Noether-Severi inequality and equality for irregular threefolds of general type (Q2671935)

The classical Severi inequality says that for a smooth projective irregular surface \(S\) of general type, \[ \text{vol}(S)\geq 2\chi(\omega_S). \] Here \(\text{vol}\) means the canoincal volume, i.e., the volume of the canonical divisor. The main result of this paper is a generalization of this result to \(3\)-folds, namely, for a smooth projective irregular \(3\)-fold \(X\) of general type, \[ \text{vol}(X)\geq \frac{4}{3}\chi(\omega_X). \] It also shows some geometric properties of \(X\) on which the equality holds. In particular, the most interesting property is that when the equality holds, then all minimal models of \(X\) are Gorenstein. Such inequality is optimal as there are many examples for which the equality holds. The second result of this paper gives a complete description of the canonical models of those \(X\) when the equality holds, roughly speaking, they are divisorial contractions of a double cover over a two-tower \(\mathbb{P}^1\)-bundle over an elliptic curve.