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

TL;DR

This paper establishes the optimal Noether-Severi inequality for irregular threefolds of general type, characterizes the equality case, and explores related topological and algebraic properties.

Contribution

It proves the sharp inequality for irregular threefolds, describes the canonical models at equality, and links the volume to the fundamental group and continuous rank.

Findings

Proved the inequality: volume ≥ (4/3) χ(ω_X) for all such threefolds.

Characterized the threefolds attaining equality, showing their fundamental group is Z^2.

Established a new inequality involving volume and the continuous rank of K_X.

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 $\mathrm{vol}(X) \ge \frac{4}{3}h^0_a(X, K_X)$ where $h^0_a(X, K_X)$ stands for the continuous rank of $K_X$, and we show that $X$ attains this equality if and only if $\mathrm{vol}(X) = \frac{4}{3}\chi(\omega_{X})$.