@MISC{Viehweg707effectiveiitaka, author = {Eckart Viehweg and De-qi Zhang}, title = {EFFECTIVE IITAKA FIBRATIONS}, year = {707} }

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Abstract

Abstract. For every n-dimensional projective manifold X of Kodaira dimension 2 we show that Φ |MKX | is birational to an Iitaka fibration for a computable positive integer M = M(b, Bn−2), where b> 0 is minimal with |bKF | ̸ = ∅ for a general fibre F of an Iitaka fibration of X, and where Bn−2 is the Betti number of a smooth model of the canonical Z/(b)-cover of the (n − 2)-fold F. In particular, M is a universal constant if the dimension n ≤ 4. Building up on the work of H. Tsuji, C.D. Hacon and J. McKernan in [HM] and independently S. Takayama in [Ta] have shown the existence of a constant rn such that Φ|mKX | is a birational map for every m ≥ rn and for every complex projective n-fold X of general type. If the Kodaira dimension κ = κ(X) < n, consider an Iitaka fibration f: X → Y, i.e. a rational map onto a projective manifold Y of dimension κ with a connected general fibre F of Kodaira dimension zero. We define the index b of F to be b = min{b ′> 0 | |b ′ KF | ̸ = ∅}, and Bn−κ to be the (n − κ)-th Betti number of a nonsingular model of the Z/(b)-cover