BRANCHED HOLOMORPHIC CARTAN GEOMETRIES AND CALABI-YAU MANIFOLDS

Indranil Biswas (1) , Sorin Dumitrescu (2)

1 TIFR - Tata Institute for Fundamental Research
2 LJAD - Laboratoire Jean Alexandre Dieudonné

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures. The key ingredient of its proof is the following result of independent interest: If E is a holomorphic vector bundle over a compact simply connected Kähler Calabi-Yau manifold, and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle equipped with the trivial connection.

Mots clés: Complex projective structure; Calabi-Yau manifolds

Domaines: Mathématiques [math]

Indranil Biswas, Sorin Dumitrescu. BRANCHED HOLOMORPHIC CARTAN GEOMETRIES AND CALABI-YAU MANIFOLDS. International Mathematics Research Notices, 2019, 2019 (23), pp.7428-7458. ⟨hal-01539196⟩