Generation by sections and $k$-ampleness

TL;DR

This paper investigates the properties of $k$-ample vector bundles over smooth projective varieties, demonstrating that the generation by global sections is unnecessary for the direct sum and tensor product to be $k$-ample.

Contribution

It proves that the condition of being generated by global sections is not required for the direct sum and tensor product of $k$-ample bundles to be $k$-ample.

Findings

Direct sum of $k$-ample bundles is $k$-ample without the global sections condition.

Tensor product of $k$-ample bundles is $k$-ample without the global sections condition.

Extends previous results by removing the generation condition.

Abstract

In the article "Submanifold of abelian varieties", A.J. Sommese proved that direct sum and tensor product of two vector bundles $E$ and $F$ over a smooth projective variety are $k$-ample if $E$ and $F$ are $k$-ample and are generated by global sections. Here we show that the latter condition is not needed.