方程家族的溶液集的几何形状。 Arakelov理论是研究此类方程组的积分解决方案的一种方法。正是阿拉克洛夫（Arakelov）的深刻见解是，研究这些解决方案是非常有帮助的，将代数几何形状以质量数（通常称为有限位置）与复数上的分析几何形状相结合。 Arakelov理论的希望一直是人们可以在有限的地方使用分析几何形状。特别是，一个人需要在这个有限的地方进行实价差异形式的概念。在1990年代，伯科维奇推出了合适的分析空间，称为伯科维奇分析空间。 2012年，查伯特·洛伊尔（Chambert-Loir）和杜克罗斯（Ducros）在伯科维奇（Berkovich）分析空间上引入了平滑的实价差异形式。 Chambert-Loir和Ducros，Gubler和Künnemann以及Liu首先显示出在Arakelov理论中应用这些差异形式。我以前的结果包括这些差异形式的庞加莱引理，这完全在刘的工作中使用。此外，在与V. Wanner的联合合作中，我们表明，关于光滑的实现差异形式的Mumford曲线的共同体学满足了这一点，以完全计算Mumford Curves的共同体。我的研究项目的目的是在一般环境中研究这些平滑的实现差异形式，并证明其同时学的结果，这些形式类似于复数上的结果。特别是，我想证明满足庞加莱二元性的曲线的同时学。 Poincaré二元性是平滑差分形式在复数上的基本特性之一。它在理论应用以及共同体的具体计算中都很有用。由于平滑价值的差异形式的定义使用热带几何形状，并且以前的工作显示了与热带几何形状中不变的直接关系，因此在热带几何形状中研究问题也将成为项目的一部分。在与K. Shaw和J. Smaka的联合工作的先前工作中，我们进一步表明，光滑的热带变化满足了庞加莱二元性。我想证明，比目前所知的满足庞加莱二元性的热带空间更多。我还想证明，某些热带空间，尤其是光滑的投射热带品种，霍奇数字中的满意度对称性。