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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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