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

• Published in: Mathematische annalen 2022-08, Vol.383 (3-4), p.943-997
• Main Authors: Chan, Tsz On Mario, Choi, Young-Jun
• Format: Article
• Language: English
• Subjects: Mathematical analysis; Mathematics; Mathematics and Statistics; Theorems
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-021-02152-3

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 “ supp D ⊂ 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.