Cohomology of real-valued differential forms on Berkovich analytic spaces

在代数几何学中，人们研究一个多项式方程族的解的集合的几何学。研究这类方程组的积分解的一种方法是阿拉克洛夫理论。这是阿拉克洛夫的伟大洞察力，研究这些解决方案，这是非常有帮助的联合收割机代数几何在素数，通常被称为有限的地方，解析几何的复数。它一直希望在阿拉克洛夫理论，人们可以使用解析几何也在有限的地方。特别是，需要在这样一个有限的地方实值微分形式的概念。在1990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在Berkovich解析空间上引入了光滑实值微分形式。Chambert-Loir和Ducros，Gubler和Künnemann以及刘展示了第一个结果在应用这些微分形式在Arakelov理论。我自己以前的结果包括这些微分形式的庞加莱引理，这在刘的工作中至关重要。此外，在与V. Wanner的联合工作中，我们证明了关于Mumford曲线的光滑实值微分形式的上同调满足庞加莱对偶，并使用它来完全计算Mumford曲线的上同调。我的研究项目的目标是在一般的背景下研究这些光滑的实值微分形式，并证明其上同调的结果，这是类似于复数的结果。特别地，我想证明曲线的上同调满足Poincaré对偶。Poincaré对偶是复数上光滑微分形式的基本性质之一。它在理论应用和具体的上同调计算中都很有用。 由于光滑实值微分形式的定义使用热带几何和以前的工作表明直接关系到热带几何中的不变量，研究热带几何中的问题也将是该项目的一部分。在所述先前的工作中，这是与K. Shaw和J. Smacka，我们进一步证明了光滑热带簇满足Poincaré对偶。 我想证明比目前已知的更多的热带空间满足庞加莱对偶。此外，我想证明，某些热带空间，特别是光滑的投影热带品种，满足对称性霍奇数。

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.