Alexander Schmitt: From the finiteness of GIT quotients to the
semistable reduction theorem for singular principal bundles

Abstract: When forming the quotient of a projective variety by a reductive group
action with Geometric Invariant Theory (GIT), one needs to fix a linearization.
The quotient will depend on this. Bialynicki-Birula and, independently,
Dolgachev and Hu have shown that only finitely many distinct quotients do arise
this way. We will first sketch the proof after ABB.

A formal generalization of GIT for the group GL_r(C) is the theory of decorated
sheaves on a projective manifold. There, the semistability concept depends on a
stability parameter which can take infinitely many values. Again, only finitely
many distinct moduli spaces will emerge. We give an elegant argument found by
Langer for this.

Finally, the speaker has succesfully applied the theory of decorated sheaves to
the construction and compactification of moduli spaces of semistable principal
bundles on a projective algebraic manifold. In this picture, the above result on
decorated sheaves gives a very nice proof for the semistable reduction theorem.