Inequalities of Noether type for 3-folds of general type. (English) Zbl 1079.14046

Chen, Meng

J. Math. Soc. Japan 56, No. 4, 1131-1155 (2004).

The Noether inequality (\(K^2_S\geq 2p_g-4\)) is a powerful instrument in the study of surfaces and the problem of finding the right analogue in dimension three is still open. In this paper the author proves the Noether type inequality \(K^3_X \geq {2\over 3}(p_g(X)-7)\) for \(X\) a smooth threefold with ample canonical class. The proof is based on an accurate study of the canonical map. Moreover the sharper inequality \(K^3_X \geq {2\over 3}(p_g(X)-5)\) is obtained for \(X\) not canonically fibred by surfaces \(S\) with \((c^2_1,p_g) =(1,2)\).

Reviewer: Luciana Picco Botta (Torino)

MSC:

14J30

\(3\)-folds

Keywords:

canonical map