Cohomology of real-valued differential forms on Berkovich analytic spaces

在代数几何中，人们研究一类多项式方程的解集的几何性质。研究这类方程组的积分解的一种方法是Arakelov理论。阿拉克洛夫的伟大见解是，为了研究这些解，将素数上的代数几何（通常称为有限位）与复数上的解析几何结合起来是非常有帮助的。在Arakelov理论中人们一直希望在有限的地方也能使用解析几何。特别地，在这样一个有限的地方，我们需要一个实值微分形式的概念。在20世纪90年代，Berkovich引入了合适的解析空间，称为Berkovich解析空间。2012年，Chambert-Loir和Ducros引入了Berkovich解析空间上的光滑实值微分形式。Chambert-Loir和Ducros， Gubler和k<s:1> nnemann以及Liu首次在Arakelov理论中应用了这些微分形式。我自己之前的结果包括关于这些微分形式的庞加莱引理，这在刘的工作中至关重要。进一步，在与V. Wanner的联合工作中，我们证明了Mumford曲线的光滑实值微分形式的上同调满足poincar<s:1>对偶性，并以此完全计算了Mumford曲线的上同调。我的研究项目的目标是在一般情况下研究这些光滑的实值微分形式，并证明它们的上同调的结果，类似于复数上的结果。特别地，我想证明曲线的上同性满足庞卡罗对偶性。庞卡罗对偶性是复数光滑微分形式的基本性质之一。它不仅在上同调的理论应用中有用，而且在上同调的具体计算中也有用。由于光滑实值微分形式的定义使用了热带几何，并且先前的工作显示了与热带几何中的不变量的直接关系，因此研究热带几何中的问题也将是该项目的一部分。在之前与K. Shaw和J. Smacka合作的工作中，我们进一步证明了光滑的热带品种满足poincarcarcars对偶性。我想展示更多的热带空间比目前已知的满足庞卡罗二象性。我还想证明某些热带空间，特别是光滑投影热带空间，满足霍奇数的对称性。

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.