The curvature condition for the singular Hermitian metric in a generalized _L_2 extension theorem on complex manifolds implies also a cohomology injectivity theorem in certain circumstances. This shows that the curvature condition is still available for the extension theorem in more general situations than before.
Cite this article
Takeo Ohsawa, On a Curvature Condition that Implies a Cohomology Injectivity Theorem of Kollár–Skoda Type. Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, pp. 565–577