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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.

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
Full Text:
DOI

### References:

[1] | J. P. DEMAILLY, Singular hermitian metrics on positive line bundles, Complex Algebraic Varieties; Proc. Bayreuth 1990, Lecture Notes in Math., 1507, Spr-inger, 1992, pp. 87-104. · Zbl 0784.32024 |

[2] | J. P. DEMAILLY, A numerical criterion for very ample line bundles, J. of Diff Geom., 37 (1993), 323-374. · Zbl 0783.32013 |

[3] | T. FUJITA, Remarks on quasi-polarized varieties, Nagoya Math. J., 115 (1989), 105-123 · Zbl 0699.14002 |

[4] | H. TSUJI, Analytic Zariski decomposition I, Existence, · Zbl 0786.14005 · doi:10.3792/pjaa.68.161 |

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