标题： d流形和导出微分几何简介 显示英文标题

标题： An introduction to d-manifolds and derived differential geometry

摘要： 这是一个作者著作“D-流形和D-轨道: 一个导出微分几何理论”的综述，可在http://people.maths.ox.ac.uk/~joyce/dmanifolds.html获取。 我们引入了一个2-范畴dMan的“d-流形”，一种新的几何对象，它们是“导出”光滑流形，在Toen和Lurie的“导出代数几何”意义上。 它们是Spivak的“导出流形”的2-范畴截断（参见arXiv:0810.5174, arXiv:1212.1153）。 流形范畴Man作为全子范畴嵌入到dMan中。 我们还定义了带有边界和角的“d-流形”的2-范畴dMan^b,dMan^c，以及这些的轨道版本dOrb,dOrb^b,dOrb^c，“d-轨道”。 为简洁起见，本综述主要集中在没有边界的d-流形上。 在arXiv:1208.4948中给出了该书更长和更详细的摘要。 微分几何的大部分内容非常漂亮地扩展到了d-流形和d-轨道上——浸入、满射、子流形、横截纤维积、定向等。 紧致的定向d-流形和d-轨道具有虚拟类。 从本质上讲，几乎所有在微分几何或复代数几何中的枚举不变量问题中使用的模空间几何结构都有截断函子到d-流形和d-轨道，包括 Banach向量丛上的Fredholm截面，Fukaya、Oh、Ohta和Ono的“Kuranishi空间”以及Hofer、Wysocki和Zehnder在辛几何中的“多面体”，以及代数几何中的C-概形和完美障碍理论。 因此，文献中的结果表明，许多重要的模空间类别是d-流形或d-轨道，包括辛几何中J-全纯曲线的模空间。 d-流形和d-轨道将在辛几何和其他领域中有所应用。 显示英文摘要

摘要： This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with corners", and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds". For brevity, this survey concentrates mostly on d-manifolds without boundary. A longer and more detailed summary of the book is given in arXiv:1208.4948. Much of differential geometry extends very nicely to d-manifolds and d-orbifolds -- immersions, submersions, submanifolds, transverse fibre products, orientations, etc. Compact oriented d-manifolds and d-orbifolds have virtual classes. There are truncation functors to d-manifolds and d-orbifolds from essentially every geometric structures on moduli spaces used in enumerative invariant problems in differential geometry or complex algebraic geometry, including Fredholm sections of Banach vector bundles over Banach manifolds, the "Kuranishi spaces" of Fukaya, Oh, Ohta and Ono and the "polyfolds" of Hofer, Wysocki and Zehnder in symplectic geometry, and C-schemes with perfect obstruction theories in algebraic geometry. Thus, results in the literature imply that many important classes of moduli spaces are d-manifolds or d-orbifolds, including moduli spaces of J-holomorphic curves in symplectic geometry. D-manifolds and d-orbifolds will have applications in symplectic geometry, and elsewhere.