Effective global generation on varieties with numerically trivial canonical class

We prove a Fujita-type theorem for varieties with numerically trivial canonical bundle using properties of semihomogeneous bundles on abelian varieties. We combine our results with work of Riess on compact hyperk"{a}hler manifolds and work of Mukai, Pareschi and Yoshioka to obtain effective global generation statements for certain moduli spaces of sheaves on abelian surfaces. Among these is the statment that if $\text{cL}$ is an ample line bundle on the Hilbert square $S^{[2]}$ of an abelian surface $S,$ then ${\text{cL}}^{\otimes m}$ is globally generated for $m\ge 3.$