We reconsider intermediate holomorphic convexity introduced by Barlet and Silva in the more general case of convexity with respect to hermitian holomorphic line bundles whose weight functions are (weakly) $q$-convex. Also we give completeness and Kahler properties of the canonical reduction.
"Convexity with respect to weakly $q$-concave line bundles and reduction." Rocky Mountain J. Math. 49 (4) 1335 - 1354, 2019. https://doi.org/10.1216/RMJ-2019-49-4-1335