An application of adjoint ideal sheaves to injectivity and extension theorems

This note reviews the authors’ approach to Fujino’s conjecture, i.e. the injectivity theorem for lc pairs on compact Kähler manifolds, via the use of adjoint ideal sheaves coupled with the associated residue computations in their previous work. Using only the techniques and results obtained from that study and under a slightly stronger positivity assumption, a “qualitative” extension result is obtained. Such extension result guarantees the existence of a global holomorphic extension $F$ of any holomorphic section $f$ on some $\sigma$-lc centres of the given lc pair. The extension $F$ can be shown to take values in the corresponding adjoint ideal sheaf, even though it comes without any $L^2$ estimate of $F$ in terms of $f$. Moreover, the proof invokes and implies no vanishing theorem in general.