Skip to main content
Log in

Positive vector bundles on complex surfaces

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

A criterion for the positivity of a semi-stable vector bundle of rank 2 on a projective surface is proved. This is of particular interest for cotangent bundles of surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. BOGOMOLOV, F.A.: Holomorphic tensors and vector bundles on projective varieties. Math. USSR Izvestija13 (1979), 499–555

    Google Scholar 

  2. FULTON, W.: Ample vectorbundles, Chern classes and numerical criteria. Inventiones math.32 (1976), 171–178

    Google Scholar 

  3. FULTON, W.,LAZARSFELD, R.: Positive polynomials for ample vector bundles. Ann. of Math.118 (1983), 35–60

    Google Scholar 

  4. GIESEKER, D.: p-ample bundles and their Chern classes. Nagoya Math. J.43 (1971), 91–116

    Google Scholar 

  5. GRAUERT, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann.146 (1962), 331–368

    Google Scholar 

  6. GRIFFITHS, Ph. A.: The extension problem in complex analysis II. Embeddings with positive normal bundle. Amer. J. Math.88 (1966), 366–446

    Google Scholar 

  7. GRIFFITHS, Ph. A.: Hermitian differential geometry, Chern classes and positive vector bundles. Global Analysis (papers in honor of K. Kodaira). Univ. of Tokyo Press & Princeton Univ. Press 1969, 185–251

  8. HIRZEBRUCH, F.: Topological methods in algebraic geometry. 3d edition. Springer: Berlin-Heidelberg-New York 1966

    Google Scholar 

  9. HARTSHORNE, R.: Ample vector bundles. Publ. Math. IHES29 (1966), 63–94

    Google Scholar 

  10. HARTSHORNE, R.: Ample subvarieties of algebraic varieties. LNM156, Springer 1970

  11. HARTSHORNE, R.: Ample vector bundles on curves. Nagoya Math. J.43 (1971), 73–89

    Google Scholar 

  12. HOSOH, T.: Ample vector bundles on a rational surface. Nagoya Math. J.59 (1975), 135–148

    Google Scholar 

  13. KAS, A.: On deformations of a certain type of irregular algebraic surface. Amer. J. Math.90 (1968), 789–804

    Google Scholar 

  14. KLEIMAN, S.: Ample vector bundles on surfaces. Proc. Amer. Math. Soc. (1969), 673–676

  15. KOBAYASHI, S.: First Chern class and holomorphic tensor fields. Nagoya Math. J.77 (1980), 5–11

    Google Scholar 

  16. KODAIRA, K.: A certain type of irregular algebraic surfaces. Journal d'Analyse Math.19 (1967), 207–215

    Google Scholar 

  17. LÜBKE, M.: Stability of Einstein-Hermitian vector bundles. Manuscripta math.42 (1983), 245–247

    Google Scholar 

  18. MARUYAMA, M.: The theorem of Grauert-Mülich-Spindler. Math. Ann.255 (1981), 317–333

    Google Scholar 

  19. MOISHEZON, B. G.: A criterion for projectivity of complete algebraic varieties. AMS transl. (2)63 (1967), 1–50

    Google Scholar 

  20. MOSTOW, G. D.,SIU, Y. T.: A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math.112 (1980), 321–360

    Google Scholar 

  21. NAKAI, Y.: A criterion of an ample sheaf on a projective scheme. Amer. J. Math.85 (1963), 14–26

    Google Scholar 

  22. LE POTIER, J.: Stabilité et amplitude sur ℙ2(C). In Vector Bundles and Differential Equations. Proc. Nice 1979. Progress in Math.7, 145–182. Birkhäuser: Boston 1980

    Google Scholar 

  23. SCHNEIDER, M.: Stabile Vektorraumbündel vom Rang 2 auf der projektiven Ebene. Nachr. Akad.Wiss.Göttingen. Math. Naturw. Klasse n°6, 1976

  24. SOMMESE, A.: On the density of ratios of Chern numbers of algebraic surfaces. Math. Ann.268 (1984), 207–221

    Google Scholar 

  25. WONG, B.: Curvature and pseudoconvexity on complex manifolds. Advances in Math.37 (1980), 99–104

    Google Scholar 

  26. WONG, B.: The uniformization of compact complex Kähler surfaces of negative curvature. J. Diff. Geom.16 (1981), 407–420

    Google Scholar 

  27. WONG, B.: A class of compact complex manifolds with negative tangent bundles. AMS Proc. of Symposia in Pure Math.41 (1984), 217–223

    Google Scholar 

  28. YAU, S. T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Pure and Appl. Math.XXXI (1978), 339–411

    Google Scholar 

  29. YAU, S. T.: Seminar on Differential Geometry. Annals of Math. Studies102. Princeton Univ. Press 1982

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to Professor Karl Stein

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schneider, M., Tancredi, A. Positive vector bundles on complex surfaces. Manuscripta Math 50, 133–144 (1985). https://doi.org/10.1007/BF01168829

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01168829

Keywords

Navigation