On the Effective Nonvanishing of Varieties of Kodaira Dimension Zero

Given a smooth projective variety $X$ of Kodaira dimension zero, we show that there exists a constant $m$ depending on two invariants of the general fiber of the Albanese map, such that $\mathrm{\mid }m{K}_X\mathrm{\mid }\mathrm{\ne }\mathrm{\varnothing }$ .