Equivariant Parabolic connections and stack of roots
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909213955981312 |
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| author | Chakraborty, Sujoy Paul, Arjun |
| author_facet | Chakraborty, Sujoy Paul, Arjun |
| contents | Let $X$ be a smooth complex projective variety equipped with an action of a linear algebraic group $G$ over $\mathbb{C}$. Let $D$ be a reduced effective divisor on $X$ that is invariant under the $G$--action on $X$. Let $s_D$ be the canonical section of $\mathcal{O}_X(D)$ vanishing along $D$. Given a positive integer $r$, consider the stack $\mathfrak{X} := \mathfrak{X}_{(\mathcal{O}_X(D),\, s_D,\, r)}$ of $r$-th roots of $(\mathcal{O}_X, s_D)$ together with the natural morphism $π: \mathfrak{X} \to X$. Under the assumption that $G$ has no non-trivial characters, we show that the $G$--action on $X$ naturally lifts to a $G$--action on $\mathfrak{X}$ such that $π$ become $G$--equivariant, and the tautological invertible sheaf $\mathscr{M}$ on $\mathfrak{X}$ admits a linearization of this $G$--action. Finally, we define the notions of $G$--equivariant logarithmic connections on $\mathfrak{X}$ and $G$--equivariant parabolic connections on $X$ with rational parabolic weights along $D$, and establish an equivalence between the category of $G$--equivariant logarithmic connections on $\mathfrak{X}$ and the category of $G$--equivariant parabolic connections on $X$ with rational parabolic weights along $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20699 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Parabolic connections and stack of roots Chakraborty, Sujoy Paul, Arjun Algebraic Geometry 14D23, 14H60, 53B15, 53C05, 14A21 Let $X$ be a smooth complex projective variety equipped with an action of a linear algebraic group $G$ over $\mathbb{C}$. Let $D$ be a reduced effective divisor on $X$ that is invariant under the $G$--action on $X$. Let $s_D$ be the canonical section of $\mathcal{O}_X(D)$ vanishing along $D$. Given a positive integer $r$, consider the stack $\mathfrak{X} := \mathfrak{X}_{(\mathcal{O}_X(D),\, s_D,\, r)}$ of $r$-th roots of $(\mathcal{O}_X, s_D)$ together with the natural morphism $π: \mathfrak{X} \to X$. Under the assumption that $G$ has no non-trivial characters, we show that the $G$--action on $X$ naturally lifts to a $G$--action on $\mathfrak{X}$ such that $π$ become $G$--equivariant, and the tautological invertible sheaf $\mathscr{M}$ on $\mathfrak{X}$ admits a linearization of this $G$--action. Finally, we define the notions of $G$--equivariant logarithmic connections on $\mathfrak{X}$ and $G$--equivariant parabolic connections on $X$ with rational parabolic weights along $D$, and establish an equivalence between the category of $G$--equivariant logarithmic connections on $\mathfrak{X}$ and the category of $G$--equivariant parabolic connections on $X$ with rational parabolic weights along $D$. |
| title | Equivariant Parabolic connections and stack of roots |
| topic | Algebraic Geometry 14D23, 14H60, 53B15, 53C05, 14A21 |
| url | https://arxiv.org/abs/2405.20699 |