Calabi's conjecture and some new results in algebraic geometry.

@article{Yau1977CalabisCA,
  title={Calabi's conjecture and some new results in algebraic geometry.},
  author={Shing-Tung Yau},
  journal={Proceedings of the National Academy of Sciences of the United States of America},
  year={1977},
  volume={74 5},
  pages={
          1798-9
        }
}
  • S. Yau
  • Published 1 May 1977
  • Mathematics
  • Proceedings of the National Academy of Sciences of the United States of America
We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For example, we prove that the only Kähler structure on a complex projective space is the standard one. 
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References

SHOWING 1-10 OF 13 REFERENCES
Characteristic Classes of Hermitian Manifolds
In recent years the works of Stiefel,1 Whitney,2 Pontrjagin,3 Steenrod,4 Feldbau,5 Ehresmann,6 etc. have added considerably to our knowledge of the topology of manifolds with a differentiable
Strong Rigidity of Locally Symmetric Spaces.
*Frontmatter, pg. i*Contents, pg. v* 1. Introduction, pg. 1* 2. Algebraic Preliminaries, pg. 10* 3. The Geometry of chi : Preliminaries, pg. 20* 4. A Metric Definition of the Maximal Boundary, pg.
Métriques riemanniennes et courbure
Nous allons etudier certains changements de metrique sur les varietes Riemanniennes, et examiner dans quelle mesure, on peut modifier les proprietes de la courbure. Un probleme fondamental de la
" Uber vierdimensionale Einstein - raume
  • 1952
Collected Works (Princeton Univ. Press, Princeton, NJ)
  • Vols. II and III
  • 1975
Equations du type Mongi-Ampere sur les varietes kdhleriennes compactes,
  • C. R. Acad. Sci. Hebd. Seances
  • 1976
" Equations du type Mongi - Ampere sur les varietes kahleriennes compactes , " C
  • R . Acad . Sci . Hebd . Seances
  • 1976
...
...