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Journal of the London Mathematical Society Advance Access originally published online on October 16, 2009
Journal of the London Mathematical Society 2009 80(3):750-770; doi:10.1112/jlms/jdp053
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© 2009 London Mathematical Society

Moduli of bundles over rational surfaces and elliptic curves I: Simply laced cases

Naichung Conan Leung

The Institute of Mathematical Sciences, Department of Mathematics
The Chinese University of Hong Kong
Shatin, N.T.
Hong Kong
P.R. China
leung@ims.cuhk.edu.hk

Jiajin Zhang

Department of Mathematics
Sichuan University
Chengdu, 610000
P.R. China
Current address:
Institute of Mathematics
Johannes Gutenberg-University Mainz
Mainz 55099
Germany
jjzhang@scu.edu.cn

It is well known that del Pezzo surfaces of degree 9 – n one-to-one correspond to flat En bundles over an elliptic curve. In this paper, we construct ADE-bundles over a broader class of rational surfaces that we call ADE-surfaces, and extend the above correspondence to all flat G-bundles over an elliptic curve, where G is any simply laced, simple, compact and simply connected Lie group. In what follows, we will construct G-bundles for a non-simply laced Lie group G over these rational surfaces, and extend the above correspondence to non-simply laced cases.

2000 Mathematics Subject Classification 14J26 (primary), 14H60 (secondary).

The first author is partially supported by a Hong Kong RGC Grant from the Hong Kong Government. The second author is supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG.

Received November 13, 2008; revised May 27, 2009; published online October 16, 2009.


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