no. 7
Regularity of push-forward of Monge–Ampère measures
Di Nezza, Eleonora1; Favre, Charles2
1 Sorbonne Université 4 Place Jussieu 75005 Paris (France)
2 CMLS, École polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex (France)
Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2965-2979.

We prove that the image under any dominant meromorphic map of the Monge–Ampère measure of a Hölder continuous quasi-psh function still possesses a Hölder potential. We also discuss the case of lower regularity.

Nous démontrons que l’image par une application méromorphe dominante d’une mesure de Monge–Ampère d’une fonction quasi-psh et hölderienne possède aussi un potentiel hölderien. Nous discutons aussi le cas de régularité plus basse.

Published online:
DOI: 10.5802/aif.3233
Classification: 32Q15, 32W20, 32Uxx
Keywords: Kähler manifolds, meromorphic map, Monge–Ampère measures
Mot clés : variétés kähleriennes, application méromorphe, mesures de Monge–Ampère

Di Nezza, Eleonora 1; Favre, Charles 2

1 Sorbonne Université 4 Place Jussieu 75005 Paris (France)
2 CMLS, École polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Di Nezza, Eleonora; Favre, Charles. Regularity of push-forward of Monge–Ampère measures. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2965-2979. doi : 10.5802/aif.3233. https://aif.centre-mersenne.org/articles/10.5802/aif.3233/

[1] Aizenbud, Avraham; Avni, Nir Representation Growth and Rational Singularities of the Moduli Space of Local Systems, Invent. Math., Volume 204 (2016) no. 1, pp. 245-316 | Zbl

[2] Bedford, Eric; Taylor, Bert A. The Dirichlet problem for the complex Monge-Ampère equation, Invent. Math., Volume 37 (1976), pp. 1-44 | Zbl

[3] Demailly, Jean-Pierre Monge–Ampère operators, Lelong numbers and intersection theory, Complex analysis and geometry (The University Series in Mathematics), Plenum Press, 1993 | Zbl

[4] Demailly, Jean-Pierre; Dinew, Sławomir; Guedj, Vincent; Hiep, Pham Hoang; Kołodziej, Sławomir; Zeriahi, Ahmed Hölder continuous solutions to Monge-Ampère equations, J. Eur. Math. Soc., Volume 16 (2014) no. 4, pp. 619-647 | Zbl

[5] Di Nezza, Eleonora Stability of Monge–Ampère energy classes, J. Geom. Anal., Volume 25 (2014) no. 4, pp. 2565-2589 | Zbl

[6] Di Nezza, Eleonora Finite Pluricomplex energy measures, Potential Anal., Volume 44 (2015) no. 1, pp. 155-167 | Zbl

[7] Dinew, Sławomir; Guedj, Vincent; Zeriahi, Ahmed Open problems in pluripotential theory, Complex Var. Elliptic Equ., Volume 61 (2016) no. 7, pp. 902-930 | Zbl

[8] Dinew, Sławomir; Zhang, Zhou On stability and continuity of bounded solutions of degenerate complex Monge–Ampère equations over compact Kähler manifolds, Adv. Math., Volume 225 (2010) no. 1, pp. 367-388 | Zbl

[9] Dinh, Tien Cuong; Nguyen, Viet Anh Characterization of Monge-Ampère measures with Hölder continuous potential, J. Funct. Anal., Volume 266 (2014), pp. 67-84

[10] Dinh, Tien Cuong; Nguyen, Viet Anh; Sibony, Nessim Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differ. Geom., Volume 84 (2010), pp. 465-488

[11] Favre, Charles Degeneration of endomorphisms of the complex projective space in the hybrid space, J. Inst. Math. Jussieu (2018), 43 pages (43 p., published online) | DOI

[12] Guedj, Vincent; Zeriahi, Ahmed Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal., Volume 15 (2005) no. 4, pp. 607-639

[13] Guedj, Vincent; Zeriahi, Ahmed The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 442-482

[14] Guedj, Vincent; Zeriahi, Ahmed Degenerate Complex Monge–Ampère Equations, EMS Tracts in Mathematics, 26, Société Mathématique de France, 2017, xxiv+472 pages | Zbl

[15] Kołodziej, Sławomir; Nguyen, Ngoc Cuong Hölder continuous solutions of the Monge-Ampère equation on compact hermitian manifolds, Ann. Inst. Fourier, Volume 68 (2018) no. 7, pp. 2951-2964

[16] Lejeune-Jalabert, Monique; Teissier, Bernard; Risler, Jean-Jacques Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse, Math., Volume 17 (2008) no. 4, pp. 781-859

[17] Peternell, Thomas Modifications, Several complex variables VII: Sheaf theoretic methods in complex analysis (Encyclopaedia of Mathematical Sciences), Volume 74, Springer, 1994, pp. 285-317 | Zbl

[18] Reiser, Andrew Pushforwards of Measures on Real Varieties under Maps with Rational Singularities (2018) (https://arxiv.org/abs/1807.00079v1)

[19] Song, Jian; Tian, Gang Canonical measures and Kähler-Ricci flow, J. Am. Math. Soc., Volume 25 (2012) no. 2, pp. 303-353

[20] Tosatti, Valentino Adiabatic limits of Ricci-flat Kähler metrics, J. Differ. Geom., Volume 84 (2010) no. 2, pp. 427-453

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