The relative pluricanonical stability for 3-folds of general type

The aim of this paper is to improve a theorem of János Kollár by a different method. For a given smooth complex projective threefold $X$ of general type, suppose the plurigenus $P_{k}(X)\ge 2$. Kollár proved that the $(11k+5)$-canonical map is birational. Here we show that either the $(7k+3)$-canonical map or the $(7k+5)$-canonical map is birational and that the $(13k+6)$-canonical map is stably birational onto its image. Suppose $P_{k}(X)\ge 3$. Then the $m$-canonical map is birational for $m\ge 10k+8$. In particular, $\phi_{12}$ is birational whenever $p_{g}(X)\ge 2$ and $\phi_{11}$ is birational whenever $p_{g}(X)\ge 3$.