An injectivity theorem on snc compact Kähler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves

Abstract

Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact Kähler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-φ_F}$. Via the use of the adjoint ideal sheaves (constructed from $φ_F$ and $D$) and the associated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on $D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related to the $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact Kähler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular Kähler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Kollár-type injectivity theorem for the cohomology on $D$ when $F$ is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact Kähler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and Păun.