Spherical amoebae and a spherical logarithm map
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913264556834816 |
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| author | Batyrev, Victor Harada, Megumi Hofscheier, Johannes Kaveh, Kiumars |
| author_facet | Batyrev, Victor Harada, Megumi Hofscheier, Johannes Kaveh, Kiumars |
| contents | Let $G$ be a connected reductive algebraic group over $\mathbb{C}$ with a maximal compact subgroup $K$. Let $G/H$ be a (quasi-affine) spherical homogeneous space. In the first part of the paper, following Akhiezer's definition of spherical functions, we introduce a $K$-invariant map $sLog_{Γ, t}: G/H \to \mathbb{R}^s$ which depends on a choice of a finite set $Γ$ of dominant weights and $s = |Γ|$. We call $sLog_{Γ, t}$ a spherical logarithm map. We show that when $Γ$ generates the highest weight monoid of $G/H$, the image of the spherical logarithm map parametrizes $K$-orbits in $G/H$. This idea of using the spherical functions to understand the geometry of the space $K \backslash G/H$ of $K$-orbits in $G/H$ can be viewed as a generalization of the classical Cartan decomposition. In the second part of the paper, we define the spherical amoeba (depending on $Γ$ and $t$) of a subvariety $Y$ of $G/H$ as $sLog_{Γ, t}(Y)$, and we ask for conditions under which the image of a subvariety $Y \subset G/H$ under $sLog_{Γ, t}$ converges, as $t \to 0$, in the sense of Kuratowski to its spherical tropicalization as defined by Tevelev and Vogiannou. We prove a partial result toward answering this question, which shows in particular that the valuation cone is always contained in the Kuratowski limit of the spherical amoebae of $G/H$. We also show that the limit of the spherical amoebae of $G/H$ is equal to its valuation cone in a number of interesting examples, including when $G/H$ is horospherical, and in the case when $G/H$ is the space of hyperbolic triangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09091 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spherical amoebae and a spherical logarithm map Batyrev, Victor Harada, Megumi Hofscheier, Johannes Kaveh, Kiumars Algebraic Geometry 14M27, 14T20 Let $G$ be a connected reductive algebraic group over $\mathbb{C}$ with a maximal compact subgroup $K$. Let $G/H$ be a (quasi-affine) spherical homogeneous space. In the first part of the paper, following Akhiezer's definition of spherical functions, we introduce a $K$-invariant map $sLog_{Γ, t}: G/H \to \mathbb{R}^s$ which depends on a choice of a finite set $Γ$ of dominant weights and $s = |Γ|$. We call $sLog_{Γ, t}$ a spherical logarithm map. We show that when $Γ$ generates the highest weight monoid of $G/H$, the image of the spherical logarithm map parametrizes $K$-orbits in $G/H$. This idea of using the spherical functions to understand the geometry of the space $K \backslash G/H$ of $K$-orbits in $G/H$ can be viewed as a generalization of the classical Cartan decomposition. In the second part of the paper, we define the spherical amoeba (depending on $Γ$ and $t$) of a subvariety $Y$ of $G/H$ as $sLog_{Γ, t}(Y)$, and we ask for conditions under which the image of a subvariety $Y \subset G/H$ under $sLog_{Γ, t}$ converges, as $t \to 0$, in the sense of Kuratowski to its spherical tropicalization as defined by Tevelev and Vogiannou. We prove a partial result toward answering this question, which shows in particular that the valuation cone is always contained in the Kuratowski limit of the spherical amoebae of $G/H$. We also show that the limit of the spherical amoebae of $G/H$ is equal to its valuation cone in a number of interesting examples, including when $G/H$ is horospherical, and in the case when $G/H$ is the space of hyperbolic triangles. |
| title | Spherical amoebae and a spherical logarithm map |
| topic | Algebraic Geometry 14M27, 14T20 |
| url | https://arxiv.org/abs/2403.09091 |