Bulletin of the Korean Mathematical Society (대한수학회보)

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MAGNETIC GEODESICS ON THE SPACE OF KÄHLER POTENTIALS

  • Sahin, Sibel (Department of Mathematics Mimar Sinan Fine Arts University)
  • Received : 2021.08.15
  • Accepted : 2022.02.04
  • Published : 2022.07.31
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Abstract

In this work, magnetic geodesics over the space of Kähler potentials are studied through a variational method for a generalized Landau-Hall functional. The magnetic geodesic equation is calculated in this setting and its relation to a perturbed complex Monge-Ampère equation is given. Lastly, the magnetic geodesic equation is considered over the special case of toric Kähler potentials over toric Kähler manifolds.

Keywords

Acknowledgement

I would like to sincerely thank the anonymous referee for his/her valuable suggestions and comments. The content and the presentation of the paper have been considerably improved thanks to his/her contributions and expertise.

References

  1. E. Calabi and X. X. Chen, The space of Kahler metrics. II, J. Differential Geom. 61 (2002), no. 2, 173-193. http://projecteuclid.org/euclid.jdg/1090351383
  2. X. X. Chen, The space of Kahler metrics, J. Differential Geom. 56 (2000), no. 2, 189-234. http://projecteuclid.org/euclid.jdg/1090347643
  3. X. X. Chen, Space of Kahler metrics. III. On the lower bound of the Calabi energy and geodesic distance, Invent. Math. 175 (2009), no. 3, 453-503. https://doi.org/10.1007/s00222-008-0153-7
  4. X. X. Chen and S. Sun, Space of Kahler metrics (V )-Kahler quantization, in Metric and differential geometry, 19-41, Progr. Math., 297, Birkhauser/Springer, Basel, 2012. https://doi.org/10.1007/978-3-0348-0257-4_2
  5. X. X. Chen and G. Tian, Geometry of Kahler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Etudes Sci. No. 107 (2008), 1-107. https://doi.org/10.1007/s10240-008-0013-4
  6. D. Coman, V. Guedj, S. Sahin, and A. Zeriahi, Toric pluripotential theory, Ann. Polon. Math. 123 (2019), no. 1, 215-242. https://doi.org/10.4064/ap180409-3-7
  7. V. Guedj, The metric completion of the Riemannian space of Kahler metrics, Preprint arXiv:1401.7857.
  8. J. Inoguchi and M. I. Munteanu, Magnetic maps, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 6, 1450058, 22 pp. https://doi.org/10.1142/S0219887814500583
  9. B. Kolev, The Riemannian space of Kahler metrics, in Complex Monge-Ampere equations and geodesics in the space of Kahler metrics, 231-255, Lecture Notes in Math., 2038, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-23669-3_6
  10. S. Semmes, Complex Monge-Ampere and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3, 495-550. https://doi.org/10.2307/2374768