On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space
| Autor: | Shizhuo Yu, Jun Peng |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
General Mathematics Frobenius splitting 14F17 20G05 53D17 Poisson distribution Mathematics - Algebraic Geometry symbols.namesake Borel subgroup Algebraic torus Algebraic group Poisson manifold FOS: Mathematics Affine space symbols Algebraically closed field Algebraic Geometry (math.AG) Mathematics |
| Zdroj: | International Mathematics Research Notices. 2021:11618-11651 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnz179 |
| Popis: | The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$, where $G$ is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic $p> 3$, $U$ is the uniradical of a Borel subgroup of $G$, and $\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$. Then we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$. |
| Databáze: | OpenAIRE |
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