Derived Moduli Spaces and Applications

这位研究人员将与他的同事们合作，发展派生变形理论的形式主义。这一理论旨在系统地解决由于代数几何中与高维变体相关的模空间通常是奇异的这一事实而产生的许多困难。其基本思想是，模问题的正确目标是某种“导出模空间”，它在适当的意义上应该是明显光滑的，并且应该带有一个差分分次结构束。通常的模空间是由导出的零次截断形式得到的，这解释了它的奇异性。在最近的工作中，Ciocan-Fontanine和Kapranov定义和研究了以差分分次方案为对象的“右派生方案范畴”，并构造了Grothendieck的Quot方案的导出版本。研究人员将把这种形式推广到更大类别的微分分次堆，并利用它来构造代数几何中其他一些重要的模空间的派生形式(如向量丛的模、稳定映射等)。作为理论的首次应用，提出了Behrend-Fantechi和Li-Tian的虚拟基础类的一个更简单和更一般的构造。这种新结构有望更好地研究Fano流形中高亏格到超曲面的稳定映射的模的情况下虚拟基类的性质，并更好地理解高亏格上的数学对称.该项目还涉及旗形的亏格为零的Gromov-Witten不变量的一些显式计算，以及在镜像对称中的应用。这是代数几何领域的研究，这是现代数学中最古老的分支之一。近年来，代数几何的方法和思想，特别是模空间的研究，被用于弦理论，这是理论物理中非常活跃的部分。弦理论的发展引发了这两个研究群体之间卓有成效的互动，并导致了许多惊人的新现象的发现和研究。镜像对称性就是一个例子。人们期望，对代数几何中模空间的更好理解将导致弦理论的更多应用。

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.