Inequalities of Noether Type for 3-folds of General Type

If X is a smooth complex projective 3-fold with ample canonical divisor K, then the inequality K ≥ 2 3 (2pg − 7) holds, where pg denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type. Introduction Given a minimal surface S of general type, we have two famous inequalities, which play crucial roles in detailed analysis of surfaces. One is the BogomolovMiyaoka-Yau inequality K S ≤ 9χ(S) ([M1], [Y1], [Y2]), while the other is the classical Noether inequality K S ≥ 2pg − 4 ≥ 2χ(X)− 6. The fundamental importance of these inequalities in mind, M. Reid asked in 1980s Question 1. What would be the right analogue of the Noether inequality in dimension three? Let X be a minimal threefold. If KX is Cartier and very ample, then K 3 X ≥ 2pg−6 by Clifford’s theorem applied to the intersection curve cut out by two general members of |KX |. In 1992, Kobayashi [Kob] studied Gorenstein canonical 3-folds and obtained an effective, but partial, upper bound ofK X in terms of pg(X) for such varieties. One of his discoveries is that too naive a generalization of the classical Noether inequality is in general false; there are a series of smooth projective 3-folds X with ample canonical divisor such that K X = 2 3 (2pg(X)− 5), (pg(X) = 7, 10, 13, · · · ). (0.1) In what follows, we show that Kobayashi’s examples indeed attain the minima of K X , provided X is smooth and KX is ample: Corollary 2. If X is a smooth complex projective 3-fold with ample canonical divisor. Then K X ≥ 2 3 (2pg(X)− 7). When X is not necessarily smooth, we have the following 2000 Mathematics Subject Classification. Primary 14J30