Derived Moduli Spaces and Applications

与他的同事合作，研究者将发展衍生变形理论的形式。这一理论的目的是解决在一个系统的方式所产生的困难的事实，即模空间代数几何relatedto高维品种通常是奇异的。其基本思想是，模问题的正确对象是某个“导出模空间”，它在适当的意义上应该是明显光滑的，并且应该带有一个微分阶化结构层。通常的模空间是从导出的版本作为度零截断得到的，这解释了它的奇异性。在最近的工作中，Ciocan-Fontanine和Kapranov定义并研究了以微分分次概型为对象的“右导出概型范畴”，并构造了Grothendieck的Quot概型的导出形式.研究者将把形式主义扩展到一个更大的微分阶化栈的范畴，并用它来构造代数几何中其他一些重要模空间的衍生版本（如向量丛的模、稳定映射等）。 作为该理论的第一个应用，本文给出了Bewed-Fantechi和Li-Tian的虚基类的一个更简单和更一般的构造。这一新的构造有望更好地适用于研究Fano流形中高亏格到超曲面的稳定映射模的虚基本类的性质，并更好地理解高亏格下的镜像对称性.该项目的另一部分是关于旗流形亏格为零的Gromov-Witten不变量的显式计算及其在镜像对称中的应用，这是现代数学最古老的分支之一--代数几何领域的研究。近年来，代数几何的方法和思想，特别是模空间的研究，已被应用于弦理论，这是理论物理中非常活跃的部分。弦理论的发展激发了两个研究群体之间富有成效的互动，并导致了许多惊人的新现象的发现和研究。镜像对称就是一个例子。人们期望，更好地理解代数几何中的模空间将导致更多的应用弦理论。

Working with his colleagues, the investigator will develop the formalism of Derived Deformation Theory. This theory aims to resolve in a systematic fashion many of the difficulties arising from the fact that moduli spaces in algebraic geometry relatedto higher dimensional varieties are typically singular. The basic idea is that the correct object to consider for a moduli problem is some ``derived moduli space'' which should be manifestly smooth in an appropriate sense and should carry a differential-graded structure sheaf. The usual moduli space is obtained from the derived version as the degree-zero truncation, and this explains its singularnature. In recent work, Ciocan-Fontanine and Kapranov defined andstudied the ``right derived category of schemes'', whose objects are differential-graded schemes, and they have constructed the derived version of Grothendieck's Quot scheme. The investigator will extend the formalism to a larger category of differential-graded stacks, and use it to construct derived versions of some other important moduli spaces in algebraic geometry (such as moduli of vector bundles, stable maps, etc.). As a first application of the theory, it is proposed to give a simpler and more general construction of the virtual fundamentalclasses of Behrend-Fantechi and Li-Tian. This new construction is expected to be better suited to investigate the properties of virtual fundamental classes in the case of moduli of stable maps of higher genus to a hypersurface in a Fano manifold and to understand mathematicallymirror symmetry at higher genus. The project has also a part dealingwith some explicit calculations of genus zero Gromov-Witten invariants of flag manifolds, and applications to mirror symmetry.This is research in the field of algebraic geometry, which is one of the oldest branches of modern mathematics. In recent years, the methods and ideas of algebraic geometry, especially the study of moduli spaces, have been employed in string theory, a very active part of theoretical physics. Developments in string theory have sparked a fruitful interaction between the two communities of researchers and have led to the discovery and study of many striking new phenomena. Mirror symmetry is one example. It is expected that a better understanding of moduli spaces in algebraic geometry will lead to more applications to string theory.