A defect relation for linear systems on compact complex manifolds (Q1071910)

In 1979 \textit{B. Shiffman} [Indiana Univ. Math. J. 28, 627-641 (1979; Zbl 0434.32024)] proved the defect relation \(\sum^{q}_{j=1}\delta_ f(D_ j)\leq 2n\) for a certain class of meromorphic maps \(f: {\mathbb{C}}^ m\to {\mathbb{P}}_ n\) of finite order. Here the \(D_ j\) are distinct hypersurfaces in \({\mathbb{P}}_ n\) such that no point is contained in the support of \(n+1\) distinct \(D_ j\) and \(f({\mathbb{C}}^ m)\not\subseteq \sup p D_ j\) for all j. Using the method of associate maps introduced by \textit{L. V. Ahlfors} [Acta Soc. Sci. Fenn. n. Ser. A3, No.4, 31 p. (1941; Zbl 0061.152)] this result is extended to meromorphic maps f such that either \(f({\mathbb{C}}^ m)\) is contained in a line of \({\mathbb{P}}_ n\) or is a projection of a ''special exponential map''. Introducing an auxiliary defect \(\tau_ f\) the author proves \(\sum^{q}_{j=1}\delta_ f(D_ j)\leq n(1+\tau_ f)\) for all meromorphic maps f. To add generality this last relation is also shown for meromorphic maps \(f: {\mathbb{C}}^ m\to X,\) where X is a compact complex n-dimensional manifold and for \(D_ 1,...,D_ q\in | L|,\) where L is a spanned line bundle.