I was reading a physics paper where it was mentioned that the basic framework of Connes' differential non-commutative geometry (or actually, a slight modification of Connes in that paper) would need some extensions in order to arrive at a theory of non-commutative complex geometry, where one now works with complex manifolds.
Are there specific technical obstacles which would make such an extension of non-commutative geometry particularly difficult or involved? What would be the main issues with trying to do this?
Best Answer
I don't think there are any obvious ``technical obstacles'' to extending noncommutative geometry to the a theory of noncommutative complex geometry. Instead I would say that there are a number of differing points of view on what form noncommutative complex geometry should take, and it's not completely clear how the different approaches relate to each other.
Let's start with Connes' theory of noncommutative geometry, in particular spectral triples, which are usually thought of as ``noncommutative Riemannian manifolds''. A basic spectral triple is an unbounded representative of a $K$-homology class of a unital $C^*$-algebra. Such unbounded representatives exist for all classes of any unital $C^*$-algebra, so in particular, they exist for the $K$-homology classes of a compact Hausdorff space. Now in general a compact Hausdorff space will may not admit a differential structure, so in the commutative case a spectral triple is a more general structure than a smooth structure. To address this Connes introduced a number of higher axioms, and then proved his reconstruction theorem, showing that spectral triples satisfying these extra axioms are equivalent to compact Riemannian manifolds. The question of how well-suited these higher axioms to the noncommutative setting is the subject of debate . . . but that is a discussion for another day.
So following this point of view, a "complex spectral triple" would be spectral triple (satisfying the higher axioms) plus "something extra". Connes' proposal was to look at positive Hochschild cocycles since in the classical case, for surfaces, such cocycles are equivalent to complex structures. This is explained in Section VI.2 of Connes' book.
The motivating noncommutative example is, as usual, the noncommutative torus $\mathbb{T}_{\theta}^2$. This is a $\theta$-deformation of a classical complex manifold (in fact a Calabi--Yau manifold) and since its structure is quite close to the classical situation, a lot of the classical complex and Kähler geometry carries over.
In another direction, there is the work of Fröhlich, Grandjean, and Recknagel:. They start with a spectral triple, and then try to build a noncommutative version of complex geometry on the associated differential graded algebra. Their approach takes globally defined classical identities in complex geometry and makes them into noncommutative axioms. For example, the Lefschetz identities and the Kähler identities play a major role in their work. Their main example is again the noncommutative torus.
In a very different approach one should mention noncommutative projective algebraic geoemtry. This is an entirely algebraic approach to noncommutative geometry based on Serre's characterisation of the quasi-coherent sheaves over a projective variety. This approach has been hugely successful in recent years, see here for a nice introduction. A connection with the differential geometric approach to noncommutative complex geometry (analogous to the classical GAGA correspondence) has been postulated, see here for example. However, we are still very far from any kind of a noncommutative GAGA.
A large and very important family of motivativing examples comes from Drinfeld--Jimbo quantum groups: the quantum flag manifolds. As shown in the seminal papers of Heckenberger and Kolb, the irreducible quantum flag manifolds admit a direct $q$-deformation of their classical de Rham complex. These complexes have a remarkable structure, $q$-deforming the classical Kähler geometry of the flag manifolds. These examples are crucial for understanding both the noncommutative geoemtry of the quantum groups, and as a forum for reconciling the various approaches to noncommutative complex geometry.
Finally, we note the approach of Pirkovskii based on topological algebras. The motivating examples here are the quantum $n$-polydisk and the quantum $n$-ball.