Some recent developments in Kähler geometry and exceptional holonomy - Simon Donaldson (UK/USA)——ICM2018的一小时报告
发文时间：2018-09-28
来源：304am永利集团
Abstract: This article is a broad-brush survey of two areas in differential geometry. While these two areas are not usually put side-by-side in this way, there are several reasons for discussing them together. First, they both fit into a very general pattern, where one asks about the existence of various differential-geometric structures on a manifold. In one case we consider a complex Kähler manifold and seek a distinguished metric, for example a Kähler-Einstein metric. In the other we seek a metric of exceptional holonomy on a manifold of dimension 7 or 8 . Second, as we shall see in more detail below, there are numerous points of contact between these areas at a technical level. Third, there is a pleasant contrast between the state of development in the fields. These questions in Kähler geometry have been studied for more than half a century: there is a huge literature with many deep and far-ranging results. By contrast, the theory of manifolds of exceptional holonomy is a wide-open field: very little is known in the way of general results and the developments so far have focused on examples. In many cases these examples depend on advances in Kähler geometry
https://eta.impa.br/dl/PL020.pdf
https://arxiv.org/abs/1808.03995
本文分为Kahler几何与例外和乐群两部分。第一部分主要讨论Fano流形上Kahler-Einstein度量的存在性与各种稳定性的关系（特别给出了YTD猜想的四种不同证明）。第二部分讨论例外和乐群G_2和Spin(7)，尤其是具有这两个和乐群的流形上的规范理论(Donaldson-Thomas理论)和校准几何。
相关附件
1-PL020 Donaldson.pdf
