Pluricanonical systems of projective varieties of general type I

Abstract

Assuming the minimal model program, we prove that there exists a positive integer $\nu_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over complex numbers, $|mK_{X}|$ gives a birational rational map from $X$ into a projective space for every $m\geq \nu_{n}$.