Skip to main content
SpringerLink
Log in

Extremal potentials and equilibrium measures for collections of Kähler classes

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Given a collection of Kähler forms and a continuous weight on a compact complex manifold we show that it is possible to define natural new notions of extremal potentials and equilibrium measures which coincide with classical notions when the collection is a singleton. We prove two regularity results and set up a variational framework. Applications to sampling of holomorphic sections are treated elsewhere.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berman, R.: From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit. Math. Z. 291, 365–394 (2019)

    Article  MathSciNet  Google Scholar 

  2. Berman, R., Boucksom, S.: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181, 337–394 (2010)

    Article  MathSciNet  Google Scholar 

  3. Berman, R., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties. J. Reine Angew. Math. 751, 27–89 (2019)

    Article  MathSciNet  Google Scholar 

  4. Berman, R., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge–Ampère equations. Publ. Math. IHES 117, 179–245 (2013)

    Article  Google Scholar 

  5. Berman, R., Boucksom, S., Witt Nyström, D.: Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math. 207, 1–27 (2011)

    Article  MathSciNet  Google Scholar 

  6. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. J. Am. Math. Soc. 22, 607–639 (2009)

    Article  Google Scholar 

  7. Guedj, V., Zeriahi, A.: The weighted Monge–Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)

    Article  MathSciNet  Google Scholar 

  8. Hultgren, J.: Mutual asymptotic Fekete sequences. arXiv:2106.04613

  9. Hultgren, J., Witt Nyström, D.: Coupled Kähler–Einstien metrics. Int. Math. Res. Not. 2019(21), 6765–6796 (2019)

    Article  Google Scholar 

  10. Kołodziej, S.: The complex Monge–Ampère equation. Acta Math. 180, 69–117 (1998)

    Article  MathSciNet  Google Scholar 

  11. Xing, Y.: Continuity of the complex Monge–Ampère operator on compact Kähler manifolds. Math. Z. 263, 331–344 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank Nick McCleerey and Tàmas Darvas for very valuable input and for reading and commenting on a draft of this paper. Special thanks to Nick McCleerey for spotting an error in the proof of Lemma 1. The author also thanks the Knut and Alice Wallenberg Foundation (Grant 2018-0357) and BSF (Grant 2016173).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, 20742-4015, MD, USA

    Jakob Hultgren

Authors
  1. Jakob Hultgren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jakob Hultgren.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hultgren, J. Extremal potentials and equilibrium measures for collections of Kähler classes. Math. Z. 301, 1555–1571 (2022). https://doi.org/10.1007/s00209-021-02964-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-021-02964-8

Keywords

Mathematics Subject Classification

Search

Navigation