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Abundance conjecture for 3-folds: Case $$\nu =1$$. (English) Zbl 0681.14019
A 3-fold X is minimal if X has at worst terminal singularities and $$K_ X$$ is nef; $$\nu (X)=1$$ means that $$K^ 2_ X\equiv 0$$ (numerical equivalence) but $$K_ X\not\equiv 0$$. The theorem is: let X be a minimal 3-fold over $${\mathbb{C}}$$ with $$\kappa$$ (X)$$\geq 0$$ and $$\nu (X)=1$$; then $$\kappa (X)=1$$, that is, X is a fibre space over a curve with general fibre a surface with $$\kappa =0$$. This generalises the characterisation of elliptic surfaces with $$\kappa =1$$ as minimal surfaces with $$\nu =1$$, which is the fundamental theorem of the classification of surfaces.
The extraordinarily clever proof involves 3 steps. (1) Let E be a connected component of a divisor in $$| mK_ X|$$ and U a tubular neighbourhood of E; then $$\pi_ 1(U\setminus E)$$ has a surjection to $${\mathbb{Z}}$$, so that cyclic covers of U branched along E exist (just as if we already knew that $$U\setminus E$$ were fibred over the punctured disc). - $$(2)\quad Replacing$$ by a suitable cyclic cover and Kulikov minimal model, one can assume that $$E\subset U$$ is a normal crossing divisor with $$K_ E\equiv 0$$, so that by known results on degenerate K3 and abelian surfaces, $$12K_ E=0$$ (analytic equivalence). - $$(3)\quad To$$ prove that $$H^ 0({\mathcal O}_{nE})\to \infty$$ with n, one can compare the divisor nE$$\subset U$$ with the versal deformation of the normal crossing surface E, for which results of R. Friedman [Ann. Math., II. Ser. 118, 75- 114 (1983; Zbl 0569.14002)] are available.
Reviewer: M.Reid

##### MSC:
 14J30 $$3$$-folds 14C20 Divisors, linear systems, invertible sheaves 14J10 Families, moduli, classification: algebraic theory
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##### References:
 [1] R. Friedman : Global smoothings of varieties with normal crossings , Annals of Math. 118 (1983) 75-114. · Zbl 0569.14002 · doi:10.2307/2006955 [2] R. Friedman and D. Morrison : The Birational Geometry of Degenerations . Birkhäuser, Boston- Basel-Stuttgart (1983). · Zbl 0493.00005 [3] Y. Kawamata : Pluricanonical systems on minimal algebraic varieties , Inv. Math. 79 (1985) 567-588. · Zbl 0593.14010 · doi:10.1007/BF01388524 · eudml:143213 [4] G. Kempf , F. Knudsen , D. Mumford and B. Saint-Donat : Toroidal embeddings I , Lect. Notes in Math. 339, Springer, Berlin -Heidelberg -New York (1973). · Zbl 0271.14017 · doi:10.1007/BFb0070318 [5] V.S. Kulikov : Degeneration of K3 and Enriques surfaces , Math. USSR Izvestija 11 (1977) 957-989. · Zbl 0387.14007 · doi:10.1070/IM1977v011n05ABEH001753 [6] J. Milnor : Singular points of complex hypersurfaces , Annals of Math. Studies 61, Princeton Univ. Press, Princeton (1968). · Zbl 0184.48405 · doi:10.1515/9781400881819 [7] Y. Miyaoka : Kodaira dimension of a minimal 3-fold , submitted to Math. Ann. · Zbl 0625.14023 · doi:10.1007/BF01458437 · eudml:164422 [8] S. Mori : Flip theorem and the existence of minimal models for 3-folds , preprint. · Zbl 0649.14023 · doi:10.2307/1990969 [9] U. Persson and H. Pinkham : Degeneration of surfaces with trivial canonical bundle , Ann. of Math. 113 (1981) 45-66. · Zbl 0426.14015 · doi:10.2307/1971133
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