Vector bundles on toric varieties

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.Comment: There is an error in one of the main conclusions of the paper. Please see the abstract for more detail

Vector bundles on toric varieties ✩ Fibrés vectoriels sur les variétés toriques

Presented by Michèle Vergne 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 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. r é s u m é Suivant un travail de Sam Payne nous étudions l'existence 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 d'un 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