Asymptotic behaviour of numerical invariants of algebraic varieties
When classifying projective varieties, a first approach is to fix certain numerical invariants and then study the set of varieties that have these numerical invariants. Typical such invariants are Chern and Hodge numbers. In this paper, the author addresses the following three questions, for a given numerical invariant: 1) Find upper bounds for the invariant in terms of dimension, codimension and degree of the variety; 2) describe the varieties for which the invariant attains maximal value; 3) study the relations between various numerical invariants. In the case of curves, Castelnuovo answered 1) and 2) for the genus, and this was generalized to higher dimensional varieties for the geometric genus by \textit{J. Harris} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 35--68 (1981; Zbl 0467.14005)]. For 3), work was done by \textit{U. Persson} [in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, 195--218 (1987; Zbl 0656.14020)] on the ``geography'' of surfaces. In the present paper the author solves what he calls the ``asymptotic version'' of the above three questions, with respect to Chern and Hodge numbers. More precisely, he shows that for varieties of high degree, those for which any one of the most important invariants -- such as class, Betti number, Chern number, middle Hodge number -- is maximal, leads to the same class of extremal varieties. Typically, as in Castelnuovo's case of curves, these varieties are hypersurfaces in a variety of minimal degree in the ambient projective space. The results in this paper builds on, and extends, some of those in [\textit{F. L. Zak}, Castelnuovo bounds for higher dimensional varieties. Compos. Math. 148 (2012) (to appear)].
scientific article; zbMATH DE number 5995803
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