On the geography of threefolds. (With an appendix by Mei-Chu Chang)
The term geography is used to study the distribution of Chern numbers of an algebraic manifold of general type. Such term was introduced by \textit{U. Persson} [Compos. Math. 43, 3-58 (1981; Zbl 0479.14018)], where he was concerned with the surfaces case. For a 3-fold of general type, since the minimal model may not be unique and may not be smooth, to well pose the geography problem one has to fix a unique model in each birational equivalence class. In the paper under review the author considers certain 3-folds of general type with some smooth minimal model and pairs \(({c_1^3\over c_1c_2}, {c_3\over c_1c_2})\) instead of triples \((c_1^3, c_1c_2, c_3)\). This was Sommese's idea in the surface case [\textit{A. J. Sommese}, Math. Ann. 268, 207-221 (1984; Zbl 0521.14016)] where he considered the ratio \({c_1^2}\over{c_2}\) instead of the pair \((c_1^2, c_2)\). Then the question is whether for any such pair, in a suitable region, there exists a 3-fold of general type with those Chern numbers. The author uses double covers of \(\mathbb{P}^1\times F_n\) branched along some special chosen configuration of divisors to construct some smooth minimal 3-folds of general type, where \(F_n\) is the Segre-Hirzebruch rational surface. In the appendix, \textit{Mei-Chu Chang} gives a ``global'' statement of Liu's theorem.
scientific article; zbMATH DE number 1036422