A note on positivity of Einstein bundles. (English) Zbl 0751.32016

The following result is obtained. Let \(X\) be a compact Kähler surface and \(E\) be a Hermitian bundle of rank 2 on \(X\) such that the first Chern form \(c_ 1(E)\) is equal to the Kähler form \(\omega\) up to a positive constant. Suppose that the first and second Chern forms satisfy \(c^ 2_ 1(E)-2c_ 2(E)=\varphi\centerdot\omega^ 2\), where \(\varphi\) is a positive function. Then \(E\) is positive. This is a differential algebraic version of an algebraic ampleness criterion of M. Schneider and A. Tancredi.


32Q20 Kähler-Einstein manifolds
32J15 Compact complex surfaces
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[1] Barth, W.; Peters, C.; Ven, A.van de, Compact complex surfaces, (Erg. der Math., 3 (1985), Springer: Springer Berlin, Heidelberg, New York), 4
[2] Donaldson, S. K., Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc., 50, 1-26 (1985) · Zbl 0529.53018
[3] Donaldson, S. K., Infinite determinants, stable bundles and curvature, Duke Math. J., 54, 231-247 (1987) · Zbl 0627.53052
[4] Griffiths, P. A., Hermitian differential geometry, Chern classes and positive vector bundles, (Global analysis, Papers in honor of K. Kodaira (1969), University of Tokyo Press and Princeton University Press), 185-251 · Zbl 0201.24001
[5] Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture Notes in Math., 156 (1970), Springer: Springer Berlin/Heidelberg/New York · Zbl 0208.48901
[6] Kobayashi, S., Differential geometry of complex vector bundles (1987), Iwanami Shoten and Princeton University Press · Zbl 0708.53002
[7] Lübke, M., Stability of Einstein-Hermitian vector bundles, Manuscripta Math., 42, 245-257 (1983) · Zbl 0558.53037
[8] Schneider, M., Complex surfaces with negative tangent bundle, Complex analysis and algebraic geometry, Lecture Notes in Math., 1194 (1986), Springer: Springer Berlin/Heidelberg/New York · Zbl 0592.14016
[9] Schneider, M.; Tancredi, A., Positive vector bundles on complex surfaces, Manuscripta Math., 50, 133-144 (1985) · Zbl 0572.32015
[10] Tsai, I.-H., Negatively curved metrics on Kodaira surfaces, Math. Ann., 285, 369-379 (1989) · Zbl 0664.53033
[11] Uhlenbeck, K. K.; Yau, S.-T., On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math., 39, 257-293 (1986) · Zbl 0615.58045
[12] Uhlenbeck, K. K.; Yau, S.-T., A note on our previous paper: On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math., Vol. XLII, 703-707 (1989) · Zbl 0678.58041
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