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Terminal Symplectic Singularities for Parabolic Base Affine Spaces

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.RT — Representation Theory

Contributed by Tom Gannon ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

Let GG be a simply connected semisimple complex algebraic group, and let UPU_P be the unipotent radical of a parabolic subgroup P⊆GP\subseteq G. We prove that the affinization T∗(G/UP)‾\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 T∗(G/UP)‾\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.

Provenance statement

This manuscript was developed through an extended, iterative exchange between the author and OpenAI’s ChatGPT Astra. The author posed the research problem and directed the investigation. ChatGPT proposed proof strategies, developed detailed arguments, drafted substantial portions of the manuscript, and assisted with revisions and proof audits. Its contributions therefore included a significant majority of the mathematical reasoning as well essentially all of the exposition. In particular, this pdf is entirely AI generated.

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Version history

  1. v1 · 14 pagesSubmitted by Tom GannonInitial depositCurrentNov 04, 2026