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A note on the theta characteristics of a compact Riemann surface

Part of: Differential topology (Co)homology theory

Published online by Cambridge University Press:  09 April 2009

Indranil Biswas
Indranil Biswas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India e-mail: indranil@math.tifr.res.in
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Abstract

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Let X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

MSC classification

Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 57R25: Vector fields, frame fields 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 415 - 424
Copyright
Copyright © Australian Mathematical Society 2004

References

[At]Atiyah, M. F., ‘Riemann surfaces and spin structures’, Ann. Sci. École Nor.Sup. 4 (1971), 47–62.CrossRefGoogle Scholar
[Mu1]Mumford, D., Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5 (Oxford University Press, London, 1970).Google Scholar
[Mu2]Mumford, D., ‘Theta characteristics of an algebraic curve’, Ann. Sci. École Nor. Sup. 4 (1971), 181–192.CrossRefGoogle Scholar
[Na]Natanzon, S. M., ‘Moduli of Riemann surface, Hurwitz-type spaces, and their superanalogs’, Russ. Mat. Surveys 54 (1999), 61–117.CrossRefGoogle Scholar