Injectivity theorems for higher direct images under proper K\"ahler morphisms on snc spaces

TL;DR

This paper proves a generalized injectivity theorem for higher direct images under proper K"ahler morphisms on snc spaces, extending Fujino's conjecture and applying harmonic integrals and residue formulas.

Contribution

It establishes a new injectivity result for higher direct images in the relative setting for lc pairs on snc spaces, generalizing previous absolute case results.

Findings

Proves Fujino's conjecture on injectivity in the relative setting.

Establishes injectivity for higher direct images of lc pairs with snc divisors.

Demonstrates applications to holomorphically convex K"ahler manifolds.

Abstract

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 K\"ahler morphism $\pi \colon X \to \Delta$ from $X$ to a complex analytic space $\Delta$, 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 $\pi$ for the lc pairs $(X, D)$ as well as $(Y, D_Y)$, where $D_Y := D \cap Y$. As an application, this result immediately implies the injectivity theorem on holomorphically convex K\"ahler 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 $\Delta$ is a point and $X$ is compact). Additionally, we make use of the Takegoshi harmonic forms to deal with the non-compactness of $X$.