## Analysis & PDE

• Anal. PDE
• Volume 11, Number 8 (2018), 2049-2087.

### Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity

#### Abstract

We establish the monotonicity property for the mass of nonpluripolar products on compact Kähler manifolds, and we initiate the study of complex Monge–Ampère-type equations with prescribed singularity type. Using the variational method of Berman, Boucksom, Guedj and Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kähler–Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for nonpluripolar products with small unbounded locus.

#### Article information

Source
Anal. PDE, Volume 11, Number 8 (2018), 2049-2087.

Dates
Revised: 4 February 2018
Accepted: 10 April 2018
First available in Project Euclid: 15 January 2019

https://projecteuclid.org/euclid.apde/1547521446

Digital Object Identifier
doi:10.2140/apde.2018.11.2049

Mathematical Reviews number (MathSciNet)
MR3812864

Zentralblatt MATH identifier
1396.32011

#### Citation

Darvas, Tamás; Di Nezza, Eleonora; Lu, Chinh H. Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity. Anal. PDE 11 (2018), no. 8, 2049--2087. doi:10.2140/apde.2018.11.2049. https://projecteuclid.org/euclid.apde/1547521446

#### References

• T. Aubin, “Équations du type Monge–Ampère sur les variétés kählériennes compactes”, Bull. Sci. Math. $(2)$ 102:1 (1978), 63–95.
• E. Bedford and J.-P. Demailly, “Two counterexamples concerning the pluri-complex Green function in $\mathbb{C}^n$”, Indiana Univ. Math. J. 37:4 (1988), 865–867.
• E. Bedford and B. A. Taylor, “The Dirichlet problem for a complex Monge–Ampère equation”, Invent. Math. 37:1 (1976), 1–44.
• E. Bedford and B. A. Taylor, “A new capacity for plurisubharmonic functions”, Acta Math. 149:1-2 (1982), 1–40.
• E. Bedford and B. A. Taylor, “Fine topology, Šilov boundary, and $(dd^c)^n$”, J. Funct. Anal. 72:2 (1987), 225–251.
• R. J. Berman, “From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit”, preprint, 2013.
• R. Berman and S. Boucksom, “Growth of balls of holomorphic sections and energy at equilibrium”, Invent. Math. 181:2 (2010), 337–394.
• R. J. Berman, S. Boucksom, V. Guedj, and A. Zeriahi, “A variational approach to complex Monge–Ampère equations”, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179–245.
• S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, “Monge–Ampère equations in big cohomology classes”, Acta Math. 205:2 (2010), 199–262.
• U. Cegrell, “Pluricomplex energy”, Acta Math. 180:2 (1998), 187–217.
• D. Coman and V. Guedj, “Quasiplurisubharmonic Green functions”, J. Math. Pures Appl. $(9)$ 92:5 (2009), 456–475.
• T. Darvas, “Weak geodesic rays in the space of Kähler potentials and the class $\mathcal{E}(X,\omega)$”, J. Inst. Math. Jussieu 16:4 (2017), 837–858.
• T. Darvas and Y. A. Rubinstein, “Tian's properness conjectures and Finsler geometry of the space of Kähler metrics”, J. Amer. Math. Soc. 30:2 (2017), 347–387.
• T. Darvas, E. Di Nezza, and C. H. Lu, “On the singularity type of full mass currents in big cohomology classes”, Compos. Math. 154:2 (2018), 380–409.
• J.-P. Demailly, “Multiplier ideal sheaves and analytic methods in algebraic geometry”, pp. 1–148 in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), edited by J. P. Demailly et al., ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.
• E. Di Nezza and C. H. Lu, “Generalized Monge–Ampère capacities”, Int. Math. Res. Not. 2015:16 (2015), 7287–7322.
