Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

Abstract

Abstract. We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.

Holomorphic Cartan geometries and Calabi–Yau manifolds

“…In view of Lemma 2.1, from (3.2) we conclude that c 2 (T M) = 0. Now the proof is completed by Lemma 1 of [6].…”

“…Sorin Dumitrescu [5] proved that if a Calabi-Yau manifold has a rigid holomorphic geometric structure of affine algebraic type, then it admits a finite covering by a torus. In [6] the second author conjectured a more general result that if a Calabi-Yau manifold M admits a holomorphic Cartan geometry, then M is covered by a torus. Our aim here is to prove this conjecture under the assumption that M is complex projective.…”

Holomorphic geometric structures on Kähler–Einstein manifolds

“…In particular, ş M η^Ω n2 2 ě 0 and equality occurs just when η " 0, which occurs just when T " 0, which occurs just when M 1 has constant sectional curvature. [34] …”

Holomorphic Cartan geometries, Calabi–Yau manifolds and rational curves

“…Since c 1 (T M) = 0, we conclude that c 2 (T M) = 0. Now from [10, p. 248, Corollary 2.2] (see also Lemma 1 of [16]) we know that M is holomorphically covered by a complex torus. This completes the proof of the theorem.…”

“…In [16], the second author conjectured that the only Calabi-Yau manifolds that bear holomorphic Cartan geometries are those covered by the complex tori. Our first theorem confirms this conjecture.…”