Effective Iitaka fibrations

Abstract

For every n n -dimensional projective manifold X X of Kodaira dimension 2 2 we show that Φ | M K X | \Phi _{|M K_X|} is birational to an Iitaka fibration for a computable positive integer M = M ( b , B n − 2 ) M = M(b, B_{n-2}) , where b > 0 b > 0 is minimal with | b K F | ≠ ∅ |bK_F| \ne \emptyset for a general fibre F F of an Iitaka fibration of X X , and where B n − 2 B_{n-2} is the Betti number of a smooth model of the canonical Z / ( b ) \mathbb {Z}/(b) -cover of the ( n − 2 ) (n-2) -fold F F . In particular, M M is a universal constant if the dimension n ≤ 4 n \le 4 .