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Complex projective 3-fold with non-negative canonical Euler-Poincaré characteristic. (English) Zbl 1149.14034
Let \(X\) be a smooth variety of general type. Then for suitable \(m\) depending only on the dimension of \(X\) [see C. D. Hacon, J. McKernan, Invent. Math. 166, No. 1, 1–25 (2006; Zbl 1121.14011) and S. Takayama, Invent. Math. 165, No. 3, 551–587 (2006; Zbl 1108.14031)], the map associated to \(| mK_X| \) is birational. The optimal value of such an \(m\) is known only for surfaces [E. Bombieri, Inst. Hautes Études Sci. Publ. Math. No. 42, 171–219 (1972; Zbl 0259.14005)]. The paper under review studies the problem for 3-folds under assumptions on the Euler characteristic. The main result states that \(m=14\), respectively \(m=8\), for smooth 3-folds with non negative, respectively positive, canonical Euler characteristic. The proof relies on a subtle technical analysis of minimal 3-folds of general type with \(p_g=1\) and \(p_2>1\).

14J30 \(3\)-folds
14E05 Rational and birational maps
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