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

这个项目的目的是研究代数簇上向量丛的模空间。代数曲面的情况在这里尤其重要，因为至少与一般情况相比，维度一即代数曲线的情况是相当容易理解的。因此，对于额外的困难，维度二即代数曲面的情况应该是典型的。在相当一般的情况下，若干年以来，所讨论的模数方案的存在是有保证的。不幸的是，更具体和详细的了解非常有限。因此，考虑其他自然属性是很有意义的，因为这些属性使问题更容易处理。在我们的例子中，我们考虑Azumaya代数在相干层上的附加作用。向量束。这些是结合代数的层，它们一般是基簇的函数域上的中心单代数。所考虑的凝聚层被假定为中心单代数的一般柄上的单模。满足这些条件的第一种情况仍然是严格可交换的。代数层恰好是基簇的结构层，凝聚层是簇上的可逆模层。新出现的模问题是完全经典的，并导致了Picard簇。他们的理论比一般向量丛的理论简单得多。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。这些模簇的一般理论实际上比一般丛的情况更直接，而且是由包括N.Hoffmann和这一建议的作者在内的几个人所做的。因此，举例来说，应用几何不变量理论不需要在变种上指定充分的除数。此外，模数方案也是自动适当的。紧凑型。因此，有一些希望，这些模变种超越纯粹存在的更好的性质也是可以更好地理解的。为了取得一些成功，当然，拥有大量理解良好的例子以能够测试更一般的猜测是一如既往的重要。此外，在理解模数问题的过程中，还必须在一些特殊的假设下研究一些有趣的变形理论问题。这个项目的一个更技术性的部分是来这里更好地理解，并在更一般的环境中找到类似的结果。

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.