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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access September 2, 2017

Criterion for connections on principal bundles over a pointed Riemann surface

Indranil Biswas EMAIL logo
From the journal Complex Manifolds


We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.

MSC 2010: 53B15; 14H60; 32A27


[1] B. Anchouche, H. Azad and I. Biswas, Harder-Narasimhan reduction for principal bundles over a compact Kähler manifold, Math. Ann. 323 (2002), 693-712.10.1007/s002080200322Search in Google Scholar

[2] 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

[3] H. Azad and I. Biswas, On holomorphic principal bundles over a compact Riemann surface admitting a flat connection, Math. Ann. 322 (2002), 333-346.10.1007/s002080100273Search in Google Scholar

[4] V. Balaji, I. Biswas and D. S. Nagaraj, Krull-Schmidt reduction for principal bundles, Jour. Reine Angew. Math. 578 (2005), 225-234.Search in Google Scholar

[5] I. Biswas and T. L. Gómez, Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom. 33 (2008), 19-46.10.1007/s10455-007-9072-xSearch in Google Scholar

[6] I. Biswas and A. J. Parameswaran, On the equivariant reduction of structure group of a principal bundle to a Levi subgroup, Jour. Math. Pures Appl. 85 (2006), 54-70.10.1016/j.matpur.2005.10.007Search in Google Scholar

[7] I. Biswas and V. Heu, Non-flat extension of flat vector bundles, Internat. Jour. Math. 26 (2015), no. 14, 1550114, 6 pp.10.1142/S0129167X15501141Search in Google Scholar

[8] I. Biswas, A. Dan, A. Paul and A. Saha, Logarithmic connections on principal bundles over a Riemann surface, arXiv:1705.00852 [math.AG].Search in Google Scholar

[9] A. Borel, Linear algebraic groups, Second edition, Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991.10.1007/978-1-4612-0941-6Search in Google Scholar

[10] P. Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin- New York, 1970.10.1007/BFb0061194Search in Google Scholar

[11] F. Digne and J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, 1991.10.1017/CBO9781139172417Search in Google Scholar

[12] P. Griffiths and J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics. Wiley-Interscience, New York, 1978.Search in Google Scholar

[13] J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, Vol. 21, Springer-Verlag, New York, Heidelberg, Berlin, 1987.Search in Google Scholar

[14] S. Lang, Algebra, Revised third edition, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002.10.1007/978-1-4613-0041-0Search in Google Scholar

[15] C. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95.10.1007/BF02699491Search in Google Scholar

[16] A. Weil, Généralisation des fonctions abéliennes, Jour. Math. Pure Appl. 17 (1938), 47-87.Search in Google Scholar

Received: 2017-6-26
Accepted: 2017-8-10
Published Online: 2017-9-2
Published in Print: 2017-8-28

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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