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The Monge-Ampère Energy Class \(E\)

  • Eleonora Di Nezza
Chapter
First Online:
Part of the Springer INdAM Series book series (SINDAMS, volume 21)

Abstract

In this short note, based on a joint work with Tamas Darvas and Chinh Lu, we introduce and investigate pluripotential tools. In particular we give a characterization of the Monge-Ampère energy class \(\mathcal{E}\) in terms of “envelopes” and we focus on some consequences.

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Notes

Acknowledgements

I would like to thank Daniele Angella, Paolo De Bartolomeis, Costantino Medori and Adriano Tomassini for organising the INDAM meeting “Complex and Symplectic Geometry” in Cortona and for the invitation to speak in that occasion. I would also like to thank Stefano Trapani for his comments and remarks on the paper [8] that gave as outcome the last section.

References

  1. 1.
    E. Bedford, B.A. Taylor, Fine topology, Silov boundary, and (dd c)n. J. Funct. Anal. 72(2), 225–251 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R.J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties. Journal für die reine und angewandte Mathematik (Crelles Journal) (2011). http://arxiv.org/abs/1111.7158
  3. 3.
    S. Boucksom, On the volume of a line bundle. Int. J. Math. 13(10), 1043–1063 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. Ecole Norm. Sup. (4) 37(1), 45–76 (2004)Google Scholar
  5. 5.
    S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes. Acta Math. 205(2), 199–262 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    T. Darvas, The Mabuchi completion of the space of Kähler potentials. Amer. J. Math. (2014, to appear) http://arxiv.org/abs/1401.7318
  7. 7.
    T. Darvas, Y.A. Rubinstein, Kiselman’s principle, the Dirichlet problem for the Monge-Ampère equation, and rooftop obstacle problems. J. Math. Soc. Jpn. 68(2), 773–796 (2016)CrossRefzbMATHGoogle Scholar
  8. 8.
    T. Darvas, E. Di Nezza, C. Lu, On the singularity type of full mass currents in big cohomology classes (2016). arXiv:1606.01527Google Scholar
  9. 9.
    J.P. Demailly, Complex Analytic and Differential Geometry. Book available at http://www-fourier.ujf-grenoble.fr/~demailly/documents.html
  10. 10.
    E. Di Nezza, Stability of Monge–Ampère energy classes. J. Geom. Anal. 25(4), 2565–2589 (2014)CrossRefzbMATHGoogle Scholar
  11. 11.
    E. Di Nezza, E. Floris, S. Trapani, Divisorial Zariski Decomposition and some properties of full mass currents (2015). http://arxiv.org/abs/1505.07638
  12. 12.
    S. Dinew, V. Guedj, A. Zeriahi, Open problems in pluripotential theory. Complex Variables Elliptic Equ. Int J. (2016). doi:http://www.tandfonline.com/doi/full/10.1080 /17476933.2015.1121481
  13. 13.
    V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Klimek, Pluripotential Theory. London Mathematical Society Monographs. New Series, vol. 6. Oxford Science Publications (Clarendon, Oxford University Press, New York, 1991)Google Scholar
  15. 15.
    J. Ross, D. Witt Nyström, Analytic test configurations and geodesic rays. J. Symplectic Geom. 12(1), 125–169 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK

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