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Special metrics in Kähler geometry

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Abstract

In this note we walk through some of the developments and contributions made in the last 50 years to the study of the complex Monge–Ampère equation in relation to the Calabi conjecture and its generalization.

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References

  1. Aubin, T.: Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2) 102(1), 63–95 (1978)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math 37(1), 1–44 (1976)

    Article  MathSciNet  Google Scholar 

  4. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1–2), 1–40 (1982)

    Article  MathSciNet  Google Scholar 

  5. Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)

    Article  Google Scholar 

  6. Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi \). J. Am. Math. Soc 28(1), 199–234 (2015)

    Article  Google Scholar 

  7. Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Am. Math. Soc. 28(1), 235–278 (2015)

    Article  Google Scholar 

  8. Darvas, T., Di Nezza, E., Lu, C.H.: Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity. Anal. PDE 11(8), 2049–2087 (2018)

    Article  MathSciNet  Google Scholar 

  9. Darvas, T., Di Nezza, E., Lu, C.H.: Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity. Math. Ann. 1–38 (2019)

  10. Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1(3), 361–409 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Sł awomir Dinew: Uniqueness in \({\cal{E}}(x,\omega )\). J. Funct. Anal. 256(7), 2113–2122 (2009)

    Article  MathSciNet  Google Scholar 

  12. Di Nezza, E., Lu, C.H.: Generalized Monge-Ampère capacities. Int. Math. Res. Not. IMRN 16, 7287–7322 (2015)

  13. Di Nezza, E., Lu, C.H.: Complex Monge-Ampère equations on quasi-projective varieties. J. Reine Angew. Math. 727, 145–167 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Guedj, V., Zeriahi, A.: EMS Tracts in Mathematics. Degenerate complex Monge-Ampère equations, vol. 26. European Mathematical Society (EMS), Zürich (2017)

    Google Scholar 

  17. Kołodziej, S.: The complex Monge-Ampère equation. Acta Math. 180(1), 69–117 (1998)

    Article  MathSciNet  Google Scholar 

  18. Nyström, D.W.: Monotonicity of non-pluripolar Monge-Ampère masses. Indiana Univ. Math. J. 68(2), 579–591 (2019)

  19. Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)

    Article  Google Scholar 

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  1. Institut de Mathématiques de Jussieu, Sorbonne Université, 4 Place Jussieu, 75005, Paris, France

    Eleonora Di Nezza

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  1. Eleonora Di Nezza
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Correspondence to Eleonora Di Nezza.

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Di Nezza, E. Special metrics in Kähler geometry. Boll Unione Mat Ital 14, 43–49 (2021). https://doi.org/10.1007/s40574-020-00234-9

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  • DOI: https://doi.org/10.1007/s40574-020-00234-9

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