Line bundles on noncommutative algebraic and arithmetic surfaces

我们想要研究非交换代数曲面和算术曲面以及这些曲面上的线丛。在这种情况下，这些非交换曲面是由Azumaya代数给出的。在代数情况下，这些曲面是经典几何对象的非交换类似物，例如K3曲面。它们具有与经典交换对应的性质相似的性质。例如，这些曲面上的线丛也按射影模格式进行分类。该方案对应于经典的Picard方案。我们想研究这些曲面的各种性质和它们的模格式。例如，我们想要了解这些非对易曲面上的Serre对偶。这有助于理解模空间的光滑性和形变理论。此外，我们还想研究这些模空间在某些情况下的辛结构。在算术情形下，我们想用Arakelov几何来研究非对易曲面和线丛。阿拉克洛夫几何是经典代数几何和复微分几何的混合体。这里的一个主要问题是，如何将Arakelov交积推广到非对易情形。另一个问题是，我们是否能赋予上同调群中的挠率一些意义。

We want to study noncommutative algebraic and arithmetic surfaces and line bundles on these surfaces. In this situation these noncommutative surfaces are given by Azumaya algebras. In the algebraic case these surfaces are noncommutative analogs of classical geometric objects, for example K3 surfaces. They have similar properties like their classical commutative counterparts. For example the line bundles on these surfaces are also classified by a projective moduli scheme. This scheme corresponds to the classical Picard scheme. We want to study miscellaneous properties of these surfaces and their moduli schemes. For example we want to understand the Serre duality on these noncommutative surfaces. This helps to understand the smoothness properties and the deformation theory of the moduli spaces. Furthermore we want to study the symplectic structure of these moduli spaces in certain situations. In the arithmetic situation we want to study the noncommutative surfaces and line bundles by using Arakelov geometry. Arakelov geometry is a mix of classical algebraic geometry and complex differential geometry. One of the main questions here is, how to generalize the Arakelov intersection product to the noncommutative situation. Another question is, if we can assign some meaning to the torsion in the cohomology groups.