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Weak positivity of Schur forms under decomposable curvature

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath.CV — Complex Variablesmath.DG — Differential Geometryquant-ph — Quantum Physics

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant Φn(H)\Phi_n(H) of a positive linear map. Du's rank-nine counterexample shows that Φn(H)≥0\Phi_n(H)\ge0 fails in general. We prove Φn(H)≥0\Phi_n(H)\ge0 for every decomposable map HH, a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of Φk(HV,U)\Phi_k(H_{V,U}) on every square compression is equivalent to Φn(H+S)≥0\Phi_n(H+S)\ge0 for every entanglement-breaking map SS. This class contains explicit indecomposable maps, but is nonconvex in every size n≥9n\ge9. We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant GPT-6 Astra (Pro). The choice of the questions worth answering was fully made by the author, as was the selection of the subset of results worth retaining. This choice of material evolved over many months, including up to the recent disproof of Φ_9 ≥ 0. None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The writing, polishing, and presentation were done with GPT-6 as well as Claude's (Fable 5.1 and Opus 5.5) assistance. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT-6 Astra (Pro)
Anthropic
Claude Fable

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Version history

  1. v1 · 37 pagesSubmitted by Suvrit SraInitial depositCurrentOct 31, 2026