Line bundles on noncommutative algebraic and arithmetic surfaces

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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.