Intermediate Pseudoconvexity of Fiber Bundles

In this paper, we investigate the pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces of genus $\ge 2$, as well as the intermediate pseudoconvexity of their complements in the associated projective space bundles. Inspired by Brunella’s work, we prove that any such ball bundle is $1$-convex, while its complement is $n$-convex, where $n$ denotes the dimension of the ball fiber, provided that the bundle admits a harmonic section with a regular point.