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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 $$p_2>1$$.

##### MSC:
 14J30 $$3$$-folds 14E05 Rational and birational maps
##### Keywords:
plurigenera; birational map; general type
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