Birational automorphisms of multidimensional algebraic manifolds
This survey summarizes the results of investigations during the two last decades in the field of birational automorphism groups of Fano manifolds. This subject, which is traditional for the Moscow school of algebraic geometry (and has been under active study only here in the second half of this century), has its origin in the works of the eminent Italian geometrician G. Fano. He proposed several very fruitful concepts but did not present them in a complete form: He did not prove any of his correctly guessed theorems. His reasoning has significant gaps and outright mistakes. Thus, the modern stage in multidimensional birational geometry begins with the correction of the latter. In 1971, \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin} published their work: Math. USSR, Sb. 15(1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86 (128), 140-166 (1971; Zbl 0222.14002), where the Fano theorem about the rigidity of smooth 3-dimensional quartics was proved. The maximal singularities method, the theory proposed in this work, appeared to be very effective for the study of the birational correspondences of multidimensional algebraic manifolds with negative canonical sheaf (of Fano manifolds). In the subsequent works by \textit{V. A. Iskovskikh} [Sov. Math., Dokl. 18, 748-750 (1977); translation from Dokl. Akad. Nauk SSSR 234, 743-745 (1977; Zbl 0406.14004), Sov. Math., Dokl. 18(1977), 948-951 (1978); translation from Dokl. Akad. Nauk SSSR 235, 509-511 (1977; Zbl 0414.14025)], birational automorphism groups were described and a birational type of several series of Fano 3-folds was found: double spaces and double quadrics of index 1, the double Veronese cone of index 2, the complete intersection of a quadric and a cubic in \(\mathbb{P}^5\). The success of the maximal singularities method in 3-dimensional birational geometry posed the question on its generalization to the multidimensional case and to the case of singular Fano manifolds. Such a generalization was found to be possible and effective: In Vestn. Mosk. Univ., Ser. I 1986, No. 2, 10-15 (1986; Zbl 0604.14026) and Invent. Math. 87, 303-329 (1987; Zbl 0613.14011), \textit{A. V. Pukhlikov} proved a theorem on the birational rigidity of smooth 4-dimensional quintics (i.e., on the coincidence of groups of birational and biregular automorphisms and the theorem on the absence of a fibre structure on Fano manifolds of lower dimension). \textit{A. V. Pukhlikov} [Math. USSSR, Ser. 63, No. 2, 457-482 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 4, 472-496 (1988; Zbl 0655.14006) and Math. USSSR, Izv. 32, No. 1, 233-243 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 229-239 (1988; Zbl 0704.14008)] used the maximal singularities method for the study of manifolds of arbitrary dimension and manifolds with sipmle singularities. The aim of this survey is to give a complete and systematic exposition of the mentioned works. The structure of these work under review is as follows. In the first chapter, we expound the ``general theory'' of the maximal singularities method. In the second chapter, we study smooth Fano manifolds of degree not exceeding 4. In the third chapter, birational automorphisms of a smooth complete intersection of a quadric and a cubic in \(\mathbb{P}^5\) are described. In the fourth chapter, birational automorphisms of smooth 4-dimensional quintics are described. The fifth chapter contains a description of the birational correspondence of 3-dimensional quartics with a double point of a general structure.
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