#### Degree

Doctor of Philosophy

Mathematics

#### Supervisor

Professor Masoud Khalkhali

#### Abstract

In this thesis, we study complex structures of quantum projective
spaces that was initiated in [19] for the quantum projective line,
$\mathbb{C}P^1_q$. In Chapters 2 and 3, which are the main parts of this thesis, we generalize the the results of [19] to the spaces $\mathbb{C}P^{2}_q$ and $\mathbb{C}P^{\ell}_q$. We consider a natural holomorphic structure on the quantum projective space already presented in [11,9],
and define holomorphic structures on its canonical quantum line bundles.
The space of holomorphic sections of these line bundles then will determine
the quantum homogeneous coordinate ring of the quantum projective space as the space of twisted polynomials.

We also introduce a twisted positive
Hochschild cocycle $2 \ell$-cocycle on $\mathbb{C}P^{\ell}_q$, by using the complex structure of $\mathbb{C}P^{\ell}_q$, and show that it is cohomologous to its fundamental class which is represented
by a twisted cyclic cocycle. This fits with the point of view of holomorphic structures in noncommutative geometry advocated in [4,5], that holomorphic structures in noncommutative geometry are represented by (extremal)
positive Hochschild cocycles within the fundamental class.

In Chapter 4, we directly verify that the main statements of Riemann-Roch formula and
Serre duality theorem hold true for $\mathbb{C}P^1_q$ and $\mathbb{C}P^2_q$.

In Chapter 5, a quantum version of the Borel-Weil theorem for $SU_q(3)$ is proved and is generalized to the case of $SU_q(n)$.

Finally, in the last chapter the noncommutative complex structure of finite spaces is investigated. The space of holomorphic functions are determined and it is also proved that there is no holomorphic structure on the nontrivial vector bundle $\mathcal E_a\oplus \mathcal E_b$ over the space of two points $X=\{a,b\}$, where dim $\mathcal E_a=2$ and dim $\mathcal E_b=1$.

COinS