论文标题
frobenius射影和阳性的阳性几何形状
Frobenius projective and affine geometry of varieties in positive characteristic
论文作者
论文摘要
本文的目的是为阳性特征的高维品种奠定基础。该理论涉及frobenius-progentive和frobenius-frobine结构，这些结构以前仅在基础品种是曲线的情况下才进行了研究。作为扩展理论的第一步，我们证明了Frobenius-Projementive和Frobenius-affine结构和研究（阳性特征版本）的各种基本特性，从S. kobayashi和Ochiai开始，该品种承认投影或仿射结构。在本文的上半年中，我们构建了具有几种普遍的本地束类型的生物对应关系。其中之一是根据Berthelot的高级差异操作员来定义的。我们还证明了Gunning的公式的积极特征版本，该版本分别为Chern类提供了必要的条件，以分别存在Frobenius-Projective或Frobenius-frobenius-frosine结构。本文的后半部分是从Frobenius-Projective和Frobenius-Affine结构的角度研究，即某些特定类别的品种，即投影和仿射空间，Abelian品种，曲线和表面。例如，结果表明，无限水平的Frobenius-Projementive结构的存在给出了Abelian品种的顺序性的特征。此外，我们证明，任何平滑的属属$> 1 $都可以承认无限水平的frobenius-progentive结构，从而导致分层基本组的表示。
The goal of the present paper is to lay the foundations for a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. This theory deals with Frobenius-projective and Frobenius-affine structures, which have been previously investigated only in the case where the underlying varieties are curves. As the first step in expanding the theory, we prove various basic properties on Frobenius-projective and Frobenius-affine structures and study (the positive characteristic version of) the classification problem, starting with S. Kobayashi and T. Ochiai, of varieties admitting projective or affine structures. In the first half of the present paper, we construct bijective correspondences with several types of generalized indigenous bundles; one of them is defined in terms of Berthelot's higher-level differential operators. We also prove the positive characteristic version of Gunning's formulas, which give necessary conditions on Chern classes for the existence of Frobenius-projective or Frobenius-affine structures respectively. The second half of the present paper is devoted to studying, from the viewpoint of Frobenius-projective and Frobenius-affine structures, some specific classes of varieties, i.e., projective and affine spaces, abelian varieties, curves, and surfaces. For example, it is shown that the existence of Frobenius-projective structures of infinite level gives a characterization of ordinariness for abelian varieties. Also, we prove that any smooth projective curve of genus $>1$ admits a Frobenius-projective structure of infinite level, which induces a representation of the stratified fundamental group.