• E. Di Nezza and C. H. Lu, “Complex Monge–Ampère equations on quasi-projective varieties”, J. Reine Angew. Math. 727 (2017), 145–167.
• S. Dinew, “An inequality for mixed Monge–Ampère measures”, Math. Z. 262:1 (2009), 1–15.
• S. Dinew, “Uniqueness in $\mathscr{E}(X,\omega)$”, J. Funct. Anal. 256:7 (2009), 2113–2122.
• S. Dinew and C. H. Lu, “Mixed Hessian inequalities and uniqueness in the class $\mathcal{E}(X,\omega,m)$”, Math. Z. 279:3-4 (2015), 753–766.
• P. Eyssidieux, V. Guedj, and A. Zeriahi, “Singular Kähler–Einstein metrics”, J. Amer. Math. Soc. 22:3 (2009), 607–639.
• B. Guan, “The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function”, Comm. Anal. Geom. 6:4 (1998), 687–703.
• Q. Guan and X. Zhou, “A proof of Demailly's strong openness conjecture”, Ann. of Math. $(2)$ 182:2 (2015), 605–616.
• V. Guedj and A. Zeriahi, “Intrinsic capacities on compact Kähler manifolds”, J. Geom. Anal. 15:4 (2005), 607–639.
• V. Guedj and A. Zeriahi, “The weighted Monge–Ampère energy of quasiplurisubharmonic functions”, J. Funct. Anal. 250:2 (2007), 442–482.
• V. Guedj and A. Zeriahi, Degenerate complex Monge–Ampère equations, EMS Tracts in Mathematics 26, European Mathematical Society, Zürich, 2017.
• V. Guedj, C. H. Lu, and A. Zeriahi, “Plurisubharmonic envelopes and supersolutions”, preprint, 2017. To appear in J. Differential Geom.
• P. H. Hiep, “The weighted log canonical threshold”, C. R. Math. Acad. Sci. Paris 352:4 (2014), 283–288.
• S. Kołodziej, “The complex Monge–Ampère equation”, Acta Math. 180:1 (1998), 69–117.
• S. Kołodziej, “The Monge–Ampère equation on compact Kähler manifolds”, Indiana Univ. Math. J. 52:3 (2003), 667–686.
• H. König and G. L. Seever, “The abstract F. and M. Riesz theorem”, Duke Math. J. 36 (1969), 791–797.
• L. Lempert, “Solving the degenerate complex Monge–Ampère equation with one concentrated singularity”, Math. Ann. 263:4 (1983), 515–532.
• D. H. Phong and J. Sturm, “The Dirichlet problem for degenerate complex Monge–Ampere equations”, Comm. Anal. Geom. 18:1 (2010), 145–170.
• D. H. Phong and J. Sturm, “Regularity of geodesic rays and Monge–Ampère equations”, Proc. Amer. Math. Soc. 138:10 (2010), 3637–3650.
• D. H. Phong and J. Sturm, “On the singularities of the pluricomplex Green's function”, pp. 419–435 in Advances in analysis: the legacy of Elias M. Stein, edited by C. Fefferman et al., Princeton Math. Ser. 50, Princeton Univ. Press, 2014.
• J. Rainwater, “A note on the preceding paper”, Duke Math. J. 36 (1969), 799–800.
• A. Rashkovskii and R. Sigurdsson, “Green functions with singularities along complex spaces”, Internat. J. Math. 16:4 (2005), 333–355.
• J. Ross and D. Witt Nyström, “Analytic test configurations and geodesic rays”, J. Symplectic Geom. 12:1 (2014), 125–169.
• J. Ross and D. Witt Nyström, “Envelopes of positive metrics with prescribed singularities”, Ann. Fac. Sci. Toulouse Math. $(6)$ 26:3 (2017), 687–728.
• D. Witt Nyström, “Monotonicity of non-pluripolar Monge–Ampère masses”, preprint, 2017.
• Y. Xing, “Continuity of the complex Monge–Ampère operator”, Proc. Amer. Math. Soc. 124:2 (1996), 457–467.
• Y. Xing, “Continuity of the complex Monge–Ampère operator on compact Kähler manifolds”, Math. Z. 263:2 (2009), 331–344.
• S. T. Yau, “On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I”, Comm. Pure Appl. Math. 31:3 (1978), 339–411.