No Access

Unipotent Schottky bundles on Riemann surfaces and complex tori

Carlos Florentino

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal

Search for more papers by this author

and

Thomas Ludsteck

IBM, Wilhelm-Fay-Strasse 30-34, 65936 Frankfurt, Germany

Search for more papers by this author

https://doi.org/10.1142/S0129167X14500566Cited by:1 (Source: Crossref)

Abstract

We study a natural map from representations of a free (respectively, free abelian) group of rank g in GL_r(ℂ), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (respectively, complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.

Keywords:

AMSC: 14H60, 14D20, 14J60

References

M. G. Atiyah, Proc. London Math. Soc. (3) 7(1), 414 (1957). Google Scholar
H. Azad and I. Biswas, Math. Ann. 322(2), 333 (2001), DOI: 10.1007/s002080100273. Crossref, Web of Science, Google Scholar
V. Balaji, Moduli Spaces and Vector Bundles, London Mathematical Society Lecture Note Series 359 (Cambridge University Press, Cambridge, 2009) pp. 2–28. Crossref, Google Scholar
I. Biswas and T. Gómez, Ann. Global Anal. Geom. 33, 19 (2008), DOI: 10.1007/s10455-007-9072-x. Crossref, Web of Science, Google Scholar
I. Biswas and S. Subramanian, Trans. Amer. Math. Soc. 356(10), 3995 (2004), DOI: 10.1090/S0002-9947-04-03567-6. Crossref, Web of Science, Google Scholar
C. Birkenhake and H. Lange , Complex Abelian Varieties , Grundlehren der Mathematischen Wissenschaften 302 ( Springer , Berlin , 2004 ) . Crossref, Google Scholar
A. Borel , Linear Algebraic Groups , 2nd edn. , Graduate Texts in Mathematics 126 ( Springer , Berlin , 1991 ) . Crossref, Google Scholar
M. Brion, Homogeneous bundles over abelian varieties, preprint (2011) , arXiv:1101.2771 . Google Scholar
H. Cartan and S. Eilenberg , Homological Algebra ( Princeton University Press , Princeton , 1956 ) . Google Scholar
C. Deninger and A. Werner, Number Fields and Function Fields — Two Parallel Worlds, Progress in Mathematics 239, eds. G. van der Geer, B. Moonen and R. Schoof (Birkhäuser, 2005) pp. 101–131. Crossref, Google Scholar
C. Deninger and A. Werner, Algebraic and Arithmetic Structures of Moduli Spaces, Advanced Studies in Pure Mathematics 58, eds. I. Nakamura and L. Weng (World Scientific, Singapore, 2010) pp. 1–26. Google Scholar
G. Faltings, Adv. Math. 198, 847 (2005), DOI: 10.1016/j.aim.2005.05.026. Crossref, Web of Science, Google Scholar
H. Farkas and I. Kra , Riemann surfaces ( GTM Springer , New York , 1992 ) . Crossref, Google Scholar
C. Florentino, Manuscripta Math. 105, 69 (2001), DOI: 10.1007/PL00005874. Crossref, Web of Science, Google Scholar
R. Gunning , Lectures on Riemann Surfaces ( Princeton Academic Press , Princeton , 1967 ) . Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 4, 5 (1960), DOI: 10.1007/BF02684778. Crossref, Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 8, 5 (1961). Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 11, 5 (1961). Google Scholar
A. Grothendieck, Éléments de géométrie algébrique: III. Étude cohomologique des faiscaeux cohérents, Seconde partie, 13 (1963) 5–91 . Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 20, 5 (1964), DOI: 10.1007/BF02684747. Crossref, Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 24, 5 (1965). Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 28, 5 (1966), DOI: 10.1007/BF02684343. Crossref, Google Scholar
A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 32, 5 (1967). Google Scholar
S. Lekaus, Vector bundles of degree zero over an elliptic curve, flat bundles and Higgs bundles over a compact Kähler manifold, Dissertation, Univ. Duisburg-Essen (2001) . Google Scholar
S. Lekaus, Trans. Amer. Math. Soc. 357, 4647 (2005), DOI: 10.1090/S0002-9947-04-03652-9. Crossref, Web of Science, Google Scholar
T. Ludsteck, Arch. Math. 92, 585 (2009), DOI: 10.1007/s00013-009-2932-9. Crossref, Web of Science, Google Scholar
T. Ludsteck, Arch. Math. 93, 323 (2009), DOI: 10.1007/s00013-009-0048-x. Crossref, Web of Science, Google Scholar
Y. Matsushima, Nagoya Math. J. 14, 1 (1959). Crossref, Google Scholar
B. Mitchell , Theory of Categories ( Academic Press , Boston, MA , 1965 ) . Google Scholar
A. Morimoto, Nagoya Math. J. 15, 83 (1959). Crossref, Google Scholar
J. Rotman , An Introduction to Homological Algebra , 2nd edn. ( Springer , 2009 ) . Crossref, Google Scholar
J.-P. Serre , Lie Algebras and Lie Groups , Lect. Notes in Mathematics 1500 ( Springer , 1992 ) . Crossref, Google Scholar
C. Simpson, Publ. Math. Inst. Hautes Études Sci. 75, 5 (1992), DOI: 10.1007/BF02699491. Crossref, Google Scholar
T. Szamuely , Galois Groups and Fundamental Groups , Cambridge Studies in Advanced Mathematics ( Cambridge University Press , Cambridge , 2009 ) . Crossref, Google Scholar