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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:
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