Moduli of Azumaya algebras, vector bundles and applications

该项目致力于研究Azumaya代数在曲面上的模空间。作为第一步，我们构造紧化。 广义Azumaya代数是曲面的导出范畴中的完美对象，具有乘法性质。事实证明，这些对象可以用来给出完全正则紧化。也有一个自然的方法来定义广义Azumaya代数的稳定性（取决于一些特定的选择）。结果是我们想称之为一个GIT栈紧化Azumaya代数的模空间。该项目建议研究这些空间，并使用它们来定义“唐纳森型不变量”。此外，模空间的几何将研究特定类型的表面，例如，椭圆曲面和K3曲面。一个复射影曲面可以看作是一个四维空间，它被赋予了许多附加结构。其中最重要的是选择切线空间上的旋转贴图;这是一个超过90度的旋转。大量的研究已经被用来对具有这种结构的四维空间进行分类。这通常是通过定义复杂的射影曲面的不变量（例如数字）来完成的，这些不变量可以用来区分它们。一个非常基本的例子是贝蒂数，它是上同调群的维数。给你一个概念，第二上同调群的元素对应于4重的二维子空间。当然，我们不是简单地列举这些;我们使用acarser等价关系（变形等价）。 这里有一个问题：这些二维子空间中有多少个具有这样的性质，即任意点处的切空间被定义了我们的4重上的复结构的旋转所保持？这样的子空间称为复曲面上的复曲线.这个问题已经被研究了很多，并且与Hodge猜想有关。然而，在这个项目中，我们走另一条路。也就是说，我们看看其他对象：我们的4-fold上的复投射丛也确定了一个2度上同调类，它们通常不是那些可以用复曲线表示的。事实证明，通过查看所有可能的复射影空间，表示给定的上同调类，我们得到一个新的空间，如果我们能理解它，告诉我们很多关于原始的4重。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。这是几何的这些模空间，将在这个项目中进行研究。在我们开始探索更多的几何性质之前，还有很多技术机械需要开发，项目的一部分将致力于开发这种机械。

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.