With a view to proving the conjecture of “dlt extension” related to the abundance conjecture, a sequence of potential candidates for replacing the Ohsawa measure in the Ohsawa–Takegoshi \(L^2\) extension theorem, called the “lc-measures”, which hopefully could provide the \(L^2\) estimate of a holomorphic extension of any suitable holomorphic section on a subvariety with singular locus, are introduced in the first half of the paper. Based on the version of \(L^2\) extension theorem proved by Demailly, a proof is provided to show that the lc-measure can replace the Ohsawa measure in the case where the classical Ohsawa–Takegoshi \(L^2\) extension works, with some improvements on the assumptions on the metrics involved. The second half of the paper provides a simplified proof of the result of Demailly–Hacon–Păun on the “plt extension” with the superfluous assumption “\({{\,\mathrm{supp}\,}}D \subset {{\,\mathrm{supp}\,}}\left( S+B\right) \)” in their result removed. Most arguments in the proof are readily adopted to the “dlt extension” once the \(L^2\) estimates with respect to the lc-measures of holomorphic extensions of sections on subvarieties with singular locus are ready.

为了证明与丰度猜想相关的“dlt 扩展”猜想，一系列用于替代大泽-武越\(L^2\)扩展定理中的大泽测度的潜在候选者，称为“ lc-测度” ，希望可以提供对具有奇异轨迹的子变种上任何合适的全纯部分的全纯扩展的\(L^2\)估计，在论文的前半部分进行了介绍。基于Demailly证明的\(L^2\)扩展定理的版本，证明了在经典的Ohsawa–Takegoshi \(L^2\)的情况下，lc-measure可以代替Ohsawa测度扩展工作，对所涉及的指标的假设进行了一些改进。论文的后半部分提供了 Demailly–Hacon–Păun 在“plt 扩展”上的结果的简化证明，其中包含多余的假设“ \({{\,\mathrm{supp}\,}}D \subset {{ \,\mathrm{supp}\,}}\left( S+B\right) \) ”在他们的结果中被删除。一旦关于具有奇异轨迹的子变体的全纯扩展的 lc 度量的\(L^2\)估计准备就绪，证明中的大多数论点都可以很容易地用于“dlt 扩展” 。