The concept of moduli goes back to B. Riemann, who shows in [68] that
the isomorphism class of a Riemann surface of genus 9 2 depends on 3g -
3 parameters, which he proposes to name "moduli". A precise formulation
of global moduli problems in algebraic geometry, the definition of
moduli schemes or of algebraic moduli spaces for curves and for certain
higher dimensional manifolds have only been given recently (A.
Grothendieck, D. Mumford, see [59]), as well as solutions in some
cases. It is the aim of this monograph to present methods which allow
over a field of characteristic zero to construct certain moduli schemes
together with an ample sheaf. Our main source of inspiration is D.
Mumford's "Geometric In- variant Theory". We will recall the necessary
tools from his book [59] and prove the "Hilbert-Mumford Criterion" and
some modified version for the stability of points under group actions.
As in [78], a careful study of positivity proper- ties of direct image
sheaves allows to use this criterion to construct moduli as
quasi-projective schemes for canonically polarized manifolds and for
polarized manifolds with a semi-ample canonical sheaf.