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The non-vanishing theorem. (English. Russian original) Zbl 0605.14006
Math. USSR, Izv. 26, 591-604 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 635-651 (1985).
The main result is the following non-vanishing theorem. Let X be a variety with routine $$(=$$ log-terminal) singularities, D a nef (numerically effective) Cartier divisor and $$A=\sum d_ iD_ i^ a$$ $${\mathbb{Q}}$$-Cartier divisor on X. Suppose that the following conditions hold: (a) The $${\mathbb{Q}}$$-divisor $$aD+A-K_ X$$ is nef and big for some $$a\in {\mathbb{Q}}$$. (b) The $$D_ i$$ are prime divisors on X, and are nonsingular, have normal crossings, and lie in the nonsingular part of X if $$d_ i<0$$. (c) Each $$d_ i>-1$$. then for all $$b\gg 0:$$ $$H^ 0(X,{\mathcal O}_ X(bD+\ulcorner A\urcorner))\neq 0$$, or, in other words, $$| bD+\ulcorner A\urcorner | \neq \emptyset$$, where for $$x\in {\mathbb{R}}$$, ”$$\ulcorner x\urcorner$$” means the smallest integer $$\geq x$$, and for a divisor $$D=\sum d_ iF_ i:\ulcorner D\urcorner =\sum \ulcorner d_ i\urcorner F_ i.$$
This theorem is motivated by and used in the extremological program of Miles Reid for the construction of minimal models.
Reviewer: V.Iliev

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14F25 Classical real and complex (co)homology in algebraic geometry 14E30 Minimal model program (Mori theory, extremal rays)
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