Tom Gannon

math.AG — Algebraic Geometry

Terminal Symplectic Singularities for Parabolic Base Affine Spaces

Contributed by Tom Gannon

Let $G$ be a simply connected semisimple complex algebraic group, and let $U_P$ be the unipotent radical of a parabolic subgroup $P\subseteq G$. We prove that the affinization $\overline{T^*(G/U_P)}$ is factorial and has terminal symplectic singularities. To establish symplectic singularities, we apply a criterion of Fu--Liu after constructing a log Fano boundary on an associated projective bundle over the full flag variety. The boundary construction uses Frobenius splitting of flag varieties and reduction modulo primes to obtain log canonicity in characteristic zero. We then prove that every first-order Poisson deformation of $\overline{T^*(G/U_P)}$ is trivial. Namikawa's deformation theory implies that the singular locus has codimension at least four, yielding terminality. No conical grading is assumed.