Generalized holomorphic Cartan geometries
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Generalized holomorphic Cartan geometries Indranil Biswas1 · Sorin Dumitrescu2 Dedicated to the memory of Stefan Papadima Received: 6 September 2018 / Revised: 18 February 2019 / Accepted: 25 February 2019 © Springer Nature Switzerland AG 2019
Abstract This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of Felix Klein, Élie Cartan and Charles Ehresmann. Then we present the concept of a branched holomorphic Cartan geometry which was introduced by Biswas and Dumitrescu (Int Math Res Not IMRN, 2017. https://doi.org/10.1093/imrn/rny003, arxiv:1706.04407). It generalizes to higher dimension the notion of a 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 (e.g. all compact complex projective manifolds admit branched holomorphic projective structures). At the same time, this new definition is rigid enough to enable us to classify branched holomorphic Cartan geometries on compact simply connected Calabi–Yau manifolds. Keywords Homogeneous spaces · Cartan geometries · Calabi–Yau manifolds Mathematics Subject Classification 53B21 · 53C56 · 53A55
IB is partially supported by a J.C. Bose Fellowship.
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Sorin Dumitrescu [email protected] Indranil Biswas [email protected]
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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
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Université Côte d’Azur, CNRS, LJAD, 06108 Nice Cedex 2, France
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I. Biswas, S. Dumitrescu
1 Introduction An important consequence of the uniformization theorem for Riemann surfaces asserts, in particular, that any Riemann surface M admits a holomorphic atlas with coordinates in CP 1 and transition maps in PSL(2, C). In other words, any Riemann surface is locally modelled on the complex projective line. This defines a (flat) complex projective structure on M. Complex projective structures on Riemann surfaces were introduced in connection with the study of the second order ordinary differential equations on complex domains and had a very major role to play in understanding the framework of the uniformization theorem [11,17]. The complex projective line CP 1 acted on by the Möbius group PSL(2, C) is a geometry in the sense of Klein’s Erlangen program in which he proposed to study all geometries in the unifying frame of the homogeneous model spaces G/H , where G is a finite dimensional Lie group and H a closed subgroup of G. Thus homogeneous spaces could be used as good models to geometrize higher dimensional topological manifolds. Indeed, following Ehresmann [14], a manifold M is locally modelled on a homogeneous space G/H , if M admits an atlas with charts in G/H satisfying the condition that the transition maps are given by elements of G using the left-translation action of G on G/H . In this way M is locally endowed with the G/H -geometry and all G-invariant geometrical features of G/H have
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