Representability of Chern-Weil forms. (English) Zbl 1402.32021

Let \(X\) be a complex projective manifold, and let \(E\) be a vector bundle of rank \(r\) over \(X\). The vector bundle \(E\) is Hartshorne-ample if its tautological line bundle is ample. If \(E\) admits a metric that is Griffiths-positive, it is easy to see that the vector bundle is Hartshorne-ample. It is expected that Hartshorne-ample bundles are always Griffiths-positive, but this is unknown even for surfaces. In this paper the author proves a weak version on the level of the associated differential forms: it is well known that for a Hartshorne-ample vector bundle over a projective surface the Chern classes \(c_1(E), c_2(E),c_1^2(E)-c_2(E)\) are positive cohomology classes. The author proves that if \(E\) is Hartshorne-ample and semistable with respect to some polarization, there exists a metric \(h\) on \(E\) such that the associated Chern-Weil forms for \(c_1(h), c_2(h),c_1^2(h)-c_2(h)\) are positive.
In the second part of the paper the author introduces a Calabi-Yang-Mills equation and solves it for a certain rank-two vector bundle over a product \(\Sigma \times \mathbb P^1\), where \(\Sigma\) is a Riemann surface.


32L05 Holomorphic bundles and generalizations
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C56 Other complex differential geometry
Full Text: DOI arXiv


[1] Alvarez-Consul, L., Garcia-Fernandez, M., Garcia-Prada, O.: Gravitating vortices and the Einstein-Bogomol’nyi equations. arXiv: 1606.07699 · Zbl 1361.53021
[2] Alvarez-Consul, L., Garcia-Fernandez, M., Garcia-Prada, O.: Gravitating vortices, cosmic strings, and the Kähler-Yang-Mills equations. arXiv: 1510.03810. (To appear in Commun. Math. Phys.) · Zbl 1361.53021
[3] Alvarez-Consul, L., Garcia-Fernandez, M., Garcia-Prada, O.: Coupled equations for Kähler metrics and Yang-Mills connections. Geom. Top. 17, 2731-2812 (2013) · Zbl 1275.32019 · doi:10.2140/gt.2013.17.2731
[4] Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169, 531-560 (2009) · Zbl 1195.32012 · doi:10.4007/annals.2009.169.531
[5] Bloch, S., Gieseker, D.: The positivity of the Chern classes of an ample vector bundle. Invent. Math. 12, 112-117 (1971) · Zbl 0212.53502 · doi:10.1007/BF01404655
[6] Bott, R., Chern, S.S.: Hermitian vector bundles and equidistribution of the zeroes of their holomorphic cross-sections. Acta. Math. 114, 71-112 (1968) · Zbl 0148.31906 · doi:10.1007/BF02391818
[7] Campana, F., Flenner, H.: A characterization of ample vector bundles on a curve. Math. Ann. 287(1), 571-575 (1990) · Zbl 0728.14033 · doi:10.1007/BF01446914
[8] Diverio, S.: Segre forms and Kobayashi-Lübke inequality. Math. Z. 283, 1033-1047 (2016) · Zbl 1347.53022 · doi:10.1007/s00209-016-1632-y
[9] Donaldson, S.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50(1), 1-26 (1985) · Zbl 0529.53018 · doi:10.1112/plms/s3-50.1.1
[10] Fulton, W., Lazarsfeld, R.: Positive polynomials for ample vector bundles. Ann. Math. 118, 35-60 (1983) · Zbl 0537.14009 · doi:10.2307/2006953
[11] García-Prada, O.: Invariant connections and vortices. Commun. Math. Phys. 156, 527-546 (1993) · Zbl 0790.53031 · doi:10.1007/BF02096862
[12] Guler. D.: Chern forms of positive vector bundles. Electronic Thesis or Dissertation. Ohio State University (2006) https://etd.ohiolink.edu/ · Zbl 0148.31906
[13] Guler, D.: On Segre forms of positive vector bundles. Can. Math. Bull. 55(1), 108-113 (2012) · Zbl 1239.53092 · doi:10.4153/CMB-2011-100-6
[14] Kazdan, J.L., Warner, F.W.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 1447 (1978)
[15] Kobayashi, S.: Differential geometry of complex vector bundles. Princeton University Press, Princeton (2014)
[16] Mourougane, C., Takayama, S.: Hodge metrics and positivity of direct images. Crelles J. 2007(606), 167-178 (2007) · Zbl 1128.14030 · doi:10.1515/CRELLE.2007.039
[17] Pingali, V.: A fully nonlinear generalized Monge-Ampère PDE on a torus. Elec. J. Diff. Eq. 2014(211), 1-8 (2014) · Zbl 1301.53073
[18] Siu, Y.T.: Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics. Birkhäuser, Basel (1987) · Zbl 0631.53004 · doi:10.1007/978-3-0348-7486-1
[19] Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39(S1), S257-S293 (1986) · Zbl 0615.58045 · doi:10.1002/cpa.3160390714
[20] Umemura, H.: Some results in the theory of vector bundles. Nagoya Math. J. 52, 97-128 (1973) · Zbl 0271.14005 · doi:10.1017/S0027763000015919
[21] Yau, S.T.: On the Ricci curvature of a compact kähler manifold and the complex Monge-Ampère equation I. Commun. Pure Appl. Math. 31(3), 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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