Symplectic singularities arising from algebras of symmetric tensors
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912590837317632 |
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| author | Fu, Baohua Liu, Jie |
| author_facet | Fu, Baohua Liu, Jie |
| contents | The algebra of symmetric tensors $S(X):= H^0(X, \sf{S}^{\bullet} T_X)$ of a projective manifold $X$ leads to a natural dominant affinization morphism $$
φ_X: T^*X \longrightarrow \mathcal{Z}_X:= \text{Spec} S(X). $$ It is shown that $φ_X$ is birational if and only if $T_X$ is big. We prove that if $φ_X$ is birational, then $\mathcal{Z}_X$ is a symplectic variety endowed with the Schouten--Nijenhuis bracket if and only if $\mathbb{P} T_X$ is of Fano type, which is the case for smooth projective toric varieties, smooth horospherical varieties with small boundary and the quintic del Pezzo threefold. These give examples of a distinguished class of conical symplectic varieties, which we call symplectic orbifold cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symplectic singularities arising from algebras of symmetric tensors Fu, Baohua Liu, Jie Algebraic Geometry 14B05, 14J42, 14J45, 14M17, 14M25 The algebra of symmetric tensors $S(X):= H^0(X, \sf{S}^{\bullet} T_X)$ of a projective manifold $X$ leads to a natural dominant affinization morphism $$ φ_X: T^*X \longrightarrow \mathcal{Z}_X:= \text{Spec} S(X). $$ It is shown that $φ_X$ is birational if and only if $T_X$ is big. We prove that if $φ_X$ is birational, then $\mathcal{Z}_X$ is a symplectic variety endowed with the Schouten--Nijenhuis bracket if and only if $\mathbb{P} T_X$ is of Fano type, which is the case for smooth projective toric varieties, smooth horospherical varieties with small boundary and the quintic del Pezzo threefold. These give examples of a distinguished class of conical symplectic varieties, which we call symplectic orbifold cones. |
| title | Symplectic singularities arising from algebras of symmetric tensors |
| topic | Algebraic Geometry 14B05, 14J42, 14J45, 14M17, 14M25 |
| url | https://arxiv.org/abs/2409.07264 |