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Characteristic forms of complex Cartan geometries. (English) Zbl 1214.53059

Let us recall that compact complex manifolds covered by noncompact Hermitian symmetric spaces share the same Chern numbers as the compact dual up to scaling. There have been established relations on all characteristic classes (not just the characteristic numbers) for normal projective connections following the Thomas theory of normal projective connections – instead of Cartan’s theory – employing sheaves instead of principal bundles. This approach has been extended to obtain relations on characteristic classes for certain holomorphic \(G\)-structures. Cartan’s theory generalizes easily to Cartan geometries, and manages abnormality without extra effort.
The author uses the Cartan theory to study relations between characteristic classes, generalizing the characteristic class results of all previously mentioned cases. The main result is:
Theorem. The ring of characteristic classes of a holomorphic Cartan geometry on a compact Kähler manifold is a quotient of the ring of characteristic forms of the model via an explicit ring morphism.
The author also derives the relations on characteristic rings of various rational homogeneous varieties to give examples, and explains how to employ these results to study various types of complex analytic differential equations.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C56 Other complex differential geometry
57R20 Characteristic classes and numbers in differential topology
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