The Noether inequality for algebraic threefolds (With an Appendix by János Kollár)

Abstract

We establish the Noether inequality for projective $3$-folds. More precisely, we prove that the inequality $$\mathrm{vol}(X)\ge \frac{4}{3}{p}_g(X)-\frac{10}{3}$$ holds for all projective $3$-folds $X$ of general type with either $p_g(X)\le 4$ or $p_g(X)\ge 21$, where $p_g(X)$ is the geometric genus and $\mathrm{vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.

v1: 48 pages, comments are welcome; v2: defn. 2.4 corrected, minor changes made to the proofs of 5.9, 5.10; v3: 34 pages, original sect. 3 and appendix were replaced by an appendix by János Kollár, which simplified the proof and improved the main result; v4: 36 pages, final version to appear in Duke Math. J