This chapter is an exposition of some of the basic ideas of Hermitian differential geometry, with applications to Chern classes and holomorphic line bundles. In Sec. 1 we shall give the basic definitions of the Hermitian analogues of the classical concepts of (Riemannian) metric, connection, and curvature. This is carried out in the context of differentiable C-vector bundles over a differentiable manifold X. More specific formulas are obtained in the case of holomorphic vector bundles (in Sec. 2) and holomorphic line bundles (in Sec. 4). In Sec. 3 is presented a development of Chern classes from the differential-geometric viewpoint. In Sec. 4 this approach to characteristic class theory is compared with the classifying space approach and with the sheaf-theoretic approach (in the case of line bundles). We prove that the Chern classes are primary obstructions to finding trivial subbundles of a given vector bundle, and, in particular, to the given vector bundle being itself trivial. In the case of line bundles, we give a useful characterization of which cohomology classes in H2(X, Z) are the first Chern class of a line bundle. Additional references for the material covered here are Chern [2], Griffiths [2], and Kobayashi and Nomizu [1].
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© 2008 Springer Science +Business Media, LLC
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Wells, R.O. (2008). Differential Geometry. In: Differential Analysis on Complex Manifolds. Graduate Texts in Mathematics, vol 65. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73892-5_3
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DOI: https://doi.org/10.1007/978-0-387-73892-5_3
Publisher Name: Springer, New York, NY
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