Noether-Severi inequality and equality for irregular threefolds of general type

Abstract

We prove the optimal Noether-Severi inequality that $\mathrm{vol}(X)\ge \frac{4}{3}\chi ({\omega }_X)$ for all smooth and irregular $3$-folds $X$ of general type over $\mathbb{C}$. For those $3$-folds $X$ attaining the equality, we completely describe their canonical models and show that the topological fundamental group ${\pi }_1(X)\simeq {\mathbb{Z}}^2$. As a corollary, we obtain for the same $X$ another optimal inequality that $\mathrm{vol}(X)\ge \frac{4}{3}{h}_a^0(X,K_X)$ where $h_a^0(X,K_X)$ stands for the continuous rank of $K_X$, and we show that $X$ attains this equality if and only if $\mathrm{vol}(X)=\frac{4}{3}\chi ({\omega }_X)$.

v3: Minor revision