The Noether inequality for algebraic 3 -folds

Abstract

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