## Miyaoka-Yau inequality for minimal projective manifolds of general type

HTML articles powered by AMS MathViewer

- by Yuguang Zhang PDF
- Proc. Amer. Math. Soc.
**137**(2009), 2749-2754 Request permission

## 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.

## Additional Information

**Yuguang Zhang**- Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China
- Address at time of publication: Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- MR Author ID: 780283
- Email: yuguangzhang76@yahoo.com
- Received by editor(s): October 20, 2008
- Received by editor(s) in revised form: November 30, 2008
- Published electronically: February 17, 2009
- Additional Notes: This work was supported by the SRC program of the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) (No. R11-2007-035-02002-0), and by the National Natural Science Foundation of China 10771143.
- Communicated by: Jon G. Wolfson
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 2749-2754 - MSC (2000): Primary 53C55, 53C44
- DOI: https://doi.org/10.1090/S0002-9939-09-09838-4
- MathSciNet review: 2497488