Symplectic singularities arising from algebras of symmetric tensors

Abstract

The algebra of symmetric tensors $S(X):=H^0(X,{\mathsf{S}}^{\bullet }{\mathsf{T}}_{\mathsf{X}})$ of a projective manifold $X$ leads to a natural dominant affinization morphism $${\phi }_X:T^{\ast }X⟶{\mathcal{Z}}_X:=\text{Spec}S(X).$$ It is shown that ${\phi }_X$ is birational if and only if $T_X$ is big. We prove that if ${\phi }_X$ 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.

39 pages. Any comments are welcome; Final version, to appear in Liouville's journal