论文标题
$ gl_n（\ Mathbb {C}）$的子视图的广义变形虫
Generalized amoebas for subvarieties of $GL_n(\mathbb{C})$
论文作者
论文摘要
本文是根据第一作者和第二作者的指导下在匹兹堡大学的三个月实习期间获得的结果的报告。第二作者和Manon和Manon中，已扩展到General Linear Group $ gl_n（\ Mathbb {C}）$的圆环$（\ Mathbb {c}^*）^n $中的子变量的概念。在本文中，我们显示了这些基质变形虫的一些基本属性，例如任何这样的变形虫都关闭，并且当多种表面为hypersurface时，其补体的连接组件是凸。我们还将Ronkin功能的概念扩展到此设置。对于Hypersurface，我们展示了如何使用牛顿多层室的概念来描述基质变形虫的渐近方向。最后，我们部分扩展了经典陈述，即变形虫会融合到热带品种。我们还讨论了一些例子。我们的矩阵变形虫应被视为Tevelev-vogiannou的球形热带化版本，用于$ gl_n（\ Mathbb {c}）$，被视为球形同质空间，用于$ gl_n（\ Mathbb {c} {c}）$ times gl_n（\ times gl_n（c）$ gl_n（\ mathbb {c}）$ national（
This paper is a report based on the results obtained during a three months internship at the University of Pittsburgh by the first author and under the mentorship of the second author. The notion of an amoeba of a subvariety in a torus $(\mathbb{C}^*)^n$ has been extended to subvarieties of the general linear group $GL_n(\mathbb{C})$ by the second author and Manon. In this paper, we show some basic properties of these matrix amoebas, e.g. any such amoeba is closed and the connected components of its complement are convex when the variety is a hypersurface. We also extend the notion of Ronkin function to this setting. For hypersurfaces, we show how to describe the asymptotic directions of the matrix amoebas using a notion of Newton polytope. Finally, we partially extend the classical statement that the amoebas converge to the tropical variety. We also discuss a few examples. Our matrix amoeba should be considered as the Archimedean version of the spherical tropicalization of Tevelev-Vogiannou for the variety $GL_n(\mathbb{C})$ regarded as a spherical homogeneous space for the left-right action of $GL_n(\mathbb{C}) \times GL_n(\mathbb{C})$.
