2024/09/21 by Tsz On Mario Chan, Chan, Tsz On Mario, Young-Jun Choi +4 · 1 citation
Mathematics · #14B05 (secondary) #32J25 (primary) 32Q15 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:14B05 #msc:32J25 #msc:32Q15
paper · pdf · doi:10.48550/arxiv.2409.14100
41 pages; v2: some errors fixed and presentation modified for readability; to appear in Ann. Inst. Fourier; v1: see our previous related work at arXiv:2307.12025
openalex publication_date 2024/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · arxiv created 2026/07/31 · arxiv updated 2026/08/03
Let X be a complex manifold, and let Y and D be two reduced simple-normal-crossing (snc) divisors on X with no common irreducible components. Given a proper locally Kähler morphism π\colon X → Δ from X to a complex analytic space Δ, we prove Fujino's conjecture on the injectivity theorem in the relative setting in a generalized form. Specifically, we establish an injectivity result for the higher direct images under π for the lc pairs (X, D) as well as (Y, DY), where DY := D ∩ Y. As an application, this result immediately implies the injectivity theorem on holomorphically convex Kähler manifolds with reduced snc divisors. The main technique in the proof consists of the theory of harmonic integrals together with residue formulae associated with adjoint ideal sheaves, which are developed from our previous work for the absolute case (where Δ is a point and X is compact). Additionally, we make use of the Takegoshi harmonic forms to deal with the non-compactness of X.