PhDNoncommutative Riemannian geometry is an area that has seen intense activity
over the past 25 years. Despite this, noncommutative complex geometry is only
now beginning to receive serious attention. The theory of quantum groups provides
a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been
investigated as noncommutative complex spaces. In a series of papers, Heckenberger
and Kolb showed that for each of these spaces, there exists a q-deformed
Dolbeault double complex.
In this thesis a comprehensive framework for noncommutative complex geometry
on quantum homogeneous spaces is introduced. The main ingredients used are covariant
differential calculi and Takeuchi's categorical equivalence for faithfully
at
quantum homogeneous spaces. A number of basic results are established, producing
a simple set of necessary and sufficient conditions for noncommutative complex
structures to exist. It is shown that when applied to the quantum projective spaces,
this theory reproduces the q-Dolbeault double complexes of Heckenberger and
Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{
Weil theory, and to produce the beginnings of a theory of noncommutative Kahler
geometry
