Hilbert-Samuel formula over adelic curves

去年，我证明了Siu不等式的一个相对版本，通过它，我们提供了关于函数场上的一个线性充足的线束的等分布定理的一个证明。在这之后，今年我将扩展等分布定理到线束很大的情况下。在“斜率有界性与等分布定理”一文中，我们讨论了HIlbert-Samuel公式、最小斜率有界性与等分布定理之间的相互作用。我先后证明了这些结果对任意特征为0的全局域上的任意线性大线束都成立。在项目的最后，我研究了半样本锥上的chi-volume在adelic曲线上的行为。证明了半样本半正阿德利克线束的Hilbert-Samuel公式。

Last year, I proved a relative version of Siu's inequality, by which we provide a reproof of Equidistribution theorem with respect to an adelic ample line bundle over a function field.After that work, this year I want to extend the equidistribution theorem to case that the line bundle is just big. In the paper "Slope boundedness and equidistribution theorem", we discussed the interplay between HIlbert-Samuel formulas, boundedness of minimal slopes, and equidistribution theorem. I successively proved those results holds for any adelic big line bundles over any global field of characteristic 0.In the end of the project, I studied the behavior of chi-volume on the semiample cone over an adelic curve. I proved a Hilbert-Samuel formula for a semiample and semipositive adelic line bundle.