2022/11/02 by Tsz On Mario Chan, Chan, Tsz On Mario
Mathematics · #Advanced Harmonic Analysis Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.CV #msc:14B05 #msc:32J25
paper · pdf · doi:10.48550/arxiv.2211.00885
published as Complex geometric analysis, 9-22. Springer Proc. Math. Stat., 481 (2025) · 10 pages; v2: minor changes and updates to some citations and references
arxiv created 2023/04/05 · arxiv updated 2026/08/03
The "qualitative" extension theorem of Demailly guarantees existence of holomorphic extensions of holomorphic sections on some subvariety under certain positive-curvature assumption, but that comes without any estimate of the extensions, especially when the singular locus of the subvariety is non-empty and the holomorphic section to be extended does not vanish identically there. Residue functions are analytic functions which connect the L2 norms on the subvarieties (or their singular loci) to L2 norms with specific weights on the ambient space. Motivated by the conjectural "dlt extension", this note discusses the possibility of retrieving the L2 estimates for the extensions in the general situation via the use of the residue functions. It is also shown in this note that the 1-lc-measure defined via the residue function of index 1 is indeed equal to the Ohsawa measure in the Ohsawa--Takegoshi L2 extension theorem.