# A transcendental approach to Koll\'ar's injectivity theorem

## TL;DR

This paper presents an analytic approach to Kollár's injectivity theorem, establishing a curvature condition that guarantees cohomology injectivity on compact Kähler manifolds without complex geometric tricks.

## Contribution

It introduces a curvature-based criterion for Kollár's injectivity theorem using harmonic forms on open sets, avoiding traditional algebraic geometric methods.

## Key findings

- Curvature condition implies Kollár's injectivity.
- Proof uses harmonic forms on Zariski open sets.
- No need for desingularizations or spectral sequences.

## Abstract

We treat Koll\'ar's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Koll\'ar type cohomology injectivity theorems. Our main theorem is formulated for a compact K\"ahler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete K\"ahler metric. We need neither covering tricks, desingularizations, nor Leray's spectral sequence.