Numerical criteria for the positivity of the difference of ample divisors
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.
intersection numbers of nef line bundles
scientific article; zbMATH DE number 795028