Pluricanonical divisors of elliptic fiber spaces (Q912940)

An elliptic fiber space f: \(V\to W\) is a proper surjective morphism of a compact complex manifold V to a compact complex manifold W, where each fiber is connected and the general fibers are smooth elliptic curves. - Iitaka showed that for any elliptic surface \(f: S\to C\) with \(\kappa(S)=1\), the m-th pluricanonical mapping gives the unique structure of the elliptic surface f: \(S\to C\) if \(m\geq 86\), and 86 is the best possible number. On the other hand, Katsura and Ueno showed that if S is an \textit{algebraic} elliptic surface defined over an algebraically closed field of characteristic \(p\geq 0\) with \(\kappa (S)=1\), then the m-th pluricanonical mapping gives the unique structure of the elliptic surface for every \(m\geq 14.\) Main theorem A. If f: \(X\to S\) is an elliptic threefold with \(\kappa (X)=2\) and \(\kappa\) (S)\(\geq 1\), then the m-th pluricanonical mapping gives the Iitaka fibration for all even integers \(m\geq 16.\) Main theorem B. Let \(\{a_ n\}_{n=1,2,...}\) be a sequence of natural numbers defined by \(a_ 1=2\), \(a_{n+1}=a_ 1a_ 2...a_ n+1\). And let \(\{b_ n\}_{n=1,2,...}\) be a sequence of natural numbers defined by \(b_ n=(n+1)(a_{n+3}-1)+2\). Then for every positive integer n, there exists an elliptic fiber space \(X^{(n+1)}\to {\mathbb{P}}^ n\) over \({\mathbb{P}}^ n\) which satisfies the following conditions: (1) \(\kappa (X^{(n+1)})=n.\) (2) \(b_ n\) is the best possible number of the Iitaka fibering of \(X^{(n+1)}\), that is, \(\dim | mK_ X| =0\) if \(m=b_ n-1\), and the m-th pluricanonical mapping gives the Iitaka fibering for all \(m\geq b_ n.\) Moreover, \(X^{(n+1)}\) cannot be bimeromorphic to any compact Kähler manifold.