We study quantum principal bundles on projective varieties using a sheaf theoretic approach. Differential calculi are introduced in this context. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. We next introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. The examples of principal (covariant) calculi on the quantum principal bundles $SL_q(2, \mathbb{C})$ and $GL_q(2, \mathbb{C})$ over the projective space $P^1(\mathbb{C})$ are presented.