Minimal 3-folds of small slope and the Noether inequality for canonically polarized 3-folds

Assume that $X$ is a smooth projective 3-fold with ample $K_X$. We study a problem of Miles Reid to prove the inequality $$K_X^3\ge \frac{2}{3}(2p_g(X)-5),$$ where $p_g(X)$ is the geometric genus. This inequality is sharp according to known examples of M. Kobayashi. We also birationally classify arbitrary minimal 3-folds of general type with small slope.

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Published 1 January 2004