Symplectic singularities arising from algebras of symmetric tensors

Baohua Fu, Jie Liu
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Abstract

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 $$ \varphi_X: T^*X \longrightarrow \mathcal{Z}_X:= \text{Spec} S(X). $$ It is shown that $\varphi_X$ is birational if and only if $T_X$ is big. We prove that if $\varphi_X$ 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.
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由对称张量代数引起的交映奇点
投影流形$X$的对称张量代数$S(X):= H^0(X, \sf{S}^{\bullet} T_X)$ 导致了一个自然的主导蔼化态量$ \varphi_X: T^*X \longrightarrow \mathcal{Z}_X:= \text{Spec} S(X)。$$ 当且仅当 $T_X$ 是大的时候,$\varphi_X$ 是双向的。我们证明,如果$\varphi_X$是双向的,那么当且仅当$\mathbb{P}T_X$是法诺类型时,$\mathcal{Z}_X$是一个具有Schouten--Nijenhuis括弧的交映体。这些给出了一类杰出的锥形交映变体的例子,我们称之为交映轨道锥。
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