Approximating Zariski decomposition of big line bundles.

*(English)*Zbl 0814.14006A 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.

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.

Reviewer: U.N.Bhosle (Bombay)

##### MSC:

14C20 | Divisors, linear systems, invertible sheaves |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

##### Keywords:

effective divisor; nef divisor; Zariski decomposition of a \(\mathbb{Q}\)- divisor; line bundle
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##### References:

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