Geometric Invariant Theory relative to a base variety
In its basic form, Geometric Invariant Theory (GIT) studies the problem of
forming appropriate quotients W//G and IP(W)//G where G is a reductive
linear algebraic group, say, over the complex numbers which acts on the
vector space W and the projective space IP(W) by means of a rational
representation \rho: G->GL(W).
In recent decades, a version of this problem relative to a projective base
manifold X has become of interest. Again, the input datum is a
representation \rho: G->GL(W) of the reductive group G on the finite
dimensional vector space W and the problem one wishes to study is the
classication of pairs (P,\sigma) where P is a principal G-bundle and
\sigma: X->P_\rho is a section of the vector bundle P_\rho with fiber W
that one associates to P and the representation \rho.
In the talk, we will give an extensive review of `classical' GIT in order
to explain the relevant phenomena and then formulate the relative problem
and give an account of known solutions.
A guiding example for the whole talk will be that of representations
associated with quivers.