On simple and stable homogeneous bundles

Simona Faini

Bollettino dell'Unione Matematica Italiana (2006)

  • Volume: 9-B, Issue: 1, page 51-67
  • ISSN: 0392-4041

Abstract

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In this work we will analyze the relation between the stability and the simplicity of a homogeneous vector bundle on a projective variety. Our main theorem shows that a homogeneous bundle is not destabilized by its homogeneous subbundles if and only if it is the tensor product of a stable homogeneous bundle and an irreducible representation. Then we give an example of a homogeneous bundle, which is simple, but not stable.

How to cite

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Faini, Simona. "On simple and stable homogeneous bundles." Bollettino dell'Unione Matematica Italiana 9-B.1 (2006): 51-67. <http://eudml.org/doc/289620>.

@article{Faini2006,
abstract = {In this work we will analyze the relation between the stability and the simplicity of a homogeneous vector bundle on a projective variety. Our main theorem shows that a homogeneous bundle is not destabilized by its homogeneous subbundles if and only if it is the tensor product of a stable homogeneous bundle and an irreducible representation. Then we give an example of a homogeneous bundle, which is simple, but not stable.},
author = {Faini, Simona},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {51-67},
publisher = {Unione Matematica Italiana},
title = {On simple and stable homogeneous bundles},
url = {http://eudml.org/doc/289620},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Faini, Simona
TI - On simple and stable homogeneous bundles
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/2//
PB - Unione Matematica Italiana
VL - 9-B
IS - 1
SP - 51
EP - 67
AB - In this work we will analyze the relation between the stability and the simplicity of a homogeneous vector bundle on a projective variety. Our main theorem shows that a homogeneous bundle is not destabilized by its homogeneous subbundles if and only if it is the tensor product of a stable homogeneous bundle and an irreducible representation. Then we give an example of a homogeneous bundle, which is simple, but not stable.
LA - eng
UR - http://eudml.org/doc/289620
ER -

References

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  1. ANCONA, V., Fibrati vettoriali su varieta razionali omogenee, Boll. UMI, 7 (1988), 299-317. 
  2. ATIYAH, M., On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France, 84 (1956), 306-317. Zbl0072.18101MR86358
  3. FAINI, S., Stabilità e semplicità dei fibrati omogenei su ℂ ⁢ ℙ 2 , tesi di laurea, Firenze, 2002. 
  4. MIGLIORINI, L., Stability of homogeneous vector bundles, Boll. UMI, Ser. VII, B 10, nr. 4 (1996), 963-990. Zbl0885.14024MR1430162
  5. OKONEK, C. - SCHNEIDER, M. - SPINDLER, H., Vector bundles on complex projective spaces, Birkhäuser, Boston, 1980. Zbl0438.32016MR561910
  6. OTTAVIANI, G., Rational homogeneous varieties, notes for the SMI-course in Cortona,1995 (available at http://www.math.unifi.it/~ottavian). 
  7. PAOLETTI, R., Stability of a class of homogeneous vector bundles on ℂ ⁢ ℙ n , Boll. UMI,Ser. VII, A 9, nr. 2 (1995), 329-343. Zbl0888.14006MR1336240
  8. RAMANAN, S., Holomorphic vector bundles on homogeneous spaces, Topology, Vol. 5, pp. 159-177, Pergamon Press, 1966. Zbl0138.18602MR190947DOI10.1016/0040-9383(66)90017-6
  9. ROHMFELD, R., Stability of homogeneous vector bundles on ℂ ⁢ ℙ n , GeometriaeDedicata, 38 (1991), 159-166. Zbl0734.14004MR1104341DOI10.1007/BF00181215
  10. ROHMFELD, R., Stabile homogenen Vektorbündel über der komplexen projectiven Ebene, PHD-thesis, Erlangen, 1989. 

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