Numerical criteria for the positivity of the difference of ample divisors
type: Item
instance of: scholarly article
title: Numerical criteria for the positivity of the difference of ample divisors
zbMATH Open document ID: 0828.14002
DOI: 10.1007/BF02572372
author: Stefano Trapani
published in: Mathematische Zeitschrift
publication date: 1995-12-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174780
review text: Let \(X\) be a compact complex projective manifold of complex dimension \(n\); let \(L\) be a nef divisor, and \(D\) an effective divisor on \(X\). In the paper it is proved that if \(L^n > n (L + b_2 (T) \{u\})^{n - 1} D\) then the Kodaira dimension of \(L - D\) is maximal. (The paper contains is in fact the proof of a slightly more general result.) Here \(T\) denotes the non-negative curvature current of a singular metric of \(D\), \(b_2(T)\) is a non-negative real number which is the largest of the Lelong numbers of \(T\) which are non-zero on a codimension one analytic subvariety of \(X\), whereas the \(1 - 1\) cohomology class \(\{u\}\) measures the ``defect of positivity'' of the tangent bundle of \(X\). [For precise definition see \textit{J. P. Demailly}, J. Algebr. Geom. 1, No. 3, 361-409 (1992; Zbl 0777.32016) and Complex algebraic varieties, Proc. Conf., Bayreuth 1990, Lect. Notes Math. 1507, 87-103 (1992; Zbl 0784.32024).] It is also shown by an example that the class \(\{u\}\) cannot be dropped in general. However, if the complete linear system of some power of \(D\) has no fixed codimension-1 components, then the theorem gives conditions, only in terms of the Chern classes of \(L\), \(D\), to insure that \(L - D\) has maximal Kodaira dimension. By using the theorem one can also obtain inequalities between intersection numbers of nef line bundles, which can be applied to the study of convex compact sets in \(\mathbb{R}^n\), as by \textit{B. Teissier} [in Semin. Differ. Geom., Ann. Math. Stud. 102, 85-105 (1982; Zbl 0494.52009)]. If the tangent bundle of \(X\) is nef (i.e., if we can choose \(\{u\} = 0)\), then by the second paper by \textit{J. P. Demailly} cited above every line bundle with maximal Kodaira dimension is ample; hence, in this case, the above inequality is a sufficient condition for \(L - D\) to be ample.
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 795028
zbMATH Keywords: defect of positivity
zbMATH Keywords: ampleness
zbMATH Keywords: nef divisor
zbMATH Keywords: effective divisor
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: Lelong numbers
zbMATH Keywords: intersection numbers of nef line bundles
MaRDI profile type: Publication
cites work: Q5287410
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description: scientific article; zbMATH DE number 795028