One of the main achievements of the algebraic geometry of the 20th century was to realize that it is fruitful not only to study algebraic varieties by themselves but also in families. That means that one should consider the space classifying all varieties with given properties. These spaces are called moduli spaces. The advantage of algebraic geometry is that many times those spaces are themsel...

We study certain principal actions on noncommutative C-algebras. Our main examples are the Zpand T-actions on the odd-dimensional quantum spheres, yielding as fixed-point algebras quantum lens spaces and quantum complex projective spaces, respectively. The key tool in our analysis is the relation of the ambient C-algebras with the Cuntz-Krieger algebras of directed graphs. A general result abou...

We survey and expand 1 on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into complex projective space (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when n = 1 is given by a non-trivial extension whi...

We introduce a generalization of Atiyah-Drinfeld-Hitchin-Manin linear algebraic data and a generalization of Atiyah-Drinfeld-Hitchin-Manin equation, which are subsequently used to construct all framed instanton bundles on complex projective spaces. Using geometric invariant theory, we prove that the moduli spaces of framed instanton sheaves is a quiasiprojective variety. We also provided a link...

We introduce a generalization of the Atiyah-Drinfeld-Hitchin-Manin linear algebraic data and a generalization of Atiyah-Drinfeld-Hitchin-Manin equation, which are subsequently used to construct all framed instanton bundles on complex projective spaces. Using geometric invariant theory, we prove that the moduli spaces of framed instanton sheaves is a quasiprojective variety. We also provide a li...