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

模空间对 Azumaya 代数上的丛进行分类——代数曲面的情况。

• 批准号: 191121449
• 负责人: Professor Dr. Ulrich Stuhler
• 金额: --
• 依托单位: Mathematisches Institut (MI)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2011
• 资助国家: 德国
• 起止时间: 2010-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/191121449?language=en
• 关键词: Moduli; spaces; classifying; bundles; over

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.

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特别重要，因为一维的情况，即代数曲线，至少与一般情况相比，是相当好理解的。因此，二维的情况，即代数曲面，应该是典型的额外困难。在相当一般的情况下，存在的模计划的问题是保证了若干年以来。不幸的是，更具体和详细的了解非常有限。因此，考虑额外的，但自然的属性是很有意义的，这使得问题更容易处理。在我们的例子中，我们考虑的附加作用的Azumaya代数上的相干层。向量丛这些是结合代数的层，一般是基簇的函数域上的中心单代数。所考虑的相干层被假定为是中心单代数的通用茎上的简单模块。满足这些条件的第一个情况仍然是严格可交换的。代数层就是基簇的结构层，凝聚层就是基簇上的可逆模层。新出现的模问题是完全经典的，并导致皮卡品种。它们的理论比一般向量丛的理论容易得多。因此，在本项目中要研究的案例应在此背景下考虑。这些模簇的一般理论实际上比一般丛的情况更直接，并且是由包括N.霍夫曼和这个建议的作者在内的几个人完成的。因此，例如，它是没有必要指定一个充分的除数的品种，以应用几何不变理论。此外，模计划是自动正确的。紧凑.因此，有一些希望，也更好地理解这些模变种超越纯粹存在的更精细的性质。为了取得一定的成功，当然，有大量的好的理解的例子来测试更一般的结构是很重要的。在理解模量问题的过程中，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，并在更一般的设置中找到类似的结果。