Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds. (7th February 2018)

• Record Type: Journal Article
• Authors: Biswas, Indranil; Dumitrescu, Sorin
• Is Part Of: International mathematics research notices. Volume 2019:Number 23(2019)
• Page Start: 7428
• Page End: 7458
• Publication Date: 2018-02-07
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny003
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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 [25 ]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a 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 any branched holomorphic projective structure. 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 and any holomorphic connection on E is trivial.