Abstract
This note concerns the global existence and convergence of the solution for Kähler-Ricci flow equation when the canonical class, K X , is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.
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Bedford, E. and Taylor, B. A., A new capacity for plurisubharmonic functions, Acta Math., 149(1-2), 1982, 1-40.
Cao, H. D., Deformation of Kaehler metrics to Kaehler-Einstein metrics on compact Kaehler manifolds, Invent. Math., 81(2), 1985, 359-372.
Cascini, P. and La Nave, G., Kähler-Ricci flow and the minimal model program for projective varieties, preprint.
Demailly, J.-P., Complex analytic and algebraic geometry, Online book: agbook.ps.gz.
Durfee, A. H., Fifteen characterizations of rational double points and simple critical points, Enseign. Math. (2), 25(1-2), 1979, 131-163.
Feldman, M., Ilmanen, T. and Knopf, D., Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Diff. Geom., 65(2), 2003, 169-209.
Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften, 224.
Kawamata, Y., The cone of curves of algebraic varieties, Ann. of Math. (2), 119(3), 1984, 603-633.
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79(3), 1985, 567-588.
Kawamata, Y., A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261(1), 1982, 43-46.
Kleiman, S. L., Toward a numerical theory of ampleness, Ann. of Math. (2), 84, 1966, 293-344.
Kobayashi, R., Einstein-Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities, Math. Ann., 272(3), 1985, 385-398.
Kolodziej, S., The complex Monge-Ampere equation, Acta Math., 180(1), 1998, 69-117.
Nakamaye, M., Stable base loci of linear series, Math. Ann., 318(4), 2000, 837-847.
Sugiyama, K., Einstein-Kähler metrics on minimal varieties of general type and an inequality between Chern numbers, Recent Topics in Differential and Analytic Geometry, 417-433; Adv. Stud. Pure Math., 18-1, Academic Press, Boston, MA, 1990.
Tian, G., Geometry and nonlinear analysis, Proceedings of the International Congress of Mathematicians (Beijing 2002), Vol. I, Higher Ed. Press, Beijing, 2002, 475-493.
Tsuji, H., Existence and degeneration of Kaehler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann., 281(1), 1988, 123-133.
Tsuji, H., Degenerated Monge-Ampere equation in algebraic geometry, Miniconference on Analysis and Applications (Brisbane, 1993), Proc. Centre Math. Appl. Austral. Nat. Univ., 33, Austral. Nat. Univ., Canberra, 1994, 209-224.
Yau, S.-T., On the Ricci curvature of a compact Kaehler manifold and the complex Monge-Ampere equation, I, Comm. Pure Appl. Math., 31(3), 1978, 339-411.
Zariski, O., The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2), 76, 1962, 560-615.
Zhang, Z., Thesis in preparation.
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(Dedicated to the memory of Shiing-Shen Chern)
* Partially supported by NSF grants and a Simons fund.
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Tian*, G., Zhang, Z. On the Kähler-Ricci Flow on Projective Manifolds of General Type. Chin. Ann. Math. Ser. B 27, 179–192 (2006). https://doi.org/10.1007/s11401-005-0533-x
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DOI: https://doi.org/10.1007/s11401-005-0533-x