Vector bundles on toric varieties

TL;DR

This paper investigates the existence of nontrivial vector bundles on toric varieties, exploring conewise linear functions and K-theory computations, but contains a correction regarding the link between K-groups and vector bundles.

Contribution

It introduces conewise linear multivalued functions on complete fans and analyzes K-theory to study vector bundles on toric 3-folds, with a correction on previous assumptions.

Findings

Every complete fan admits a nontrivial conewise linear multivalued function.

The Grothendieck group of certain toric 3-folds is large, indicating nontrivial vector bundles.

Either a toric 3-fold has a nontrivial line bundle or a finite cover with large Grothendieck group exists.

Abstract

CORRECTION. One of the main results in this paper contains a fatal error. We cannot conclude the existence of nontrivial vector bundles on X from the nontriviality of its K-group. The K-group that is computed here is the Grothendieck group of perfect complexes and not vector bundles. Since the varieties are not quasi-projective, existence of nontrivial perfect complexes says nothing about the existence of nontrivial vector bundles. We thank Sam Payne for drawing our attention to the error and Christian Haesemeyer for explanations about the K-theory. Abstract: Following Sam Payne's work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Corti\~nas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group.