J. Aust. Math. Soc.  76 (2004), 415-423
A note on the theta characteristics of a compact Riemann surface

Indranil Biswas
  School of Mathematics
  Tata Institute of Fundamental Research
  Homi Bhabha Road
  Bombay 400005
  India
  indranil@math.tifr.res.in


Abstract
Let $X$ be a compact connected Riemann surface and $\xi$ a square root of the holomorphic cotangent bundle of $X$. Sending any line bundle $L$ over $X$ of order two to the image of $\dim H^0(X, \xi\otimes L) - \dim H^0(X,\xi)$ in ${\mathbb Z}/2{\mathbb Z}$ 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 $X$.
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