Moduli spaces classifying bundles over Azumaya algebras-the case of algebraic surfaces.

本课题的目的是研究代数变量上向量束的模空间。代数曲面的情况在这里特别重要，因为一维的情况，即代数曲线，是相当容易理解的，至少与一般情况相比是这样。因此，在二维的情况下，也就是代数曲面应该是典型的附加困难。在相当一般的情况下，所讨论的模格式的存在可以保证多年。不幸的是，更具体和详细的理解是非常有限的。因此，考虑额外的、但自然的属性是很有意义的，这会使问题更容易处理。在本例中，我们考虑Azumaya代数对相干轴的附加作用。向量包。这些是结合代数的集合，它们一般是基变量函数域上的中心简单代数。所考虑的相干束被假定为一般茎上的简单模块，而一般茎是中心简单代数。满足这些条件的第一种情况仍然是严格可交换的。代数的束就是基簇的结构束，相干束是基簇上的可逆模束。出现的模问题是完全经典的，并导致皮卡德变分。他们的理论比一般向量束的理论简单得多。因此，在这个项目中想要研究的案例，应该在这个背景下考虑。这些模变体的一般理论实际上比一般束的情况更直接，由包括N.Hoffmann和本建议的作者在内的几个人完成。例如，在应用几何不变量理论时，没有必要指定一个充足的除数。同时，对模格式进行了自动调整。紧凑。因此，也有一些希望，这些模块化品种的更精细的性质，超越纯粹的存在是更好地理解。当然，要取得一些成功，重要的是要有大量易于理解的例子，以便能够测试更一般的猜想。在理解模量问题的过程中，必须在一些特殊的假设下研究一些有趣的变形理论问题。这个项目更技术性的部分是来这里更好地理解并在更一般的设置中找到类似的结果。

The aim of this project is the study of moduli spaces of vector bundles over algebraic varieties. The case of an algebraic surfaces is particular important here as the case of dimension one,that is algebraic curves, is rather well understood, at least in comparison to the general case.Therefore the case of dimension two, that is algebraic surfaces should be typical for the additional difficulties. The existence of the moduli schemes in question is guaranteed since a number of years under quite general circumstances. Unfortunately a more concrete and detailed understanding is very limited. Therefore it makes good sense to consider additional, but natural properties, which make the problems easier to handle. In our case we consider the additional action of an Azumaya algebra on the coherent sheaves resp. vector bundles. These are sheaves of associative algebras, which generically are central simple algebras over the function field of the base variety. The coherent sheaves considered are assumed to be simple modules over the generic stalk that is the central simple algebra. The first case satisfying these conditions is still strictly commutative. The sheaf of algebras is just the structure sheaf of the base variety, the coherent sheaves are invertible module sheaves over the variety. The emerging moduli problem is completely classical and leads to the Picard varieties. Their theory is much easier than the theory for general vector bundles. Therefore the cases which one wants to study in this project, should be considered in this context. The general theory of these moduli varieties in fact is more direct than the case of general bundles and was done by several people including N.Hoffmann and the author of this proposal. So for example it is not necessary to specify an ample divisor on the variety to apply geometric invariant theory. Also the moduli scheme are automatically proper resp. compact. Therefore there is some hope, that also the finer properties of these modular varieties beyond pure existence are better understandable. To have some success it is of course as always important to have a good number of well understood examples to be able to test more general conjectures. Also on the way to understanding the moduli problem some interesting deformation theoretic questions had to be studied under somewhat special assumptions. A more technical part of this project is to come here to a better understanding and to find similiar results in a more general setting.