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Holomorphic Cartan geometries, Calabi-Yau manifolds and rational curves. (English) Zbl 1187.53028

Let \(G\) be a complex Lie group with a closed subgroup \(H\). A Cartan geometry of type \(G/H\) on a compact Kähler manifold \(M\) is a holomorphic \(H\)-bundle over \(M\) with a holomorphic \(\mathfrak{g}\)-valued \(1\)-form \(\theta\) on it satisfying certain conditions. The notion is modeled on the quotient bundle \(G\to G/H\) with \(\theta=g^{-1}dg\) known as the tautological Cartan geometry.
McKay conjectured recently that the only Calabi-Yau manifolds admitting a Cartan geometry are those étale covered by a complex torus. The main result of this note is a proof of this conjecture via the Bogomolov inequality for semistable sheaves. In addition, the authors show that a Cartan geometry on a projective and rationally connected \(M\) is holomorphically isomorphic to the tautological one.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
14M17 Homogeneous spaces and generalizations
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
32Q25 Calabi-Yau theory (complex-analytic aspects)
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References:

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