Vector bundles on toric varieties

Saman Gharib; Kalle Karu

doi:10.1016/j.crma.2011.12.013

Geometry

Vector bundles on toric varieties

Presented by: Michèle Vergne

Saman Gharib ¹; Kalle Karu ¹

¹ Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 209-212.

Abstract
Résumé

Following Sam Payneʼs 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 Cortiñas, 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.

Suivant un travail de Sam Payne nous étudions lʼexistence de fibrés vectoriels non triviaux sur une variété torique. Notre premier résultat établit que tout éventail complet admet une fonction, non triviale, qui est linéaire et multi-valuée sur chaque cône. Une telle fonction peut potentiellement être la classe de Chern dʼun fibré vectoriel torique. Nous utilisons alors un résultat de Cortiñas, Haesemeyer, Walker et Weibel pour montrer que le groupe de Grothendieck (non équivariant) de la variété torique de dimension 3 étudiée par Payne est grand et ainsi la variété a un fibré vectoriel non trivial. Par un calcul similaire nous montrons que pour toute variété torique X de dimension 3, soit X a un fibré en droites non trivial, soit il existe un morphisme torique, surjectif, fini de Y sur X, où Y a un grand groupe de Grothendieck.

Received: 2011-11-29
Accepted: 2011-12-22
Published online: 2012-02-01

DOI: 10.1016/j.crma.2011.12.013

Author's affiliations:

Saman Gharib ¹; Kalle Karu ¹

¹ Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

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Saman Gharib; Kalle Karu. Vector bundles on toric varieties. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 209-212. doi : 10.1016/j.crma.2011.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.12.013/

References
Cited by

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Cited by Sources:

^☆ This work was partially supported by NSERC Discovery and USRA grants.

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