Acta Mathematica

Monge–Ampère equations in big cohomology classes

Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi

Full-text: Open access

Abstract

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L1+ε-density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

Article information

Source
Acta Math., Volume 205, Number 2 (2010), 199-262.

Dates
Received: 23 January 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892500

Digital Object Identifier
doi:10.1007/s11511-010-0054-7

Mathematical Reviews number (MathSciNet)
MR2746347

Zentralblatt MATH identifier
1213.32025

Rights
2010 © Institut Mittag-Leffler

Citation

Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed. Monge–Ampère equations in big cohomology classes. Acta Math. 205 (2010), no. 2, 199--262. doi:10.1007/s11511-010-0054-7. https://projecteuclid.org/euclid.acta/1485892500


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References

  • A ubin, T., Équations du type Monge–Ampère sur les variétés kähleriennes compactes. C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Aiii, A119–A121.
  • — Réduction du cas positif de l’équation de Monge–Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité. J. Funct. Anal., 57 (1984), 143–153.
  • B edford, E. & T aylor, B. A., A new capacity for plurisubharmonic functions. Acta Math., 149 (1982), 1–40.
  • — Fine topology, Shilov boundary, and (ddc)n. J. Funct. Anal., 72 (1987), 225–251.
  • B enelkourchi, S., G uedj, V. & Z eriahi, A., A priori estimates for weak solutions of complex Monge–Ampère equations. Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 81–96.
  • B erman, R. & B oucksom, S., Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math., 181 (2010), 337–394.
  • B irkar, C., C ascini, P., H acon, C. D. & M cK ernan, J., Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 23 (2010), 405–468.
  • B łocki, Z., On the regularity of the complex Monge–Ampère operator, in Complex Geometric Analysis in Pohang (1997), Contemp. Math., 222, pp. 181–189. Amer. Math. Soc., Providence, RI, 1999.
  • B łocki, Z. & K ołodziej, S., On regularization of plurisubharmonic functions on manifolds. Proc. Amer. Math. Soc., 135 (2007), 2089–2093.
  • B oucksom, S., On the volume of a line bundle. Internat. J. Math., 13 (2002), 1043–1063.
  • — Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup., 37 (2004), 45–76.
  • B oucksom, S., D emailly, J.-P., P ăun, M. & P eternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. To appear in J. Algebraic Geom.
  • B oucksom, S., F avre, C. & J onsson, M., Differentiability of volumes of divisors and a problem of Teissier. J. Algebraic Geom., 18 (2009), 279–308.
  • C egrell, U., Pluricomplex energy. Acta Math., 180 (1998), 187–217.
  • — The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier (Grenoble), 54 (2004), 159–179.
  • D emailly, J.-P., Regularization of closed positive currents and intersection theory. J. Algebraic Geom., 1 (1992), 361–409.
  • — A numerical criterion for very ample line bundles. J. Differential Geom., 37 (1993), 323–374.
  • Complex Analytic and Algebraic Geometry. Book available at
  • D emailly, J.-P. & P ăun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. of Math., 159 (2004), 1247–1274.
  • D emailly, J.-P., Peternell, T. & Schneider, M., Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom., 3 (1994), 295–345.
  • D inew, S., An inequality for mixed Monge–Ampère measures. Math. Z., 262 (2009), 1–15.
  • — Uniqueness in $ \mathcal{E}\left( {X,\omega } \right) $. J. Funct. Anal., 256 (2009), 2113–2122.
  • E in, L., L azarsfeld, R., M ustata, M., N akamaye, M. & P opa, M., Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble), 56 (2006), 1701–1734.
  • E vans, L. C., Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math., 35 (1982), 333–363.
  • E yssidieux, P., G uedj, V. & Z eriahi, A., Singular Kähler–Einstein metrics. J. Amer. Math. Soc., 22 (2009), 607–639.
  • G uedj, V. & Z eriahi, A., Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal., 15 (2005), 607–639.
  • — The weighted Monge–Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal., 250 (2007), 442–482.
  • K iselman, C. O., Sur la définition de l’opérateur de Monge–Ampère complexe, in Complex Analysis (Toulouse, 1983), Lecture Notes in Math., 1094, pp. 139–150. Springer, Berlin–Heidelberg, 1984.
  • K ollár, J. & M ori, S., Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.
  • K ołodziej, S., The complex Monge–Ampère equation. Acta Math., 180 (1998), 69–117.
  • — The complex Monge–Ampère equation and pluripotential theory. Mem. Amer. Math. Soc., 178:840 (2005).
  • L azarsfeld, R., Positivity in Algebraic Geometry. I, II. Ergebnisse der Mathematik und ihrer Grenzgebiete, 49. Springer, Berlin–Heidelberg, 2004.
  • M abuchi, T., K-energy maps integrating Futaki invariants. Tohoku Math. J., 38 (1986), 575–593.
  • R ainwater, J., A note on the preceding paper. Duke Math. J., 36 (1969), 799–800.
  • S ibony, N., Quelques problèmes de prolongement de courants en analyse complexe. Duke Math. J., 52 (1985), 157–197.
  • S iu, Y.-T., Invariance of plurigenera. Invent. Math., 134 (1998), 661–673.
  • — A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring. Preprint, 2006. arXiv:math/0610740v1 [math.AG].
  • S ong, J. & T ian, G., Canonical measures and Kähler–Ricci flow. Preprint, 2008. arXiv:0802.2570v1 [math.DG].
  • S ugiyama, K., Einstein–Kähler metrics on minimal varieties of general type and an inequality between Chern numbers, in Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., 18, pp. 417–433. Academic Press, Boston, MA, 1990.
  • T rudinger, N. S., Regularity of solutions of fully nonlinear elliptic equations. Boll. Un. Mat. Ital. A, 3 (1984), 421–430.
  • T suji, H., Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann., 281 (1988), 123–133.
  • — Finite generation of canonical rings. Preprint, 1999. arXiv:math/9908078v8 [math.AG].
  • — Dynamical constructions of Kähler–Einstein metrics. Preprint, 2006. arXiv:math/0606626v3 [math.AG].
  • Y au, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Comm. Pure Appl. Math., 31 (1978), 339–411.
  • Z eriahi, A., Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions. Indiana Univ. Math. J., 50 (2001), 671–703.