  The following paper is around parts of the first named author's thesis. We
discuss (what we call) a Hilbert bundle approach to complex Brunn-Minkowski
theory and obtain a general monotonicity theorem. As two applications, we prove
a generalization of Guan's sharp strong openness theorem and a sharp
Ohsawa-Takegoshi extension theorem. A second proof of Guan-Zhou's strong
openness theorem using a Donnelly-Fefferman estimate is also given.
