论文标题
混合曲线的模量II：热带和混合laplacians
Moduli of hybrid curves II: Tropical and hybrid Laplacians
论文作者
论文摘要
本文是我们在曲线及其模量空间的混合几何形状上工作的续集。我们介绍了混合拉普拉斯式的概念，制定了混合泊松方程，并为拉普拉斯操作员的收敛赋予了数学含义，又为riemann表面上的泊松方程提供了解决方案。然后，作为本文的主要定理，我们获得了Arakelov绿色函数在靠近其模量空间边界的Riemann表面上的渐近描述。这是根据合适的杂化绿色功能概念来完成的。 作为我们方法的副产品，我们获得了独立利益的其他结果。特别是，我们引入了风扇和多面体空间的更高等级的规范压实，并使用它们来定义高级热带曲线的模量空间。此外，我们在较高等级的非Archimedean，Hybrid和Tame分析中发展了函数理论的第一步。此外，我们在相应的模量空间中的极限热带曲线上建立了Laplace操作员在度量图上向热带拉普拉斯操作员的收敛性，从而在度量图上有了新的观点。 我们对Arakelov绿色功能的结果灵感来自几位作者的作品，尤其是Faltings，De Jong，Wentworth和Wolpert的作品，并解决了由Riemann表面的Arakelov几何形状引起的长期开放问题。靠近模量空间边界的混合分层行为预计将是一个广泛的现象，将在我们即将进行的工作中进行探索。
The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions. As a byproduct of our approach, we obtain other results of independent interest. In particular, we introduce higher rank canonical compactifications of fans and polyhedral spaces and use them to define the moduli space of higher rank tropical curves. Moreover, we develop the first steps of a function theory in higher rank non-Archimedean, hybrid, and tame analysis. Furthermore, we establish the convergence of the Laplace operator on metric graphs toward the tropical Laplace operator on limit tropical curves in the corresponding moduli spaces, leading to new perspectives in operator theory on metric graphs. Our result on the Arakelov Green function is inspired by the works of several authors, in particular those of Faltings, de Jong, Wentworth and Wolpert, and solves a long-standing open problem arising from the Arakelov geometry of Riemann surfaces. The hybrid layered behavior close to the boundary of moduli spaces is expected to be a broad phenomenon and will be explored in our forthcoming work.
