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Stability conditions on twisted derived categories I: Existence and full numerical support

Primarily human-written textHuman understanding: all partsmath.AG — Algebraic Geometry

Contributed by Zhirui Li ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We prove that for every nonempty smooth projective variety XX over an arbitrary field and every Brauer class α\alpha, the twisted derived category Db(X,α)\mathrm{D}^b(X,\alpha) admits a Bridgeland stability condition with full numerical support. Based on Chunyi Li's construction and its extension to arbitrary fields, we establish uniform phase control under tensoring by an Azumaya algebra and calibrate the resulting slicing to the induced central charge. We further show that the constructed stability condition is geometric and satisfies a mass-Hom bound, which persists throughout its connected component.

Author comments: 26 pages. Comments and suggestions are welcome.

Provenance statement

The initial construction of stability conditions on twisted derived categories was proposed by ChatGPT 6 Astra. After independently examining the construction and its proofs, the author further asked the AI about the resulting stability conditions satisfied additional properties, such as geometricity and mass--Hom bound. The AI subsequently provided arguments for geometricity, the mass–Hom bound, and the moduli-space application. ChatGPT 6 Astra and Gemini 4 Argon were also used to assist with mathematical verification and exposition. The author independently examined and verified all mathematical statements and proofs, rewrote almost all AI-generated proofs, and takes full responsibility for the mathematical content and any remaining errors. A detailed AI disclosure is included in the manuscript.

Tools used

OpenAI
ChatGPTVersion 6 Astra
Google
GeminiVersion 4 Argon

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Version history

  1. v1 · 26 pagesSubmitted by Zhirui Li26 pages. Comments and suggestions are welcome.CurrentNov 12, 2026