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

Abstract

Abstract We prove the optimal Noether–Severi inequality that vol ⁡ ( X ) ≥ 4 3 ⁢ χ ⁢ ( ω X ) {\operatorname{vol}(X)\geq\frac{4}{3}\chi(\omega_{X})} for all smooth and irregular 3-folds X of general type over ℂ {\mathbb{C}} . For those 3-folds X attaining the equality, we completely describe their canonical models and show that the topological fundamental group π 1 ⁢ ( X ) ≃ ℤ 2 {\pi_{1}(X)\simeq\mathbb{Z}^{2}} . As a corollary, we obtain for the same X another optimal inequality that vol ⁡ ( X ) ≥ 4 3 ⁢ h a 0 ⁢ ( X , K X ) {\operatorname{vol}(X)\geq\frac{4}{3}h^{0}_{a}(X,K_{X})} where h a 0 ⁢ ( X , K X ) {h^{0}_{a}(X,K_{X})} stands for the continuous rank of K X {K_{X}} , and we show that X attains this equality if and only if vol ⁡ ( X ) = 4 3 ⁢ χ ⁢ ( ω X ) {\operatorname{vol}(X)=\frac{4}{3}\chi(\omega_{X})} .