发表机构
University of Cyprus; Imperial College London（塞浦路斯大学; 帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明四元数维数至 14 的正四元数 Kähler 流形满足 LeBrun--Salamon 猜想，通过分析等距群秩和特征数正性，结合多种曲率矩与 Hodge--Riemann 正性，逐步扩展证明至 n=14。
AI 中文摘要
我们证明了四元数维数 n ≤ 14 的正四元数 Kähler 流形的 LeBrun--Salamon 猜想。为此，我们首先证明任何最小反例 M 的等距群 Isom(M) 的秩至多为 1，因此其维数至多为 3。另一方面，我们考虑一个特征数，当 n 为偶数时它等于 dim Isom(M) - (n+3)，当 n 为奇数时等于 dim Isom(M) - (n+1)，并证明当 n ≤ 14 时该指标是若干正量之和，从而得出矛盾。用于证明正性的工具集分阶段扩大。投影曲率矩足以处理 n ≤ 8；半正定曲率矩将论证扩展到 n ≤ 10；带符号的标量轨道矩提供了 n=11,12 所需的新不等式。单独的标量轨道矩在 n=13 时失效；将其与上同调类平方的 Hodge--Riemann 正性相结合，可将证明扩展到 n=13,14。我们给出了精确的分离障碍，以解释方法的变化。
英文摘要
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.
Comments40 pages: 27 pages plus appendices, including a discussion of Lean partial verification available online; comments very welcome
