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Approximating Zariski decomposition of big line bundles. (English) Zbl 0814.14006
A Zariski decomposition of a $$\mathbb{Q}$$-divisor $$D$$ is a decomposition $$D {\overset {\textstyle\sim} \approx b} N + E$$ where $$N$$ is a nef $$\mathbb{Q}$$-divisor and $$E$$ is an effective $$\mathbb{Q}$$-divisor with negative definite intersection matrix such that $$N.E_ i = 0$$ for all components $$E_ i$$, of $$E$$. Existence of a Zariski decomposition is not known in general (except for surfaces). The author proves the following theorem which may be regarded as an approximate Zariski decomposition.
Let $$L$$ be a line bundle on a variety $$V$$ of dimension $$n$$. Suppose that $h^ \circ (V,tL) \geq dt^ n/n! + \varphi(t)$ for infinitely many positive integers $$t$$, $$\varphi(t) =$$ a function such that $$\varphi(t)/t^ n \to 0$$ as $$n \to \infty$$ and $$d$$ is a positive real number. Then for any $$\varepsilon > 0$$, there exists a birational morphism $$f : M \to V$$, together with an effective $$\mathbb{Q}$$-divisor $$E$$ on $$M$$ such that $$f^* L = H + E$$, $$H$$ a semiample $$\mathbb{Q}$$-bundle with $$H^ n > d - \varepsilon$$.
Note that if $$H^ \circ(tf^*L) = H^ \circ(tH)$$ for all $$t> 0$$ such that $$tE$$ is a $$\mathbb{Z}$$-divisor, then $$f^*L = H + E$$ is a Zariski decomposition of $$L$$ and one has $$H^ \circ(V,tL) = dt^ n/n! +$$ (lower order terms), $$d = H^ n$$. The theorem (coupled with other techniques) has applications to inequalities on intersection numbers of line bundles, no applications are discussed here.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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##### References:
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