# Generalized holomorphic Cartan geometries

@article{Biswas2019GeneralizedHC, title={Generalized holomorphic Cartan geometries}, author={Indranil Biswas and Sorin Dumitrescu}, journal={European Journal of Mathematics}, year={2019}, pages={1-20} }

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…

## 8 Citations

### CARTAN GEOMETRIES ON COMPLEX MANIFOLDS OF ALGEBRAIC DIMENSION ZERO

- MathematicsÉpijournal de Géométrie Algébrique
- 2019

International audience
We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes…

### Frobenius-Ehresmann structures and Cartan geometries in positive characteristic

- MathematicsIndagationes Mathematicae
- 2022

### Deformation theory of holomorphic Cartan geometries, II

- MathematicsComplex Manifolds
- 2022

Abstract In this continuation of [4], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal…

### Holomorphic G-Structures and Foliated Cartan Geometries on Compact Complex Manifolds

- MathematicsSurveys in Geometry I
- 2022

This is a survey paper dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case: holomorphic foliations with…

### Holomorphic Bundles Trivializable by Proper Surjective Holomorphic Map

- Mathematics
- 2020

Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $\varphi^* E$ is trivial for some surjective holomorphic map $\varphi$, to $M$, from some…

### Stability and holomorphic connections on vector bundles over LVMB manifolds

- Mathematics
- 2019

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero…

## 36 References

### Branched holomorphic Cartan geometries and Calabi-Yau manifolds

- Mathematics
- 2017

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…

### Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

- Mathematics
- 1988

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex…

### On Uniformization of Complex Manifolds: The Role of Connections

- Mathematics
- 2015

The classical uniformization theorem for Riemann surfaces and its recent extensions can be viewed as introducing special pseudogroup structures, affine or projective structures, on higher-dimensional complex manifolds, modeled on the theory as developed for R Siemann surfaces.

### Branched structures on Riemann surfaces

- Mathematics
- 1972

Following results of Gunning on geometric realizations of projective structures on Riemann surfaces, we investigate more fully certain generalizations of such structures. We define the notion of a…

### Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups

- Mathematics
- 2004

Preface: why projective? 1. Introduction 2. The geometry of the projective line 3. The algebra of the projective line and cohomology of Diff(S1) 4. Vertices of projective curves 5. Projective…

### Complex analytic connections in fibre bundles

- Mathematics
- 1957

Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic…

### Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

- Mathematics
- 2007

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple…

### On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

- Mathematics
- 1978

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the…

### On the relations between branched structures and affine and projective bundles on Riemann surfaces

- Mathematics
- 1971

A classification for analytic branched G-structures on a Riemann surface M is provided by means of a map (G into the moduli spaces of flat G-bundles on M. (G = GA(1, C) or PL(1, C).) Conditions are…