发表机构
University of California, Riverside（加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了各类几乎单复代数群的Helly数，将其推广到一般约化群，并应用于给出环面主G-丛等变分裂的光滑环面簇的扇形判据，明确了射影空间上该分裂性质成立的条件。
AI 中文摘要
对于半单秩为m的几乎单复代数群G，我们证明：当有限个抛物子群的每个至多m+2个成员的子族都包含公共极大环面时，该有限子族必含公共极大环面。我们确定了A、B、C、G₂、F₄、E₇、E₈型几乎单群对应的Helly数h(G)，并得到D型和E₆型的Helly数相差1的界。对于正半单秩的一般约化群，其Helly数为各几乎单因子Helly数的最大值。我们将这些结果应用于建立一个扇形理论判据，用于刻画所有环面主G-丛等变分裂的光滑环面簇。特别地，当d≥2时，该分裂性质在射影空间ℙᵈ上成立当且仅当d≥h(G)。
英文摘要
For an almost simple complex algebraic group $G$ of semisimple rank $m$, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most $m+2$ members does. We determine the resulting Helly number $(h(G))$ for almost simple groups of types $A$, $B$, $C$, $G_2$, $F_4$, $E_7$, and $E_8$, and obtain bounds differing by one in types $D$ and $E_6$. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal $G$-bundle splits equivariantly. In particular, for $d \ge 2$, this splitting property holds on $\mathbb{P}^d$ if and only if $d \ge h(G)$
