×

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.

MSC:

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

References:

[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
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.