Moduli of Azumaya algebras, vector bundles and applications

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This project is devoted to the study of moduli spaces of Azumaya algebrasover surfaces. As a first step we construct compactifications. Ageneralized Azumaya algebra is a perfect object in the derivedcategory of the surface, endowed with a multiplication. It turns outthat these objects can be used to give completely canonicalcompactifications. There is also a natural way to define stability ofgeneralized Azumaya algebras (depending on some auxiliarychoices). The result is what we would like to call a GIT stackcompactifying the moduli space of Azumaya algebras. The projectproposes to study these spaces and to use them to define ``Donaldsontype invariants''. In addition the geometry of the moduli spaces willbe studied for particular types of surfaces, e.g., elliptic surfacesand K3 surfaces. A complex projective surface can be viewed as a 4 dimensional spacewhich is endowed with a lot of additional structure. The mostimportant of these is a choice of a rotation map on the tangentspaces; it is a rotation over 90 degrees. A lot of research has beendone to classify four dimensional spaces which are endowed with such astructure. This is usually done by defining invariants (for examplenumbers) of complex projective surfaces which can be used to tell themapart. A very basic example are the Betti numbers, which aredimensions of cohomology groups. To give you an idea, an element ofthe second cohomology group corresponds to a 2 dimensional subspace ofthe 4-fold. Of course we are not simply enumerating these; we use acoarser equivalence relation (deformation equivalence). Here is a question: How many of these 2 dimensional subspaces havethe property that the tangent space at any point is preserved by therotation that defines the complex structure on our 4-fold? Such asubspace is called a complex curve on the complex surface. Thisquestion has been much studied, and is related to the Hodgeconjecture. However, in this project we go the other way. Namely, welook at other objects: Complex projective bundles over our 4-folddetermine a degree 2 cohomology class as well, and they are typicallynot those which can be represented by complex curves. It turns outthat by looking at all possible complex projective bundlesrepresenting the given cohomology class we get a new space which, ifwe can understand it, tells us a lot about the original 4-fold. Allkinds of new invariants of the original complex projective surface canbe defined in terms of these moduli spaces. It is the geometry ofthese moduli spaces that will be studied in this project. There is alot of techincal machinery that has to be developed before we canbegin the exploration of more geometrical properties and part of theproject will be devoted to developing this machinery.