Global Generation of Adjoint Line Bundles on Projective $5$-folds

TL;DR

This paper proves that under certain positivity conditions on an ample line bundle on a smooth projective 5-fold, the adjoint bundle is globally generated, extending the understanding of line bundle generation in higher dimensions.

Contribution

It establishes new criteria for the global generation of adjoint line bundles on 5-dimensional smooth projective varieties with specific intersection properties.

Findings

Global generation of adjoint bundles under given conditions

Extension of known results to 5-folds

Explicit numerical bounds for ampleness and intersection numbers

Abstract

Let $X$ be a smooth projective variety of dimension $5$ and $L$ be an ample line bundle on $X$ such that $L^5>7^5$ and $L^d\cdot Z\geq 7^d$ for any subvariety $Z$ of dimension $1\leq d\leq 4$. We show that $\mathcal{O}_X(K_X+L)$ is globally generated.