In this talk, we will discuss an optimal $L^2$ extension theorem for holomorphic sections of the canonical bundle twisted with a singular Hermitian line bundle on weakly pseudoconvex K\"{a}hler manifolds, and the pseudoeffectivity of twisted relative pluricanonical bundles for K\"{a}hler fibrations. In our proofs, a crucial step is to find and prove a generalization of Siu's lemma, and Guan-Zhou's solution of Demailly's strong openness conjecture also plays an important role. These are joint works with Professor Xiangyu Zhou.
