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

Effective global generation on varieties with numerically trivial canonical class