The sharp lower bound for the volume of 3-folds of general type with \chi(\Co{X})=1

TL;DR

This paper establishes the exact minimal volume for smooth projective 3-folds of general type with Euler characteristic one, proving a sharp lower bound of 1/420 and providing an explicit example attaining it.

Contribution

It proves the sharp lower bound for the canonical volume of certain 3-folds of general type and constructs an example achieving this bound.

Findings

The minimal volume for the class is 1/420.

An explicit example of a 3-fold with volume 1/420 is provided.

The bound is proven to be sharp.

Abstract

Let $V$ be a smooth projective 3-fold of general type. Denote by $K^{3}$, a rational number, the self-intersection of the canonical sheaf of any minimal model of $V$. One defines $K^{3}$ as a canonical volume of $V$. The paper is devoted to proving the sharp lower bound $K^{3}\ge {1/420}$ which can be reached by an example: $X_{46}\subseteq \mathbb{P}(4,5,6,7,23)$.