2025/03/20 by Yoshinori Namikawa, Namikawa, Yoshinori, Yuji Odaka +1 · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2503.15791
openalex publication_date 2025/03/20 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of (ℂ^*)r, which is canonical. Under mild assumptions, we actually prove such singularity germ is the cone vertex over a contact orbifold with weak Kähler-Einstein metric, forcing r=1. In particular, it admits a (canonical) good ℂ^*-action, which also extends to (canonical) actions of ℍ^*⊃ SU(2). These settle Kaledin's conjecture conditionally but in a substantially stronger form by establishing the canonicity, the extensibility of the action, for instance. Our key idea is to use the Donaldson-Sun theory on local Kähler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. For general symplectic singularities, we prove the same assertions, assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)Kähler metrics. Conversely, our results can also be used to study the local behavior of such metrics around the germ.