---
abstract: 'We introduce the notion of mixed-$\omega$-sheaves and use it for the study of a relative version of Fujita’s freeness conjecture.'
address: 'Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Osamu Fujino
date: '2019/7/29, version 0.40'
title: 'On mixed-$\omega$-sheaves'
---

Introduction {#z-sec1}
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Let us recall Fujita’s famous freeness conjecture on adjoint bundles. Note that everything is defined over $\mathbb C$, the complex number field, in this paper.

\[z-conj1.1\] Let $X$ be a smooth projective variety with $\dim X=n$ and let $\mathcal L$ be any ample invertible sheaf on $X$. Then $\omega_X\otimes \mathcal L^{\otimes l}$ is generated by global sections for every $l\geq n+1$.

Although there have already been many related results, Conjecture \[z-conj1.1\] is still open. As a generalization of Conjecture \[z-conj1.1\], Popa and Schnell proposed the following conjecture, which is a relative version of Fujita’s freeness conjecture.

\[z-conj1.2\] Let $f:X\to Y$ be a surjective morphism between smooth projective varieties with $\dim Y=n$. Let $\mathcal L$ be any ample invertible sheaf on $Y$. Then, for every positive integer $k$, the sheaf $$f_*\omega^{\otimes k}_X\otimes \mathcal L^{\otimes l}$$ is generated by global sections for $l\geq k(n+1)$.

We can find some interesting results on 