Varieties with Free Tangent Sheaf

每个几何对象都带有附着到对象的点的切线空间。这些应该被认为是所讨论的几何对象在给定点附近的最佳可能的线性近似。所有这些切线空间的集合是另一个几何对象，即所谓的切丛。在代数几何中，对象是用多项式方程定义的，称为射影方案，允许奇点是很自然的；然后人们谈论切线束。切层的性质对于理解射影格式的结构是非常重要的。在这个研究项目中，我们研究了切层尽可能简单的射影方案的几何和算法，即自由层，或满足相关条件的射影方案。这一点在复数上是很好理解的，其中条件是所谓的阿贝尔品种的特征。然而，与之形成鲜明对比的是，到目前为止，这些计划在具有积极特征的地面上的结构仍然是一个令人震惊的谜团。我们试图解开其中的一些谜团。

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.