The non-vanishing theorem. (English. Russian original) Zbl 0605.14006

Shokurov, V. V.

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)

Keywords:

Mori extremal rays; nef Cartier divisor; non-vanishing theorem