---
abstract: 'We prove a non-vanishing theorem of the cohomology $H^0$ of the adjoint divisor $K_X + \lceil L \rceil$ where $\lceil L \rceil$ is the round up of a nef and big ${\bold Q}$-divisor $L$.'
address:
- 'Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan'
- 'Mathematics Division, National Center for Theoretical Science at Taipei'
- 'Institute of Mathematics, Fudan University, Shanghai, 200433, P. R. China'
- 'Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore'
author:
- Jungkai Alfred Chen
- Meng Chen
- 'De-Qi Zhang'
title: 'A nonvanishing theorem for Q-divisors'
---

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 Introduction
=============

We work over the complex number field [**C**]{}. The motivation of this note is to find an effective version of the famous non-vanishing theorem of Kawamata and Shokurov (see [@KMM], [@Sh]). We propose the following:

\[mainconj\] Let $X$ be a nonsingular projective variety. Let $L$ be a [**Q**]{}-divisor on $X$ satisfying the conditions below:

[ ]{}

$L$ is nef and big,

$K_X + L$ is nef, and

either $L$ is a Cartier integral divisor, or $L$ is effective.

Then $H^0(X, K_X + \lceil L \rceil) \ne 0$, 