zbMATH — the first resource for mathematics

Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. (English) Zbl 0617.32044
A theorem of P. Gauduchon states that an arbitrary hermitian metric on a compact complex surface has a conformal rescaling such that the associated Kähler form is then \({\bar \partial}\partial\)-closed. Given such a form, the degree of a holomorphic line bundle can be defined in the usual way and with that, the notion of stability in the sense of Mumford and Takemoto for torsion-free sheaves. It is proved here that an indecomposable holomorphic vector bundle on the surface is stable iff it admits an irreducible Hermitian-Einstein connection, where ”stable” and ”Hermitian-Einstein” are both with respect to a given positive \({\bar \partial}\partial\)-closed (1,1)-form. This generalizes a result of Donaldson, who proved this theorem in the case of algebraic surfaces in \({\mathbb{P}}_ N\) equipped with a Kähler metric whose Kähler form is cohomologous to that of the Fubini-Study metric.

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53C05 Connections, general theory
32J15 Compact complex surfaces
Full Text: DOI EuDML
[1] Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Phil. Trans. Roy. Soc. Lond. Ser. A308, 524-615 (1982) · Zbl 0509.14014
[2] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. Roy. Soc. Lond. Ser. A362, 425-461 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[3] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin Heidelberg New York: Springer 1984 · Zbl 0718.14023
[4] Donaldson, S.K.: A new proof of a theorem of Narasimhan and Seshadri. J. Differ. Geom.18, 269-277 (1983) · Zbl 0504.49027
[5] Donaldson, S.K.: Anti-self-dual Yang-Mills connections over complex algebraic varieties and stable vector bundles. Proc. Lond. Math. Soc.50, 1-26 (1985) · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1
[6] Donaldson, S.K.: La topologie différentielle des surfaces complexes. C. R. Acad. Sci. Paris301, 317-320 (1985) · Zbl 0584.57010
[7] Gauduchon, P.: Le thèorème de l’excentricité nulle. C. R. Acad. Sci. Paris285, 387-390 (1977) · Zbl 0362.53024
[8] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd ed. Berlin Heidelberg New York: Springer 1983 · Zbl 0562.35001
[9] Grauert, H., Remmert, R.: Coherent analytic sheaves. Berlin Heidelberg New York: Springer 1984 · Zbl 0537.32001
[10] Griffiths, P.A., Harris, J.: Principles if algebraic geometry. New York: Wiley 1987
[11] Hitchin, N.J.: Math. Rev.81e, 1822 (1981)
[12] Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles over curves. Math. Ann.212, 215-248 (1975) · Zbl 0324.14006 · doi:10.1007/BF01357141
[13] Kobayashi, S.: First Chern class and holomorphic tensor fields. Nagoya Math. J.77, 5-11 (1980) · Zbl 0432.53049
[14] Kobayashi, S.: Curvature and stability of vector bundles. Proc. Japan Acad. Ser. A. Math. Sci.58, 158-162 (1982) · Zbl 0546.53041 · doi:10.3792/pjaa.58.158
[15] Lübke, M.: Chernklassen von Hermite-Einstein Vektor-Bündeln. Math. Ann.260, 133-141 (1982) · Zbl 0481.53058 · doi:10.1007/BF01475761
[16] Lübke, M.: Stability of Einstein-Hermitian vector bundles. Manuscr. Math.42, 245-257 (1983) · Zbl 0558.53037 · doi:10.1007/BF01169586
[17] Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math.82, 540-564 (1965) · Zbl 0171.04803 · doi:10.2307/1970710
[18] Okonek, C., Schneider, M., Spindler, H.: Vector bundles over complex projective space. Boston: Birkhäuser 1980 · Zbl 0438.32016
[19] Sedlacek, S.: A direct method for minimizing the Yang-Mills functional. Commun. Math. Phys.86, 515-528 (1982) · Zbl 0506.53016 · doi:10.1007/BF01214887
[20] Serre, J.-P.: Prolongement de faisceaux analytiques cohérents. Ann. Inst. Fourier16, 363-374 (1966) · Zbl 0144.08003
[21] Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11-30 (1982) · Zbl 0491.58032 · doi:10.1007/BF01947068
[22] Uhlenbeck, K.K.: Connection withL p bounds on curvature. Commun. Math. Phys83, 31-42 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069
[23] Uhlenbeck, K.K., Yau, S.-T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure App. Math.39, 257-293 (1986) · Zbl 0615.58045 · doi:10.1002/cpa.3160390714
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.