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Remarks on the abundance conjecture. (English) Zbl 1365.14019
Let $$\pi :X\to U$$ be a projective morphism of varieties over $$\mathbb C$$ and $$(X,\Delta )$$ a log canonical pair. The abundance conjecture says that if $$K_X+\Delta$$ is $$\pi$$-nef (so that $$(K_X+\Delta )\cdot C\geq 0$$ for any curve $$C$$ contained in fibers of $$\pi$$), then $$K_X+\Delta$$ is $$\pi$$-semiample i.e. there exists a morphism $$f:X\to Y$$ over $$U$$ such that $$K_X+\Delta \sim _{\mathbb R }f^* D$$ for some $$\mathbb R$$-divisor $$D$$ which is ample over $$U$$. This is one of the most important conjectures in higher dimensional birational geometry. It is known to hold if $$\dim X=3$$.
The main result of the paper under review is to show that the abundance conjecture holds for $$n$$-dimensional varieties $$X$$ such that $$K_X+\Delta$$ is $$\pi$$-big assuming that it holds in full generality for $$(n-1)$$-dimensional varieties. In particular the abundance conjecture holds for $$4$$-folds $$X$$ such that $$K_X+\Delta$$ is $$\pi$$-big. Recall that $$K_X+\Delta$$ is $$\pi$$-big if it is $$\mathbb R$$-linearly equivalent to the sum of a $$\pi$$-ample $$\mathbb R$$-divisor and an effective $$\mathbb R$$-divisor.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J35 $$4$$-folds
##### Keywords:
abundance conjecture
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