Varieties with Free Tangent Sheaf

具有自由切线束的品种

• 批准号: 536205323
• 负责人: Professor Dr. Stefan Schröer
• 金额: --
• 依托单位: Lehrstuhl für Algebraische Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/536205323?language=en
• 关键词: Varieties; Free; Tangent; Sheaf

Each geometric object comes with tangent spaces, attached to the points of the object. These should be thought of the best possible linear approximation for the geometric object in question, near a given point. The collection of all such tangent spaces is another geometric object, the so-called tangent bundle. In algebraic geometry, where the objects are defined in terms of polynomial equations and called projective schemes, it is natural to allow singularities; one then speaks about the tangent sheaf. Properties of the tangent sheaf are of fundamental importance in understanding the structure of projective schemes. In this research project we study the geometry and arithmetic of projective schemes where the tangent sheaf is as simply as it could be, namely a free sheaf, or satisfies related conditions. This is well-understood over the complex numbers, where the conditions characterize the so-called abelian varieties. In marked contrast, however, the structure of these schemes over ground fields of positive characteristics remains a startling mystery to date. We seek to unravel some of these mysteries.

每个几何对象都带有切线空间，附加到对象的点。这些应该被认为是给定点附近所讨论的几何对象的最佳可能线性近似。所有这些切空间的集合就是另一个几何对象，即所谓的切丛。在代数几何中，对象是用多项式方程定义的，称为射影格式，因此很自然地允许奇点；然后我们谈到切线束。切束的性质对于理解射影格式的结构至关重要。在这个研究项目中，我们研究射影格式的几何和算术，其中切束尽可能简单，即自由束，或满足相关条件。这在复数上是很好理解的，复数的条件表征了所谓的阿贝尔簇。然而，与此形成鲜明对比的是，迄今为止，这些具有积极特征的地面领域的方案结构仍然是一个令人震惊的谜团。我们试图解开其中一些谜团。