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

Perling, Markus; Schröer, Stefan

doi:10.1090/tran/6813

Vector bundles on proper toric 3-folds and certain other schemes
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by Markus Perling and Stefan Schröer PDF

Trans. Amer. Math. Soc. 369 (2017), 4787-4815 Request permission

Abstract:

We show that a proper algebraic $n$-dimensional scheme $Y$ admits non-trivial 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.

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