Differential Calculi on Quantum Principal Bundles over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. Our main class of examples is given by quantum principal bundles over quantum flags. Given a first order differential calculus on the total space of the bundle, covariant under the action of the quantum structure group, if the base is affine we construct calculi on the base and on the structure Hopf algebra. If the base is projective we construct calculi on the sheaf of localizations, these induce canonical differential structures on the base sheaf and on the structure Hopf algebra and vice versa. The example of the quantum principal bundle $\mathcal{O}_q(\mathrm{SL}_2)$ over $\mathrm{P}^1(\mathbb{C})$ is discussed in detail.