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An approach to the abundance conjecture for 3-folds. (English) Zbl 0722.14026
The author approaches the following question: Is every minimal model good? (“abundance conjecture”).
To be more precise, let X be a projective minimal n-fold (i.e. a projective n-fold over \({\mathbb{C}}\) having only terminal singularities and nef canonical divisor \(K_ X)\) then the numerical Kodaira dimension \(\nu\) (X) of X is by definition the largest integer a with \(K^ a_ X\cdot H^{n-a}\neq 0\) (H is an ample divisor on X). In general one has \(n\geq \nu (X)\geq \kappa (X)\) and X is called “good” if Kodaira dimension and numerical Kodaira dimension are equal. Equivalently, X is good if some multiple of \(K_ X\) is generated by global sections [see Y. Kawamata, Invent. Math. 79, 567-588 (1985; Zbl 0593.14010)]. The abundance conjecture states that every minimal model has abundant canonical class, which means that some multiple of \(K_ X\) is globally generated. The present paper deals with the case \(n=3\), where the minimal model conjecture is known to be true [S. Mori, J. Am. Math. Soc. 1, No.1, 117-253 (1988; Zbl 0649.14023)].
If X is as above and of dimension three then:
- a theorem of Miyaoka states that \(\kappa\) (X)\(\geq 0\). This gives the abundance conjecture in the case \(\nu (X)=0,\)
- the case \(\nu (X)=1\) is treated by Y. Miyaoka [Compos. Math. 68, No.2, 203-220 (1988; Zbl 0681.14019)],
- one easily deduces from the Kawamata-Viehweg vanishing theorem [cf. Y. Kawamata, Math. Ann. 261, 43-46 (1982; Zbl 0476.14007) and E. Viehweg, J. Reine Angew. Math. 335, 1-8 (1982; Zbl 0485.32019)], that in case \(\nu (X)=3\) the abundance conjecture holds.
The author investigates the remaining case \(\nu (X)=2\) and proves the following theorem: If X is a projective minimal Gorenstein 3-fold such that there exists a multicanonical divisor having a component being not birationally equivalent to a ruled surface, then the abundance conjecture holds for X.

MSC:
14J30 \(3\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
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