Symplectic singularities arising from algebras of symmetric tensors

TL;DR

This paper explores the conditions under which the algebra of symmetric tensors on a projective manifold leads to symplectic singularities, linking geometric properties of the manifold to symplectic structures on associated varieties.

Contribution

It establishes a characterization of when the affinization of the cotangent bundle is birational and symplectic, introducing symplectic orbifold cones as a new class of conical symplectic varieties.

Findings

finization morphism is birational iff the tangent bundle is big.

 the morphism is birational, the associated variety is symplectic if nd only if ertain geometric conditions hold.

Examples include toric, horospherical varieties, and the quintic del Pezzo threefold.

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.