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The Monge-Ampère Energy Class \(E\)

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Complex and Symplectic Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 21))

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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.

In memory of Paolo de Bartolomeis

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References

  1. E. Bedford, B.A. Taylor, Fine topology, Silov boundary, and (dd c)n. J. Funct. Anal. 72(2), 225–251 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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. S. Boucksom, On the volume of a line bundle. Int. J. Math. 13(10), 1043–1063 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. Ecole Norm. Sup. (4) 37(1), 45–76 (2004)

    Google Scholar 

  5. S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes. Acta Math. 205(2), 199–262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  8. T. Darvas, E. Di Nezza, C. Lu, On the singularity type of full mass currents in big cohomology classes (2016). arXiv:1606.01527

    Google Scholar 

  9. J.P. Demailly, Complex Analytic and Differential Geometry. Book available at http://www-fourier.ujf-grenoble.fr/~demailly/documents.html

  10. E. Di Nezza, Stability of Monge–Ampère energy classes. J. Geom. Anal. 25(4), 2565–2589 (2014)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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. J. Ross, D. Witt Nyström, Analytic test configurations and geodesic rays. J. Symplectic Geom. 12(1), 125–169 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Correspondence to Eleonora Di Nezza .

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Di Nezza, E. (2017). The Monge-Ampère Energy Class \(E\). In: Angella, D., Medori, C., Tomassini, A. (eds) Complex and Symplectic Geometry. Springer INdAM Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-62914-8_4

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