Abstract We ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.

中文翻译：

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半齐次向量丛的连续CM-正则性

摘要 我们询问射影簇 X 上向量丛的 CM (Castelnuovo-Mumford) 正则性何时是数值的，并解决 X 是阿贝尔簇的情况。我们证明了阿贝尔簇 X 上半齐次向量丛的连续 CM 正则性是陈数据的分段常数函数，我们还使用泛型消失理论来获得任何向量的连续 CM 正则性的尖锐上限束在 X 上。从这些结果我们得出结论，许多半齐次丛的连续 CM 正则性——包括许多当 X 是雅可比矩阵时的 Verlinde 丛——既是数值型的，也是极值的。