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.

该项目致力于研究Azumaya代数表面的模量空间。作为第一步，我们构建压缩。 固定化azumaya代数是表面衍生法中的完美对象，并具有乘法。事实证明，这些对象可用于给出完全规范的串联。也有一种自然的方法来定义综合性azumaya代数的稳定性（取决于某些辅助操作）。结果就是我们想称呼Azumaya代数的模量空间的git stackCompactify。研究这些空间并使用它们来定义``donaldsontype''''''的项目植物。此外，模量空间的几何形状将研究特定类型的表面，例如椭圆表面和K3表面。复杂的投影表面可以看作是4维空间，并具有许多其他结构。其中最重要的是横切空间上旋转图的选择。这是90度以上的旋转。大量研究已经对四个维空间进行了分类，这些空间具有诸如Ascructure。这通常是通过定义复杂的投影表面的不变（用于审查的）来完成的，这些表面可用于告诉themapart。一个非常基本的例子是贝蒂数字，这些数字是共同学组的范围。为了给您一个想法，第二个同胞组的元素对应于4倍的2维子空间。当然，我们不仅仅是列举这些内容。我们使用较大的等价关系（变形等效）。 这是一个问题：这两个维度子空间中有多少个具有属性，这些属性在任何点都通过在我们4倍上定义复杂结构的方法保留了任何点？如ubspace称为复杂表面上的复杂曲线。研究问题已经进行了大量研究，并且与hodgeconjecture有关。但是，在这个项目中，我们会以其他方式进行。也就是说，对其他物体进行欢迎：在我们的4倍第二个学位的同时组中，复杂的投影束也是如此，它们通常不是可以用复杂曲线代表的那些。通过查看所有可能的复杂的投影捆绑表述，我们将获得一个新的空间，如果我们能理解，它会告诉我们很多有关原始4倍的信息。原始复杂射击表面的新不变式的杂物可以根据这些模量空间定义。该项目将研究这些模量空间的几何形状。在我们可以探索更多几何特性之前，必须开发大量的技术机械，并且该项目的一部分将专门用于开发这种机械。