Moduli of Azumaya algebras, vector bundles and applications

Azumaya 代数模、向量丛和应用

• 批准号: 0245203
• 负责人: Aise de Jong
• 金额: $ 29.42万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245203&HistoricalAwards=false
• 关键词: Moduli; Azumaya; algebras; vector; bundles

This project is devoted to the study of moduli spaces of Azumaya algebrasover surfaces. As a first step we construct compactifications. Ageneralized Azumaya algebra is a perfect object in the derivedcategory of the surface, endowed with a multiplication. It turns outthat these objects can be used to give completely canonicalcompactifications. There is also a natural way to define stability ofgeneralized Azumaya algebras (depending on some auxiliarychoices). The result is what we would like to call a GIT stackcompactifying the moduli space of Azumaya algebras. The projectproposes to study these spaces and to use them to define ``Donaldsontype invariants''. In addition the geometry of the moduli spaces willbe studied for particular types of surfaces, e.g., elliptic surfacesand K3 surfaces. A complex projective surface can be viewed as a 4 dimensional spacewhich is endowed with a lot of additional structure. The mostimportant of these is a choice of a rotation map on the tangentspaces; it is a rotation over 90 degrees. A lot of research has beendone to classify four dimensional spaces which are endowed with such astructure. This is usually done by defining invariants (for examplenumbers) of complex projective surfaces which can be used to tell themapart. A very basic example are the Betti numbers, which aredimensions of cohomology groups. To give you an idea, an element ofthe second cohomology group corresponds to a 2 dimensional subspace ofthe 4-fold. Of course we are not simply enumerating these; we use acoarser equivalence relation (deformation equivalence). Here is a question: How many of these 2 dimensional subspaces havethe property that the tangent space at any point is preserved by therotation that defines the complex structure on our 4-fold? Such asubspace is called a complex curve on the complex surface. Thisquestion has been much studied, and is related to the Hodgeconjecture. However, in this project we go the other way. Namely, welook at other objects: Complex projective bundles over our 4-folddetermine a degree 2 cohomology class as well, and they are typicallynot those which can be represented by complex curves. It turns outthat by looking at all possible complex projective bundlesrepresenting the given cohomology class we get a new space which, ifwe can understand it, tells us a lot about the original 4-fold. Allkinds of new invariants of the original complex projective surface canbe defined in terms of these moduli spaces. It is the geometry ofthese moduli spaces that will be studied in this project. There is alot of techincal machinery that has to be developed before we canbegin the exploration of more geometrical properties and part of theproject will be devoted to developing this machinery.

本课题致力于研究Azumaya代数曲面上的模空间。作为第一步，我们构造紧凑化。广义Azumaya代数是曲面可导范畴中的一个完美对象，具有乘法性质。事实证明，这些对象可以用来给出完全规范的紧凑化。定义广义Azumaya代数的稳定性也有一种自然的方法(取决于某些辅助选择)。其结果就是我们所说的压缩Azumaya代数的模空间的Git堆栈。该项目建议研究这些空间，并用它们来定义‘’Donaldson类型不变量‘’。此外，还将研究特定类型的曲面的模空间的几何，例如椭圆曲面和K3曲面。一个复杂的射影曲面可以看作是一个具有许多附加结构的四维空间。其中最重要的是选择切线空间上的旋转贴图；这是超过90度的旋转。对于具有这种结构的四维空间，已经有很多研究对其进行分类。这通常是通过定义复杂射影曲面的不变量(例如枚举数)来实现的，这些不变量可以用来区分它们。一个非常基本的例子是Betti数，它是上同调群的维度。为了给你一个概念，第二上同调群的一个元素对应于四重的二维子空间。当然，我们并不是简单地列举这些；我们使用的是更复杂的等价关系(形变等价)。这里有一个问题：这些二维子空间中有多少具有这样的性质，即通过定义我们四重空间上的复杂结构的旋转，在任何一点上的切线空间是保持的？这样的子空间称为复杂曲面上的复杂曲线。这个问题已经得到了很多研究，并且与霍奇斯猜想有关。然而，在这个项目中，我们走的是另一条路。也就是说，我们来看看其他对象：我们的4重上射丛也决定了一个2度上同调类，它们通常不是那些可以用复杂曲线表示的。结果表明，通过查看表示给定上同调类的所有可能的复射射丛，我们得到了一个新的空间，如果我们能够理解它，它可以告诉我们许多关于原来的4倍的信息。利用这些模空间可以定义原复射影曲面的各种新的不变量。本课题将研究的就是这些模空间的几何。在我们可以开始探索更多的几何性质之前，必须开发许多技术机械，该项目的一部分将致力于开发这种机械。