This text is an attempt to document some of these constructions, many of which have only appeared in journal form. The emphasis here is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.
Among Manin’s numerous achievements are the proof of the functional analogue of the Mordell conjecture, the theory of Gauss--Manin connection, the proof with V. Iskovskikh of the nonrationality of smooth quartic threefolds, the theory of $p$-adic automorphic functions, and the theory of quantum computations.
Contributors in the first volume include:
S. K. Donaldson
B. van Geemen
G. van der Geer
Contributors: C.P. Boyer, J. Cheeger, X. Dai, K. Galicki, P. Gauduchon, N. Hitchin, L. Katzarkov, J. Kollár, C. LeBrun, P. Rukimbira, S.R. Simanca, J. Sparks, C. van Coevering, and W. Ziller.
Throughout the book, the reader will find a good number of inspirating problems related to the topics covered.
This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Moreover, the authors have added a part dedicated to arithmetical cohomology and noncommutative geometry, a report on point counts on varieties with many rational points, the recent polynomial time algorithm for primality testing, and some others subjects.
From the reviews of the 2nd edition:
"... For my part, I come to praise this fine volume. This book is a highly instructive read ... the quality, knowledge, and expertise of the authors shines through. ... The present volume is almost startlingly up-to-date ..." (A. van der Poorten, Gazette, Australian Math. Soc. 34 (1), 2007)