Table of Contents
Geometry
Volume 2014, Article ID 243236, 14 pages
http://dx.doi.org/10.1155/2014/243236
Research Article

Vanishing Theorems on Compact Hyper-kähler Manifolds

Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Received 3 June 2013; Revised 14 December 2013; Accepted 16 December 2013; Published 16 February 2014

Academic Editor: Reza Saadati

Copyright © 2014 Qilin Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Beauville, “Variétés Kähleriennes dont la première classe de Chern est nulle,” Journal of Differential Geometry, vol. 18, no. 4, pp. 755–782, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. A. Bogomolov, “On the decomposition of Kähler manifolds with trivial canonical class,” Mathematics of the USSR, vol. 22, no. 4, pp. 580–583, 1974. View at Publisher · View at Google Scholar
  3. M. Verbitsky, “Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds,” Compositio Mathematica, vol. 143, no. 6, pp. 1576–1592, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Y. T. Siu, “Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems,” Journal of Differential Geometry, vol. 17, no. 1, pp. 55–138, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Fujiki, “On the de Rham cohomology group of a compact Kähler symplectic manifold,” in Algebraic Geometry, vol. 10 of Advanced Studies in Pure Mathematics, pp. 105–165, Mathematical Society of Japan, Tokyo, Japan, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 1978. View at MathSciNet
  7. M. Verbitsky, “Hypercomplex structures on Kähler manifolds,” Geometric & Functional Analysis, vol. 15, no. 6, pp. 1275–1283, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Alesker and M. Verbitsky, “Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry,” The Journal of Geometric Analysis, vol. 16, no. 3, pp. 375–399, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Obata, “Affine connections on manifolds with almost complex, quaternion or Hermitian structure,” Japanese Journal of Mathematics, vol. 26, pp. 43–79, 1955. View at Google Scholar · View at MathSciNet
  10. A. Newlander and L. Nirenberg, “Complex analytic coordinates in almost complex manifolds,” Annals of Mathematics, vol. 65, pp. 391–404, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J.-P. Demailly, “Multiplier ideal sheaves and analytic methods in algebraic geometry,” in School on Vanishing Theorems and Effective Results in Algebraic Geometry, pp. 1–149, The Abdus Salam ICTP, Trieste, Italy, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. A. Bogomolov, “Hamiltonian Kählerian manifolds,” Soviet Mathematics Doklady, vol. 19, pp. 1462–1465, 1978, Translated from Doklady Akademii Nauk SSSR, vol. 243, pp. 1101–1104, 1978. View at Google Scholar · View at MathSciNet
  13. Q. Yang, “(k,s)-positivity and vanishing theorems for compact Kähler manifolds,” International Journal of Mathematics, vol. 22, no. 4, pp. 545–576, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. B. Shiffman and A. J. Sommese, Vanishing Theorems on Complex Manifolds, vol. 56 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 1985. View at MathSciNet
  15. G. Gigante, “Vector bundles with semidefinite curvature and cohomology vanishing theorems,” Advances in Mathematics, vol. 41, no. 1, pp. 40–56, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Akizuki and S. Nakano, “Note on Kodaira-Spencer's proof of Lefschetz theorems,” Proceedings of the Japan Academy, vol. 30, pp. 266–272, 1954. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet