Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access February 20, 2022

Deformation theory of holomorphic Cartan geometries, II

Indranil Biswas, Sorin Dumitrescu and Georg Schumacher
From the journal Complex Manifolds

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 deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.

MSC 2010: 32G13; 53C55

References

[1] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc.85 (1957), 181–207.10.1090/S0002-9947-1957-0086359-5Search in Google Scholar

[2] N. Bergeron and T. Gelander, A note on local rigidity, Geom. Dedicata107 (2004), 111–131.10.1023/B:GEOM.0000049122.75284.06Search in Google Scholar

[3] I. Biswas and S. Dumitrescu, Generalized holomorphic Cartan geometries, European J. Math. 6 (special issue dedicated to the memory of Stefan Papadima) (2020), 661–680.10.1007/s40879-019-00327-6Search in Google Scholar

[4] I. Biswas, S. Dumitrescu and G. Schumacher, Deformation theory of holomorphic Cartan geometries, Indag. Math.31 (2020), 512–524.10.1016/j.indag.2020.03.008Search in Google Scholar

[5] R. D. Canary, D. B. A. Epstein and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), 3–92, London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, Cambridge, 1987.Search in Google Scholar

[6] T. Chen, The associated map of the nonabelian Gauss–Manin connection, Cent. Eur. Jour. Math.10 (2012), 1407–1421.10.2478/s11533-011-0110-3Search in Google Scholar

[7] C. Ehresmann, Sur les espaces localement homogènes, L’Enseign. Math.35 (1936), 317–333.Search in Google Scholar

[8] E. Ghys, Déformations des structures complexes sur les espaces homogènes de SL(2, ℂ), Jour. Reine Angew. Math.468 (1995), 113–138.10.1515/crll.1995.468.113Search in Google Scholar

[9] W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math.54 (1984), 200–225.10.1016/0001-8708(84)90040-9Search in Google Scholar

[10] W. M. Goldman and J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math.67 (1988), 43–96.10.1007/BF02699127Search in Google Scholar

[11] R. C. Gunning, On uniformization of complex manifolds: the role of connections, Princeton Univ. Press, 1978.Search in Google Scholar

[12] M. S. Raghunathan, Vanishing theorems for cohomology groups associate to discrete subgroups of semi-simple Lie groups, Osaka Math. Jour.3 (1966), 243–256, corrections ibid. 16, (1979), 295–299.Search in Google Scholar

[13] R. W. Sharpe, Di˙erential Geometry : Cartan’s Generalization of Klein’s Erlangen Program, Graduate Text Math., 166, Springer-Verlag, New York, Berlin, Heidelberg, 1997.Search in Google Scholar

[14] H.-P. de Saint Gervais, Uniformization of Riemann Surfaces. Revisiting a hundred year old theorem, E.M.S., 2016.10.4171/145Search in Google Scholar

[15] Y. Wakabayashi, Frobenius-Ehresmann structures and Cartan geometries in positive characteristic, arXiv:2109.02826.Search in Google Scholar

Received: 2021-11-29
Accepted: 2022-01-19
Published Online: 2022-02-20

© 2022 Indranil Biswas et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Scroll Up Arrow