Regularity of push-forward of Monge–Ampère measures
[Régularité par poussé en avant des mesures de Monge–Ampère]
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2965-2979.

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.

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.

Publié le :
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)
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Di Nezza, Eleonora; Favre, Charles. Regularity of push-forward of Monge–Ampère measures. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2965-2979. doi : 10.5802/aif.3233. http://archive.numdam.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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