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Seshadri constants for vector bundles
Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jpaa.2020.106559
Mihai Fulger , Takumi Murayama

We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose tangent bundle has positive Seshadri constant at a point is isomorphic to a projective space. We conjecture that the Fano condition can be removed. Among other results in this direction, we prove the conjecture for surfaces. Second, we restate a classical conjecture on the nef cone of self-products of curves in terms of semistability of higher conormal sheaves, which we use to identify new nef classes on self-products of curves. Third, we prove that our Seshadri constants can be used to control separation of jets for direct images of pluricanonical bundles, in the spirit of a relative Fujita-type conjecture of Popa and Schnell.

中文翻译:

向量丛的 Seshadri 常数

我们在相对设置中为线束引入 Seshadri 常数。他们推广了射影簇上线丛的经典 Seshadri 常数及其对由 Beltrametti-Schneider-Sommese 和 Hacon 研究的向量丛的扩展。与经典理论有相似之处。特别是,我们给出了一个 Seshadri 类型的充足性标准,我们将 Seshadri 常数与射流分离和渐近基位点联系起来。我们给出了新版 Seshadri 常数的三个应用。首先,Mori 的一个著名结果可以重申为任何 Fano 流形,其切丛在一点上具有正的 Seshadri 常数,与射影空间同构。我们推测可以去除 Fano 条件。在这个方向的其他结果中,我们证明了曲面的猜想。第二,我们根据高同常滑轮的半稳定性重申了关于曲线自积的 nef 锥的经典猜想,我们用它来识别曲线自积的新 nef 类。第三,本着 Popa 和 Schnell 的相对藤田类型猜想的精神,我们证明了我们的 Seshadri 常数可用于控制喷流的分离,以获得多波束的直接图像。
更新日期:2021-04-01
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