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The weighted Monge-Ampère energy of quasiplurisubharmonic functions. (English) Zbl 1143.32022
Over the last few decades, complex Monge-Ampére equations have played a crucial role in Kähler geometry and complex dynamics. While, in the smooth case, the existence of a solution was successfully established by S. T. Yau [Comm. Pure Appl. Math. 31, 339-411 (1978; Zbl 0369.53059)], it is a very useful problem to study the behaviour of such equation in the singular case.
In the paper under review, the authors establish very interesting results in this direction. More specifically, let \(X\) be a compact Kähler manifold of dimension \(n\), with Kähler form \(\omega\). A function \(\phi\) is called \(\omega\)-plurisubharmonic if the current \(\omega_\phi=\omega+d d^c\phi\) is positive. Given a Radon measure \(\mu\) on \(X\), \(\phi\) is a solution of the complex Monge-Ampére equation if \(\omega_\phi^n=\mu\). The authors study a new class \(\mathcal E(X,\omega)\) of \(\omega\)-plurisubharmonic functions for which \(\omega_\phi^n\) is well defined. In particular, this is the largest class on which the comparison principle is valid. If \(X\) is a compact Riemann surface, then the set \(\mathcal E(X,\omega)\) corresponds to the set of \(\omega\)-subharmonic functions whose Laplacian does not charge polar sets. In general, it is shown that the Monge-Ampére equation \(\omega_\phi^n=\mu\) admits a solution \(\phi\in \mathcal E(X,\omega)\) if and only if \(\mu\) does not charge pluripolar sets.
The authors also obtain some uniqueness properties, generalizing Calabi’s result to the singular case, and derive some applications to complex dynamics and to the existence of singular Kähler-Einstein metrics.

MSC:
32W20 Complex Monge-Ampère operators
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[1] Aubin, T., Équations du type monge – ampère sur LES variétés kählériennes compactes, Bull. sci. math. (2), 102, 1, 63-95, (1978) · Zbl 0374.53022
[2] Bedford, E.; Diller, J., Energy and invariant measures for birational surface maps, Duke math. J., 128, 2, 331-368, (2005) · Zbl 1076.37031
[3] Bedford, E.; Taylor, B.A., The Dirichlet problem for a complex monge – ampère equation, Invent. math., 37, 1, 1-44, (1976) · Zbl 0315.31007
[4] Bedford, E.; Taylor, B.A., Variational properties of the complex monge – ampère equation I. Dirichlet principle, Duke math. J., 45, 2, 375-403, (1978) · Zbl 0401.35093
[5] Bedford, E.; Taylor, B.A., A new capacity for plurisubharmonic functions, Acta math., 149, 1-2, 1-40, (1982) · Zbl 0547.32012
[6] Bedford, E.; Taylor, B.A., Fine topology, silov boundary, and \((d d^c)^n\), J. funct. anal., 72, 2, 225-251, (1987) · Zbl 0677.31005
[7] Blocki, Z., Uniqueness and stability for the monge – ampère equation on compact Kaehler manifolds, Indiana univ. math. J., 52, 6, 1697-1701, (2003) · Zbl 1054.32024
[8] Blocki, Z., On the definition of the monge – ampère operator in \(\mathbb{C}^2\), Math. ann., 328, 3, 415-423, (2004) · Zbl 1060.32018
[9] Blocki, Z., The domain of definition of the complex monge – ampère operator, Amer. J. math., 128, 2, 519-530, (2006) · Zbl 1102.32018
[10] Calabi, E., On Kähler manifolds with vanishing canonical class, (), 78-89 · Zbl 0080.15002
[11] Cegrell, U., Pluricomplex energy, Acta math., 180, 2, 187-217, (1998) · Zbl 0926.32042
[12] Cegrell, U.; Kolodziej, S.; Zeriahi, A., Subextension of plurisubharmonic functions with weak singularities, Math. Z., 250, 1, 7-22, (2005) · Zbl 1080.32032
[13] Demailly, J.-P., Monge – ampère operators, Lelong numbers and intersection theory, (), 115-193 · Zbl 0792.32006
[14] Demailly, J.-P.; Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of math. (2), 159, 3, 1247-1274, (2004) · Zbl 1064.32019
[15] Diller, J.; Favre, C., Dynamics of bimeromorphic maps of surfaces, Amer. J. math., 123, 6, 1135-1169, (2001) · Zbl 1112.37308
[16] J. Diller, R. Dujardin, V. Guedj, Dynamics of rational endomorphisms of projective surfaces, preprint, 2006
[17] Eyssidieux, P.; Guedj, V.; Zeriahi, A., Singular Kähler – einstein metrics, preprint, arXiv: · Zbl 1215.32017
[18] Guedj, V., Ergodic properties of rational mappings with large topological degree, Ann. of math. (2), 161, 3, 1684-1703, (2005)
[19] Guedj, V.; Zeriahi, A., Intrinsic capacities on compact Kähler manifolds, J. geom. anal., 15, 4, 607-639, (2005) · Zbl 1087.32020
[20] Guedj, V.; Zeriahi, A., Monge – ampère operators on compact Kähler manifolds, preprint, arXiv: · Zbl 1157.32033
[21] Kolodziej, S., The complex monge – ampère equation, Acta math., 180, 1, 69-117, (1998) · Zbl 0913.35043
[22] Kolodziej, S., The monge – ampère equation on compact Kähler manifolds, Indiana univ. math. J., 52, 3, 667-686, (2003) · Zbl 1039.32050
[23] Rainwater, J., A note on the preceding paper, Duke math. J., 36, 799-800, (1969) · Zbl 0201.45801
[24] Rao, M.M.; Ren, Z.D., Theory of Orlicz spaces, Monogr. textbooks pure appl. math., vol. 146, (1991), Dekker New York, xii+449 pp · Zbl 0724.46032
[25] Sibony, N., Dynamique des applications rationnelles de \(\mathbf{P}^k\), (), pp. ix – xii, 97-185 · Zbl 1020.37026
[26] Tian, G., Canonical metrics in Kähler geometry, Lectures math. ETH Zürich, (2000), Birkhäuser Basel · Zbl 0978.53002
[27] Xing, Y., Continuity of the complex monge – ampère operator, Proc. amer. math. soc., 124, 2, 457-467, (1996) · Zbl 0849.31010
[28] Yau, S.T., On the Ricci curvature of a compact Kähler manifold and the complex monge – ampère equation. I, Comm. pure appl. math., 31, 3, 339-411, (1978) · Zbl 0369.53059
[29] Zeriahi, A., The size of plurisubharmonic lemniscates in terms of hausdorff – riesz measures and capacities, Proc. London math. soc. (3), 89, 1, 104-122, (2004) · Zbl 1058.31005
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