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

We prove the optimal Noether-Severi inequality that vol(X)≥43χ(ωX) for all smooth and irregular 3-folds X of general type over C. For those 3-folds X attaining the equality, we completely describe their canonical models and show that the topological fundamental group π1(X)≃ℤ2. As a corollary, we obtain for the same X another optimal inequality that vol(X)≥43ha0(X,KX) where ha0(X,KX) stands for the continuous rank of KX, and we show that X attains this equality if and only if vol(X)=43χ(ωX).