Rank one summands of Frobenius pushforwards of line bundles on G/P
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908616412364800 |
|---|---|
| author | Rączka, Feliks |
| author_facet | Rączka, Feliks |
| contents | Let $X=G/P$ be a partial flag variety, where $G$ is a semi-simple, simply connected algebraic group defined over an algebraically closed field $K$ of positive characteristic. Let $\mathsf{F}\colon X\to X$ be the absolute Frobenius morphism. Given a line bundle $\mathscr{L}$ on $X$ and an integer $r\geq1$, we describe all line bundles that are direct summands of the pushforward $\mathsf{F}_{*}^{r}\mathscr{L}$. For $\mathscr{L}$ corresponding to a dominant weight, we also compute, for $r$ sufficiently large, the multiplicity of $\mathscr{O}_{X}$ as a summand of $\mathsf{F}_{*}^{r}\mathscr{L}$. As an application we answer a question of Gros-Kaneda about the multiplcity of $\mathscr{L}(-ρ)$ as a direct summand of $\mathsf{F}_{*}\mathscr{O}_{G/B}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rank one summands of Frobenius pushforwards of line bundles on G/P Rączka, Feliks Algebraic Geometry Representation Theory 14G17, 14M15, 14F06, 20G05, 13A35 Let $X=G/P$ be a partial flag variety, where $G$ is a semi-simple, simply connected algebraic group defined over an algebraically closed field $K$ of positive characteristic. Let $\mathsf{F}\colon X\to X$ be the absolute Frobenius morphism. Given a line bundle $\mathscr{L}$ on $X$ and an integer $r\geq1$, we describe all line bundles that are direct summands of the pushforward $\mathsf{F}_{*}^{r}\mathscr{L}$. For $\mathscr{L}$ corresponding to a dominant weight, we also compute, for $r$ sufficiently large, the multiplicity of $\mathscr{O}_{X}$ as a summand of $\mathsf{F}_{*}^{r}\mathscr{L}$. As an application we answer a question of Gros-Kaneda about the multiplcity of $\mathscr{L}(-ρ)$ as a direct summand of $\mathsf{F}_{*}\mathscr{O}_{G/B}$. |
| title | Rank one summands of Frobenius pushforwards of line bundles on G/P |
| topic | Algebraic Geometry Representation Theory 14G17, 14M15, 14F06, 20G05, 13A35 |
| url | https://arxiv.org/abs/2507.20759 |