Symplectic singularities arising from algebras of symmetric tensors

2024/09/11 by Baohua Fu, Jie Liu, Fu, Baohua +1
Engineering · Mathematics · Physics and Astronomy · #14B05 #14J42 #14J45 #14M17 #14M25 #Algebraic Geometry (math.AG) #Elasticity and Material Modeling #FOS: Mathematics #Nonlinear Waves and Solitons #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2409.07264

openalex publication_date 2024/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The algebra of symmetric tensors S(X):= H⁰(X, \sfS^\bullet T_X) of a projective manifold X leads to a natural dominant affinization morphism φ_X: T^*X \longrightarrow Z_X:= 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 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.

Related

• The algebra of symmetric tensors on smooth projective varieties
• Symmetric tensors on the intersection of two quadrics and Lagrangian fibration
• Symmetries and vanishing theorems for symplectic varieties
• Footnotes to the birational geometry of primitive symplectic varieties
• The affine closure of cotangent bundles of horospherical spaces