Symplectic singularities arising from algebras of symmetric tensors

The algebra of symmetric tensors $S\left(X\right)≔H^0(X,S^{•}{T}_X)$ of a projective manifold X leads to a natural dominant affinization morphism${\phi }_X:T^{⁎}X⟶Z_X≔\mathrm{Spec}S\left(X\right).$ 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 $Z_X$ is a symplectic variety endowed with the Schouten–Nijenhuis bracket if and only if $P{T}_X$ 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.