Advertisement

Fibrés semi-positifs et semi-négatifs sur une variété Kählérienne compacte

  • 21 Accesses

  • 3 Citations

Summary

Let M be a kaehlerian compact manifold and E\(\xrightarrow{r}\) M a holomorphic line bundle. In the first part two vanishing theorems for Hv,0(E) and H0,q(E) are obtained. In the second part the following theorems are proved: 1) Let E\(\xrightarrow{r}\) M be a holomorphic vector bundle weakly semi-definite negative. If the Ricci tensor of M is ≥0 on M and >0 in a point, then Hv,0(E)=0, p >0. 2) Let M' be a kaehlerian manifold with negative bisectional holomorphic curvature. Let M be a compact complex submanifold. There is no holomorphic vector field of M' over M and there is no locally trivial strict deformation of M. Intuitively this means that among the complex submanifolds close to M (if there is any) there exists one with a complex structure different from that of M.

Bibliographie

  1. [1]

    S. Bochner,Vectors fields and Ricci curvature, Bull. Amer. Math. Soc.,52 (1946), pp. 776–797.

  2. [2]

    S. S. Chern,Complex manifolds without potential theory, Van Nostrand, Princeton, 1967.

  3. [3]

    P. Gauduchon,Sur les formes à valeurs dans un fibré vectoriel holomorphe, C. R. Acad. Sc. Paris,273 (1971), pp. 398–401.

  4. [4]

    J. Girbau,Sur les théorèmes d'annulation de Kodaira, C. R. Acad. Sc. Paris,272 (1971), pp. 740–742 et273 (1971), pp. 461–462.

  5. [5]

    P. A. Griffiths,Hermitian differential geometry, J. Math. Mech.,14 (1965), pp. 117–140.

  6. [6]

    P. A. Griffiths,Positive vectors bundles. Global analysis, Princeton Math., series n. 29 (1969), pp. 185–281.

  7. [7]

    K. Kodaira,On a differential geometric method in the theory of analytic stacks, Proc. N. Ac. Sc. USA,39 (1953), pp. 1268–1273.

  8. [8]

    K. Kodaira,On cohomology groups of compact analytic varieties, Proc. N. Ac. Sc. USA,39 (1953), pp. 865–868.

  9. [9]

    K. Kodaira -D. C. Spencer,On deformations on complex analytic structures, Ann. of Math.,63 (1958), pp. 328–466.

  10. [10]

    A. Lichnerowicz,Variétés kählériennes et première classe de Chern, J. of Diff. Geom.,1 (1967), pp. 195–223.

  11. [11]

    A. Lichnerowicz,Applications harmoniques et variétes kählériennes, Inst. Naz. di Alta Math., Symposia Math. III, pp. 341–402.

  12. [12]

    S. Nakano,On complex analytic vector bundles, J. Math. Soc. Japan,7 (1955), pp. 1–12.

  13. [13]

    E. Vesentini,Osservazioni sulle strutture fibrate analitiche, Atti Ac. Naz. Lincei,23 (1957), pp. 231–241 et24 (1957), pp. 505–512.

Download references

Author information

Additional information

Entrata in Redazione l'll novembre 1972.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Girbau, J. Fibrés semi-positifs et semi-négatifs sur une variété Kählérienne compacte. Annali di Matematica 101, 171 (1974). https://doi.org/10.1007/BF02417103

Download citation