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.


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