This edition of the book has been extended to take account of
one of these developments， one which was just hinted at in the
second edition. A close and very fruitful relationship has been
discovered between geometric invariant theory for quasi projective
complex varieties and the moment map in Symplectic geometry， and a
chapter has been added describing this relationship and some of its
applications. In an infinite-dimensional setting the moment map
links geometric invariant theory and Yang-Mills theory， which has
of course been the focus of much attention among mathematicians
over the last fifteen years.
In style this extra chapter is closer to the appendices added in
the second edition than to the original text. In particular no
proofs are given where satisfactory references exist.