光滑环面Fano簇的切丛的斜率稳定性

中文摘要 AI 辅助

我们通过精确计算Klyachko准则，对三维至六维的所有8630个光滑环面Fano簇的切丛的反典范斜率稳定性进行了分类，并确定了所有严格半稳定情形下的多稳定性（等价于关于反典范凯勒形式的厄米-爱因斯坦度量的存在性）。该普查中的每个凯勒-爱因斯坦簇都具有多稳定切丛，但其逆命题却不成立：在109个具有稳定切丛的五维簇中，有102个并非凯勒-爱因斯坦簇。该普查筛选出了一种具有高皮卡秩的构造，我们将其一般化引入：以射影直线的乘积为底、由A₂的根参数化的根扭环面dP-纤维丛。每个非空的、由非零根扭构成且和为零的多重集都会在任意维度上生成一个稳定切丛。在这些零和扭中，当且仅当该多重集在根六边形的取反或三阶旋转下不变时，所得簇为凯勒-爱因斯坦簇。因此，对于每个n≥4，都存在皮卡秩为n+2的稳定环面Fano簇；根扭和为零并不强制其具有凯勒-爱因斯坦性质，而另一组不平衡的族则表明，和为零并非稳定性的必要条件。对于根扭族，斜率不等式可简化为A₂矩六边形上的积分；正层分解和精确的一维卷积估计确立了稳定性，而六边形在极端指数处的一阶矩的严格排序则给出了凯勒-爱因斯坦分类。

英文摘要

We construct smooth toric Fano $n$-folds of Picard number $n+2$ whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every $n\ge4$. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of $A_2$. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all $8{,}630$ varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.

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26 pages