Varieties with Free Tangent Sheaf

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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.