论文标题
k ksemistable fano品种的高度上的尖锐界限
Sharp bounds on the height of K-semistable Fano varieties I, the toric case
论文作者
论文摘要
受藤田（Fujita）的代数几何结果的启发，即复杂的投影空间在所有可吻合的复杂的Fano品种中具有最大程度的程度，我们推测，当x是x是x上最大的k kemeSissiS ksemistrized Metrized Metrized Metrized Metrized arithmetize arithmetic fano品种x，这是X型的投射空间，而不是全体会，而含有fibubini st. fubibini stricy Metric Metric Metric。当n小于或等于6时，我们的主要结果确定了感谢您的福特·范诺品种的规范积分模型（较高维度扩展到该程度的猜想“间隙假设”）。翻译成感谢您的kähler几何形状，这种结果在唐纳森引入的复曲面上产生了一个尖锐的下限，该结合被定义为曲叶横chi功能的最小值。此外，我们将猜想重新制定为Odaka模块化高度的最佳下限。在任何维度n中，都显示了如何通过X的程度来控制Kähler-Einstein Model X的高度。在本文的续集中，我们的高度猜想是通过利用一个更一般的Goolegarithmic设置来为任何投射的对角线fano hypersurface建立。
Inspired by Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety X of relative dimension n is maximal when X is the projective space over the integers, endowed with the Fubini-Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when n is less than or equal to 6 (the extension to higher dimensions is conditioned on a conjectural "gap hypothesis" for the degree). Translated into toric Kähler geometry this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. We furthermore reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension n it is shown how to control the height of the canonical toric model X, with respect to the Kähler-Einstein metric, by the degree of X. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.
