Positivity properties of toric vector bundles

Milena Hering[1]; Mircea Mustaţă[2]; Sam Payne[3]

  • [1] University of Connecticut Department of Mathematics 196 Auditorium Road U-3009 Storrs CT 06269-3009 (USA)
  • [2] University of Michigan Department of Mathematics East Hall Ann Arbor, MI 48109 (USA)
  • [3] Stanford University Department of Mathematics Bldg 380 Stanford, CA 94305 (USA)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 2, page 607-640
  • ISSN: 0373-0956

Abstract

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We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles ℳ L that arise as the kernel of the evaluation map H 0 ( X , L ) ⊗ 𝒪 X → L , for ample line bundles L . We give examples of twists of such bundles that are ample but not globally generated.

How to cite

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Hering, Milena, Mustaţă, Mircea, and Payne, Sam. "Positivity properties of toric vector bundles." Annales de l’institut Fourier 60.2 (2010): 607-640. <http://eudml.org/doc/116283>.

@article{Hering2010,
abstract = {We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles $\mathcal\{M\}_L$ that arise as the kernel of the evaluation map $H^0(X,L) \otimes \mathcal\{O\}_X \rightarrow L$, for ample line bundles $L$. We give examples of twists of such bundles that are ample but not globally generated.},
affiliation = {University of Connecticut Department of Mathematics 196 Auditorium Road U-3009 Storrs CT 06269-3009 (USA); University of Michigan Department of Mathematics East Hall Ann Arbor, MI 48109 (USA); Stanford University Department of Mathematics Bldg 380 Stanford, CA 94305 (USA)},
author = {Hering, Milena, Mustaţă, Mircea, Payne, Sam},
journal = {Annales de l’institut Fourier},
keywords = {Toric variety; toric vector bundle; toric varieties},
language = {eng},
number = {2},
pages = {607-640},
publisher = {Association des Annales de l’institut Fourier},
title = {Positivity properties of toric vector bundles},
url = {http://eudml.org/doc/116283},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Hering, Milena
AU - Mustaţă, Mircea
AU - Payne, Sam
TI - Positivity properties of toric vector bundles
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 2
SP - 607
EP - 640
AB - We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles $\mathcal{M}_L$ that arise as the kernel of the evaluation map $H^0(X,L) \otimes \mathcal{O}_X \rightarrow L$, for ample line bundles $L$. We give examples of twists of such bundles that are ample but not globally generated.
LA - eng
KW - Toric variety; toric vector bundle; toric varieties
UR - http://eudml.org/doc/116283
ER -

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