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

TL;DR

This paper proves a Kollár-type injectivity theorem for lc pairs on compact Kähler spaces using harmonic integrals and adjoint ideal sheaves, confirming Fujino's conjecture and offering an alternative proof to Cao and Pe2un.

Contribution

It introduces a novel approach combining harmonic integrals and adjoint ideal sheaves to establish injectivity theorems on singular Kähler spaces.

Findings

Proves a Kollár-type injectivity theorem for lc pairs on Kähler spaces.

Confirms Fujino's conjecture on injectivity for compact Kähler lc pairs.

Provides an alternative proof to Cao and Pe2un's recent result.

Abstract

Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact K\"ahler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-\varphi_F}$. Via the use of the adjoint ideal sheaves (constructed from $\varphi_F$ 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 K\"ahler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular K\"ahler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Koll\'ar-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 K\"ahler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and P\u{a}un.