Weak positivity of Schur forms under decomposable curvature

Suvrit Sra

Research Preprints · Complex geometry

Weak positivity of Schur forms under decomposable curvature

Suvrit Sra · TU Munich, Department of Mathematics, School of CIT

Version v1 · released 8 October 2026 · 37 pages · Schur forms under decomposable curvature

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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 $\Phi_n(H)$ of a positive linear map. Du's rank-nine counterexample shows that $\Phi_n(H)\ge0$ fails in general. We prove $\Phi_n(H)\ge0$ for every decomposable map $H$, 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 $\Phi_k(H_{V,U})$ on every square compression is equivalent to $\Phi_n(H+S)\ge0$ for every entanglement-breaking map $S$. This class contains explicit indecomposable maps, but is nonconvex in every size $n\ge9$. 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.

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Version	Released	Length	File	Proof	arXiv
v1	8 October 2026	37 pp	PDF	.ots	—

Cite

@misc{sra2026qperm-v1,
  author       = {Suvrit Sra},
  title        = {{Weak positivity of Schur forms under decomposable curvature}},
  year         = {2026},
  month        = oct,
  howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qperm/v1}},
  note         = {Version v1, released 8 October 2026},
}

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