Pluricanonical systems of projective varieties of general type

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 {\bf C}, $\mid m{K}_X\mid$ gives a birational rational map from $X$ into a projective space for every $m\ge {\nu }_n$. This theorem gives an affirmative answer to Severi’s conjecture. The key ingredients of the proof are the theory of AZD which was originated by the aurhor and the subadjunction formula for AZD’s of logcanoncial divisors.