Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

Julien Keller; Christina Tønnesen-Friedman

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1673-1687
  • ISSN: 2391-5455

Abstract

top
We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.

How to cite

top

Julien Keller, and Christina Tønnesen-Friedman. "Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections." Open Mathematics 10.5 (2012): 1673-1687. <http://eudml.org/doc/269378>.

@article{JulienKeller2012,
abstract = {We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.},
author = {Julien Keller, Christina Tønnesen-Friedman},
journal = {Open Mathematics},
keywords = {Scalar curvature; Projective bundle; Kähler manifold; Polarization; Stability; Coupled equations; Moment map; Ruled manifold; polarization; moment map; coupled equations; ruled manifold},
language = {eng},
number = {5},
pages = {1673-1687},
title = {Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections},
url = {http://eudml.org/doc/269378},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Julien Keller
AU - Christina Tønnesen-Friedman
TI - Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1673
EP - 1687
AB - We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.
LA - eng
KW - Scalar curvature; Projective bundle; Kähler manifold; Polarization; Stability; Coupled equations; Moment map; Ruled manifold; polarization; moment map; coupled equations; ruled manifold
UR - http://eudml.org/doc/269378
ER -

References

top
  1. [1] Alvarez-Consul L., Garcia-Fernandez M., Garcia-Prada O., Coupled equations for Kähler metrics and Yang-Mills connections, preprint available at http://arxiv.org/abs/1102.0991 Zbl1275.32019
  2. [2] Apostolov V., Calderbank D.M.J., Gauduchon P., Hamiltonian 2-forms in Kähler geometry I. General theory, J. Differential Geom., 2006, 73(3), 359–412 Zbl1101.53041
  3. [3] Apostolov V., Calderbank D.M.J., Gauduchon P., Tønnesen-Friedman C.W., Hamiltonian 2-forms in Kähler geometry II. Global classification, J. Differential Geom., 2004, 68(2), 277–345 Zbl1079.32012
  4. [4] Apostolov V., Calderbank D.M.J., Gauduchon P., Tønnesen-Friedman C.W., Hamiltonian 2-forms in Kähler geometry III. Extremal metrics and stability, Invent. Math., 2008, 173(3), 547–601 http://dx.doi.org/10.1007/s00222-008-0126-x Zbl1145.53055
  5. [5] Apostolov V., Tønnesen-Friedman C., A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces, Bull. London Math. Soc., 2006, 38(3), 494–500 http://dx.doi.org/10.1112/S0024609306018480 Zbl1122.53042
  6. [6] Besse A.L., Einstein Manifolds, Ergeb. Math. Grenzgeb., 10, Springer, Berlin, 1987 
  7. [7] Donaldson S.K., Symmetric spaces, Kähler geometry and Hamiltonian dynamics, In: Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196, American Mathematical Society, Providence, 1999, 13–33 Zbl0972.53025
  8. [8] Donaldson S.K., Kronheimer P.B., The Geometry of Four-Manifolds, Oxford Math. Monogr., Clarendon Press/Oxford University Press, New York, 1990 Zbl0820.57002
  9. [9] Fujiki A., Remarks on extremal Kähler metrics on ruled manifolds, Nagoya Math. J., 1992, 126, 89–101 Zbl0772.53044
  10. [10] Garcia-Fernandez M., Coupled Equations for Kähler Metrics and Yang-Mills Connections, PhD thesis, Universidad Autónoma de Madrid, Madrid, 2009, preprint available at http://arxiv.org/abs/1102.0985 
  11. [11] Guan D., Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends, Trans. Amer. Math. Soc., 1995, 347(6), 2255–2262 Zbl0853.53047
  12. [12] Guan D., On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles, Math. Res. Lett., 1999, 6(5–6), 547–555 Zbl0968.53050
  13. [13] Guan D., Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one. III, Internat. J. Math., 2003, 14(3), 259–287 http://dx.doi.org/10.1142/S0129167X03001806 Zbl1048.32014
  14. [14] Koiso N., Sakane Y., Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds, In: Curvature and Topology of Riemannian Manifolds, Katata, August 26–31, 1985, Lecture Notes in Math., 1201, Springer, Berlin, 1986, 165–179 http://dx.doi.org/10.1007/BFb0075654 
  15. [15] Tønnesen-Friedman C.W., Extremal Kähler metrics on minimal ruled surfaces, J. Reine Angew. Math., 1998, 502, 175–197 http://dx.doi.org/10.1515/crll.1998.086 Zbl0921.53033

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.