This project is devoted to the study of moduli spaces of Azumaya algebras over surfaces. As a first step we construct compactifications. A generalized Azumaya algebra is a perfect object in the derived category of the surface, endowed with a multiplication. It turns out that these objects can be used to give completely canonical compactifications. There is also a natural way to define stability of generalized Azumaya algebras (depending on some auxiliary choices). The result is what we would like to call a GIT stack compactifying the moduli space of Azumaya algebras. The project proposes to study these spaces and to use them to define ``Donaldson type invariants''. In addition the geometry of the moduli spaces will be studied for particular types of surfaces, e.g., elliptic surfaces and K3 surfaces.
A complex projective surface can be viewed as a 4 dimensional space which is endowed with a lot of additional structure. The most important of these is a choice of a rotation map on the tangent spaces; it is a rotation over 90 degrees. A lot of research has been done to classify four dimensional spaces which are endowed with such a structure. This is usually done by defining invariants (for example numbers) of complex projective surfaces which can be used to tell them apart. A very basic example are the Betti numbers, which are dimensions of cohomology groups. To give you an idea, an element of the second cohomology group corresponds to a 2 dimensional subspace of the 4-fold. Of course we are not simply enumerating these; we use a coarser equivalence relation (deformation equivalence). Here is a question: How many of these 2 dimensional subspaces have the property that the tangent space at any point is preserved by the rotation that defines the complex structure on our 4-fold? Such a subspace is called a complex curve on the complex surface. This question has been much studied, and is related to the Hodge conjecture. However, in this project we go the other way. Namely, we look at other objects: Complex projective bundles over our 4-fold determine a degree 2 cohomology class as well, and they are typically not those which can be represented by complex curves. It turns out that by looking at all possible complex projective bundles representing the given cohomology class we get a new space which, if we can understand it, tells us a lot about the original 4-fold. All kinds of new invariants of the original complex projective surface can be defined in terms of these moduli spaces. It is the geometry of these moduli spaces that will be studied in this project. There is a lot of techincal machinery that has to be developed before we can begin the exploration of more geometrical properties and part of the project will be devoted to developing this machinery.
