Regularity properties of the degenerate Monge-Ampère equations on compact Kähler manifolds

Abstract

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampére equations on compact Kähler manifolds.

Keywords

Monge-Ampère operator / Estimates / Pluripotential theory

References

[1] Aubin T.. Some Nonlinear Problems in Riemannian Geometry, 1998, New York: Springer-Verlag

[2] Cascini, P. and La Nave, G., Kähler-Ricci Flow and the Minimal Model Program for Projective Varieties, preprint. arXiv: math.AG/0603064

[3] Demailly J.-P.. Regularization of closed positive currents and intersection theory. J. Alg. Geom., 1992, 1: 361-409

[4] Demailly J.-P., Kollár J.. Semicontinuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Ec. Norm. Sup., 2001, 34: 525-556

[5] Demailly J.-P., Păun M.. Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2), 2004, 159(3): 1247-1274

[6] Demailly, J.-P. and Pali, N., Degenerate complex Monge-Ampère equations over compact Kähler manifolds, preprint. arXiv: 07105109V2

[7] Eyssidieux, Ph., Guedj, V. and Zeriahi, A., Singular Kähler-Einstein metrics, J. Amer. Math. Soc., to appear. arXiv: math.AG/0603431

[8] Kolodziej S.. The complex Monge-Ampère equation. Acta Math., 1998, 180: 69-117

[9] Kolodziej S., Tian G.. A uniform L ^∞ estimate for complex Monge-Ampère equations. Math. Ann., 2008, 342(4): 773-787

[10] Siu Y.-T.. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 1974, 27: 53-156

[11] Siu Y.-T.. Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein metrics, 1987, Basel: Birkhäuser Verlag

[12] Skoda H.. Sous-ensembles analytiques d’ordre fini ou infini dans ℂⁿ. Bull. Soc. Math. France, 1972, 100: 353-408

[13] Tian G., Zhang Z.. On the Kähler-Ricci flow of projective manifolds of general type. Chin. Ann. Math., 2006, 27B(2): 179-192

[14] Tsuji H.. Existence and degeneration of Kähler-Einstein metrics on minimal algebraic manifolds of general type. Math. Ann., 1988, 281(1): 123-133

[15] Yau S.-T.. On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation. Comm. Pure Appl. Math., 1978, 31: 339-411