Monatsh Math (2017) 184:171–173 DOI 10.1007/s00605-017-1080-8 © Springer-Verlag GmbH Austria 2017 Noguchi, J., Winkelmann J., Nevanlinna Theory in Several Complex Variables and Diophantine Approximation (Grundlehren der mathematischen Wissenschaften Vol. 350). XIV, 416 pp., Springer, Tokyo Heidelberg New York Dordrecht London, 2014. EUR 10,449. This intriguing monograph describes the state of the art of large parts of Nevanlinna theory in several complex variables. After reviewing the one-dimensional case the authors show the First Main Theorem for meromorphic mappings from the multidi- mensional complex space into a compact complex manifold with respect to holomor- phic line bundles and coherent ideal sheaves of the structure sheaf. Then they cover several cases in which the Second Main Theorem has been established, including a version for maps to abelian and semi-abelian varieties, which is due to the authors and K. Yamanoi. Further chapters are devoted to applications to Kobayashi hyperbolicity, Nevanlinna theory over function ﬁelds (including an analogue of the abc-conjecture), and Diophantine approximation over number ﬁelds in the spirit of Vojta’s dictionary. As can be seen from this description this volume is too advanced for newcomers to the ﬁeld. It will, however, be an important reference for graduate students and researchers in Nevanlinna
Monatshefte f�r Mathematik – Springer Journals
Published: Aug 8, 2017
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