Birational Automorphisms of Higher-dimensional Algebraic Varieties

The present survey covers the known results on the groups of birational automorphisms, rationality problem and birational classii-cation for Fano brations. 0. Birational geometry starts with M.NN other's paper 45] on Cremona transformations. The problems of birational geometry of algebraic varieties, that is, birational classiication, the rationality and unirationality problems, structure of the group of birational automorphisms, formed a subject of exclusive attention for the Ital-of all | G.Fano, laid the foundation of the modern birational geometry, outlined solutions to certain hard problems, gave surprisingly exact forecasts and suggested some crucial ideas. The modern period of birational geometry started with Yu.I.Manin's papers on geometry of surfaces over non-closed elds 38,39]. The breakthrough into higher dimensions was made in 1970 in the papers of V.A.Iskovskikh and Yu.I.Manin 29] and H.Clemens and Ph.Griiths 7], where the L uroth problem got its negative solution (both the techniques and the nal results of these papers were absolutely independent of each other). In 29] Iskovskikh and Manin, using certain classical ideas of NN other and Fano, developed a new method of study of birational correspondences between algebraic varieties (which have no nontrivial diierential-geometric birational invariants) | the method of maximal singulari-ties. The results which were obtained by means of this method in 70s were summed up 15 years ago in 24,25]. Since that day, a considerable progress has been made in the eld. It is worth noting that, although we have now new approaches and concepts 34,35], this method up to this day is the most eeective tool in birational geometry. The contemporary state of the theory form the subject of the present survey. 1. The aim of birational geometry is birational classiication of algebraic varieties. In the most general sense, for two given varieties V , V 0 we should be able to say, whether their function elds k(V) and k(V 0) are isomorphic, and if yes, how such Documenta Mathematica Extra Volume ICM 1998 II 97{107