The Lebrun--Salamon conjecture is true in dimension up to 56

Abstract

We prove the LeBrun--Salamon conjecture for positive quaternionic Kähler manifolds of quaternionic dimension n <= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which equals dim Isom(M) - (n+3) if n is even and dim Isom(M) - (n+1) if n is odd, and show that this index is a sum of positive quantities when n <= 14, yielding a contradiction. The toolset used to show positivity enlarges in stages. Projection curvature moments suffice through n=8; positive-semidefinite curvature moments extend the argument through n=10; and signed scalar orbital moments give the new inequalities needed for n=11,12. Scalar orbital moments alone fail at n=13; combining them with Hodge--Riemann positivity for squares of cohomology classes extends the proof to n=13,14. We give exact separating obstructions explaining the changes of method.

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BibTeX

@misc{serrano2026lebrunsalamon,
  title = {{The Lebrun--Salamon conjecture is true in dimension up to 56}},
  author = {Jordi Daura Serrano and Aron Gohr and Marie-Amélie Lawn and Travis Schedler},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06336},
  eprint = {2610.06336},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06336},
}

RIS

TY  - UNPB
TI  - The Lebrun--Salamon conjecture is true in dimension up to 56
AU  - Jordi Daura Serrano
AU  - Aron Gohr
AU  - Marie-Amélie Lawn
AU  - Travis Schedler
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06336
UR  - https://arxiv.org/abs/2610.06336
AB  - We prove the LeBrun--Salamon conjecture for positive quaternionic Kähler manifolds of quaternionic dimension n <= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which equals dim Isom(M) - (n+3) if n is even and dim Isom(M) - (n+1) if n is odd, and show that this index is a sum of positive quantities when n <= 14, yielding a contradiction. The toolset used to show positivity enlarges in stages. Projection curvature moments suffice through n=8; positive-semidefinite curvature moments extend the argument through n=10; and signed scalar orbital moments give the new inequalities needed for n=11,12. Scalar orbital moments alone fail at n=13; combining them with Hodge--Riemann positivity for squares of cohomology classes extends the proof to n=13,14. We give exact separating obstructions explaining the changes of method.
ER  -

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