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Positivity properties of Schur classes

  • Matt Larson
  • Alan Stapledon
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Abstract

Results of Fulton-Lazarsfeld and Ross-Toma show that Schur classes of ample and nef vector bundles have remarkable positivity properties. We generalize these results to a purely algebraic setting. We show that if a projective bundle ring, a ring which resembles the cohomology ring of a projective bundle over a smooth complex projective variety, has the Kähler package with respect to a suitable cone, then the associated Schur classes satisfy a version of the Hodge-Riemann relations. This result is new even for the projectivization of an ample vector bundle over a smooth complex projective variety. We apply this result to prove that Schur coefficients of matroids are nonnegative.

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Cite

BibTeXRIS
BibTeX
@misc{larson2026positivity,
  title = {{Positivity properties of Schur classes}},
  author = {Matt Larson and Alan Stapledon},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13025},
  eprint = {2609.13025},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13025},
}
RIS
TY  - UNPB
TI  - Positivity properties of Schur classes
AU  - Matt Larson
AU  - Alan Stapledon
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.13025
UR  - https://arxiv.org/abs/2609.13025
AB  - Results of Fulton-Lazarsfeld and Ross-Toma show that Schur classes of ample and nef vector bundles have remarkable positivity properties. We generalize these results to a purely algebraic setting. We show that if a projective bundle ring, a ring which resembles the cohomology ring of a projective bundle over a smooth complex projective variety, has the Kähler package with respect to a suitable cone, then the associated Schur classes satisfy a version of the Hodge-Riemann relations. This result is new even for the projectivization of an ample vector bundle over a smooth complex projective variety. We apply this result to prove that Schur coefficients of matroids are nonnegative.
ER  -