The Noether inequality for algebraic threefolds (With an Appendix by J\'{a}nos Koll\'{a}r)

TL;DR

This paper proves an optimal inequality relating volume and geometric genus for certain algebraic threefolds, extending the understanding of their classification and geometric properties.

Contribution

It establishes the Noether inequality for projective threefolds of general type with specific geometric genus ranges, filling a gap in algebraic geometry.

Findings

Proves the inequality ${ m vol}(X) \\geq \\frac{4}{3}p_g(X)-\\frac{10}{3}$ for certain threefolds.

Shows the inequality is optimal based on known examples.

Extends the classification theory of algebraic threefolds.

Abstract

We establish the Noether inequality for projective $3$-folds. More precisely, we prove that the inequality $${\rm vol}(X)\geq \tfrac{4}{3}p_g(X)-{\tfrac{10}{3}}$$ holds for all projective $3$-folds $X$ of general type with either $p_g(X)\leq 4$ or $p_g(X)\geq 21$, where $p_g(X)$ is the geometric genus and ${\rm vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.