Inequality of Noether Type for Gorenstein Minimal 3-folds of General Type

TL;DR

This paper establishes an optimal inequality relating the canonical volume and Euler characteristic for Gorenstein minimal 3-folds of general type, extending Noether-type inequalities to higher dimensions.

Contribution

It proves a new optimal inequality for Gorenstein minimal 3-folds of general type, linking their canonical volume and Euler characteristic.

Findings

Proves the inequality: $K_X^{3} \,\geq\, \frac{4}{3}\chi(\omega_X) - 2$.

Establishes the inequality as optimal for this class of 3-folds.

Extends classical inequalities to higher-dimensional algebraic varieties.

Abstract

Let $X$ be a Gorenstein minimal $3$-fold of general type. We prove the optimal inequality: $$K_X^{3}\geq \frac{4}{3}\chi(\omega_X)-2,$$ where $\chi(\omega_X)$ is the Euler-Poincar$\acute{\text{e}}$ characteristic of the dualizing sheaf $\o_{X}$.