Vector bundles on proper toric 3-folds and certain other schemes

TL;DR

The paper demonstrates the existence of nontrivial vector bundles of rank n on certain proper algebraic schemes, including all proper toric threefolds, even when these schemes are non-projective, with some bundles having arbitrarily large top Chern number.

Contribution

It establishes conditions under which proper algebraic schemes, especially toric varieties, admit nontrivial vector bundles of rank equal to their dimension, expanding understanding beyond projective cases.

Findings

Proper toric threefolds admit nontrivial rank three vector bundles.

Existence of vector bundles with arbitrarily large top Chern number.

Applicable to higher-dimensional toric varieties with specific convexity properties.

Abstract

We show that a proper algebraic n-dimensional scheme Y admits nontrivial vector bundles of rank n, even if Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional locus in only finitely many points. Moreover, there are such vector bundles with arbitrarily large top Chern number. Applying this to toric varieties, we infer that every proper toric threefold admits such vector bundles of rank three. Furthermore, we describe a class of higher-dimensional toric varieties for which the result applies, in terms of convexity properties around rays.