×

Positive vector bundles on complex surfaces. (English) Zbl 0572.32015

A criterion for the positivity of a semi-stable rank 2 vector bundle on a projective surface is given. As application the following result is obtained. Let X be a complex projective surface. If \(c_ 1(X)^ 2-2c_ 2(X)>0\), \(c_ 2(X)>0\) and \(T^*\!_ X/C\) is positive for all integral curves \(C\subset X\), then \(T^*_ X\) is positive.
Reviewer: A.Pankov

MSC:

32L05 Holomorphic bundles and generalizations
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] BOGOMOLOV, F.A.: Holomorphic tensors and vector bundles on projective varieties. Math. USSR Izvestija13 (1979), 499-555 · Zbl 0439.14002 · doi:10.1070/IM1979v013n03ABEH002076
[2] FULTON, W.: Ample vectorbundles, Chern classes and numerical criteria. Inventiones math.32 (1976), 171-178 · Zbl 0341.14004 · doi:10.1007/BF01389960
[3] FULTON, W.,LAZARSFELD, R.: Positive polynomials for ample vector bundles. Ann. of Math.118 (1983), 35-60 · Zbl 0537.14009 · doi:10.2307/2006953
[4] GIESEKER, D.: p-ample bundles and their Chern classes. Nagoya Math. J.43 (1971), 91-116 · Zbl 0221.14010
[5] GRAUERT, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann.146 (1962), 331-368 · Zbl 0173.33004 · doi:10.1007/BF01441136
[6] GRIFFITHS, Ph. A.: The extension problem in complex analysis II. Embeddings with positive normal bundle. Amer. J. Math.88 (1966), 366-446 · Zbl 0147.07502 · doi:10.2307/2373200
[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 · Zbl 0138.42001
[9] HARTSHORNE, R.: Ample vector bundles. Publ. Math. IHES29 (1966), 63-94 · Zbl 0173.49003
[10] HARTSHORNE, R.: Ample subvarieties of algebraic varieties. LNM156, Springer 1970 · Zbl 0208.48901
[11] HARTSHORNE, R.: Ample vector bundles on curves. Nagoya Math. J.43 (1971), 73-89 · Zbl 0218.14018
[12] HOSOH, T.: Ample vector bundles on a rational surface. Nagoya Math. J.59 (1975), 135-148 · Zbl 0332.14006
[13] KAS, A.: On deformations of a certain type of irregular algebraic surface. Amer. J. Math.90 (1968), 789-804 · Zbl 0202.51702 · doi:10.2307/2373484
[14] KLEIMAN, S.: Ample vector bundles on surfaces. Proc. Amer. Math. Soc. (1969), 673-676 · Zbl 0176.18502
[15] KOBAYASHI, S.: First Chern class and holomorphic tensor fields. Nagoya Math. J.77 (1980), 5-11 · Zbl 0432.53049
[16] KODAIRA, K.: A certain type of irregular algebraic surfaces. Journal d’Analyse Math.19 (1967), 207-215 · Zbl 0172.37901 · doi:10.1007/BF02788717
[17] LÜBKE, M.: Stability of Einstein-Hermitian vector bundles. Manuscripta math.42 (1983), 245-247 · Zbl 0558.53037 · doi:10.1007/BF01169586
[18] MARUYAMA, M.: The theorem of Grauert-Mülich-Spindler. Math. Ann.255 (1981), 317-333 · doi:10.1007/BF01450706
[19] MOISHEZON, B. G.: A criterion for projectivity of complete algebraic varieties. AMS transl. (2)63 (1967), 1-50 · Zbl 0186.26203
[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 · Zbl 0453.53047 · doi:10.2307/1971149
[21] NAKAI, Y.: A criterion of an ample sheaf on a projective scheme. Amer. J. Math.85 (1963), 14-26 · Zbl 0112.13102 · doi:10.2307/2373180
[22] LE POTIER, J.: Stabilité et amplitude sur ?2(C). In Vector Bundles and Differential Equations. Proc. Nice 1979. Progress in Math.7, 145-182. Birkhäuser: Boston 1980
[23] SCHNEIDER, M.: Stabile Vektorraumbündel vom Rang 2 auf der projektiven Ebene. Nachr. Akad.Wiss.Göttingen. Math. Naturw. Klasse n{\(\deg\)}6, 1976 · Zbl 0349.14007
[24] SOMMESE, A.: On the density of ratios of Chern numbers of algebraic surfaces. Math. Ann.268 (1984), 207-221 · doi:10.1007/BF01456086
[25] WONG, B.: Curvature and pseudoconvexity on complex manifolds. Advances in Math.37 (1980), 99-104 · Zbl 0453.32003 · doi:10.1016/0001-8708(80)90029-8
[26] WONG, B.: The uniformization of compact complex Kähler surfaces of negative curvature. J. Diff. Geom.16 (1981), 407-420 · Zbl 0518.32019
[27] WONG, B.: A class of compact complex manifolds with negative tangent bundles. AMS Proc. of Symposia in Pure Math.41 (1984), 217-223 · Zbl 0552.32026
[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 · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[29] YAU, S. T.: Seminar on Differential Geometry. Annals of Math. Studies102. Princeton Univ. Press 1982 · Zbl 0471.00020
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.