Extension with log-canonical measures and an improvement to the plt extension of Demailly--Hacon--Paun

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-Paun on the "plt extension" with the superfluous assumption "$\operatorname{supp} D \subset \operatorname{supp}(S+B)$" 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.