Finite Generation and Structure of Invariant Jets under Non-Reductive Reparametrization
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| Format: | Preprint |
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2020
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| _version_ | 1866911276690571264 |
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| author | Rahmati, Mohammad Reza |
| author_facet | Rahmati, Mohammad Reza |
| contents | We study invariant jet differentials in the framework of complex hyperbolicity, focusing on the algebra of invariants for the non--reductive reparametrization group $G_k = \mathbb{C}^{\ast} \ltimes U_k$. The paper develops a uniform, representation--theoretic, and graded--algebraic strategy for the $ρ$--action of $G_k$ on $J_k\mathbb{C}^n$, establishing in particular the finite generation of the invariant jet algebra central to the Green--Griffiths--Demailly program. Specifically, we prove that the $\mathbb{C}^{\ast}$--graded algebra of unipotent invariants $\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ is finitely generated for all $n,k$; equivalently, the fiber ring of invariant jet differentials is a finitely generated positively graded $\mathbb{C}$--algebra, so that its projective spectrum $\operatorname{Proj}\,\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ exists and coincides with the Demailly--Semple tower. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_08310 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Finite Generation and Structure of Invariant Jets under Non-Reductive Reparametrization Rahmati, Mohammad Reza Representation Theory We study invariant jet differentials in the framework of complex hyperbolicity, focusing on the algebra of invariants for the non--reductive reparametrization group $G_k = \mathbb{C}^{\ast} \ltimes U_k$. The paper develops a uniform, representation--theoretic, and graded--algebraic strategy for the $ρ$--action of $G_k$ on $J_k\mathbb{C}^n$, establishing in particular the finite generation of the invariant jet algebra central to the Green--Griffiths--Demailly program. Specifically, we prove that the $\mathbb{C}^{\ast}$--graded algebra of unipotent invariants $\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ is finitely generated for all $n,k$; equivalently, the fiber ring of invariant jet differentials is a finitely generated positively graded $\mathbb{C}$--algebra, so that its projective spectrum $\operatorname{Proj}\,\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ exists and coincides with the Demailly--Semple tower. |
| title | Finite Generation and Structure of Invariant Jets under Non-Reductive Reparametrization |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2012.08310 |