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


Authors: Markus Perling and Stefan Schröer
Journal: Trans. Amer. Math. Soc. 369 (2017), 4787-4815
MSC (2010): Primary 14J60, 14M25
DOI: https://doi.org/10.1090/tran/6813
Published electronically: November 28, 2016
MathSciNet review: 3632550
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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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Additional Information

Markus Perling
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, 33501 Bielefeld, Germany
Email: perling@math.uni-bielefeld.de

Stefan Schröer
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, 40204 Düsseldorf, Germany
Email: schroeer@math.uni-duesseldorf.de

DOI: https://doi.org/10.1090/tran/6813
Received by editor(s): November 4, 2014
Received by editor(s) in revised form: July 16, 2015
Published electronically: November 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society