Miyaoka-Yau inequality for minimal projective manifolds of general type

Zhang, Yuguang

doi:10.1090/S0002-9939-09-09838-4

Miyaoka-Yau inequality for minimal projective manifolds of general type
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by Yuguang Zhang

Proc. Amer. Math. Soc. 137 (2009), 2749-2754

DOI: https://doi.org/10.1090/S0002-9939-09-09838-4

Published electronically: February 17, 2009

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Abstract:

In this short paper, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using Kähler-Ricci flow.

References

Thierry Aubin, Équations du type Monge-Ampère sur les variétés kähleriennes compactes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, Aiii, A119–A121. MR 433520
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
P. Cascini, P. La Nave, Kähler-Ricci flow and the minimal model program for projective varieties. arXiv:math/0603064.
S. Y. Cheng and S.-T. Yau, Inequality between Chern numbers of singular Kähler surfaces and characterization of orbit space of discrete group of $\textrm {SU}(2,1)$, Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984) Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 31–44. MR 833802, DOI 10.1090/conm/049/833802
Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007. Geometric aspects. MR 2302600, DOI 10.1090/surv/135
Fuquan Fang, Yuguang Zhang, and Zhenlei Zhang, Non-singular solutions to the normalized Ricci flow equation, Math. Ann. 340 (2008), no. 3, 647–674. MR 2357999, DOI 10.1007/s00208-007-0164-5
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Ryoichi Kobayashi, Einstein-Kähler $V$-metrics on open Satake $V$-surfaces with isolated quotient singularities, Math. Ann. 272 (1985), no. 3, 385–398. MR 799669, DOI 10.1007/BF01455566
Yoichi Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225–237. MR 460343, DOI 10.1007/BF01389789
Gang Tian and Zhou Zhang, On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. MR 2243679, DOI 10.1007/s11401-005-0533-x
Hajime Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), no. 1, 123–133. MR 944606, DOI 10.1007/BF01449219
Hajime Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology 27 (1988), no. 4, 429–442. MR 976585, DOI 10.1016/0040-9383(88)90022-5
Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 451180, DOI 10.1073/pnas.74.5.1798
Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304
Z. Zhang, Scalar Curvature Bound for Kähler-Ricci Flows over Minimal Manifolds of General Type. arXiv:math/0801.32481.