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

We establish the Noether inequality for projective 3-folds. More precisely, we prove that the inequality vol(X)≥43p_g(X)-103 holds for all projective 3-folds X of general type with either p_g(X)≤ 4 or p_g(X)≥ 21, where p_g(X) is the geometric genus and vol(X) is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.