On the birational automorphisms of varieties of general type. (English) Zbl 1281.14036

It is known that varieties of general type have a finite automorphism group [H. Matsumura, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 151–155 (1963; Zbl 0134.16601)]. A celebrated result of Hurwitz bounds the order of the automorphism group of a curve \(C\) of general type by \(84(g(C)-1)\). More recently G. Xiao [J. Algebr. Geom. 4, No. 4, 701–793 (1995; Zbl 0841.14011)] proved an analog of Hurwitz theorem for surfaces of general type. Note that the genus of a smooth curve can be interpreted in terms of volume of the canonical class. It is therefore natural to ask if it is possible to bound the order of the automorpshism group of a variety of general type, say \(X\), with a constant depending only on the dimension and \(\mathrm{vol}(X,K_X)\). This impressive paper answers completely to this question and proves the excistence of a universal constant c, depending only on the dimension, such that for any projective variety of general type \(X\) the dimension of the birational automorphism group has at most \(c\cdot \mathrm{vol}(X,K_X)\) elements. This generalization is highly non trivial and uses all the tools and techniques of Minimal Model Program, in particular there is a very subtle part on Descending Chain Conditions, that in the authors words could provide an affirmative answer to the more general Kollár’s DCC conjectures [J. Kollár, Contemp. Math. 162, 261–275 (1994; Zbl 0860.14014)]. The paper is very well written.


14J50 Automorphisms of surfaces and higher-dimensional varieties
14E30 Minimal model program (Mori theory, extremal rays)
14B05 Singularities in algebraic geometry
Full Text: DOI arXiv


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