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## Riemann’s theorem on theta

Is there a low-tech way to compute the chern class of the theta bundle, i.e. the bundle associated to theta divisor? Here, the theta divisor is the analytic set of holomorphic line-bundles in $Pic_{g-1}(\Sigma)$ which possesses a non-trivial holomorphic section. By low-tech I mean without much knowledge about algebraic geometry or homology theory. The reason why I ask is, that this gives a (in my opinion) nicer proof of Riemann's theorem, because any two holomorphic line bundles with the same chern class are translates on a torus.

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 Could you clarify what you mean by "compute the chern class..."? A trivial answer is that it's the fundamental class of the theta divisor, but I assume you don't mean that. What form of answer do you want? – Donu Arapura Jan 9 2011 at 13:47 One could characterize this class, up to scalars, as the unique class in $H^2 (Pic)=\Lambda^2 H^1(\Sigma)$ invariant under the mapping class group action of $Sp(2g;\mathbb{Z})$. Since $\Theta$ exists for all curves, $c_1(\Theta)$ must be invariant under monodromies in $M_g$, and must therefore be $Sp$-invariant. – Tim Perutz Jan 9 2011 at 15:28 No offense Tim, but I think your question is an example of 'one mans meat is another mans poison'. Personally I'm comfortable with what Donu said, but I find the terminology of your response high-tech. I also believe that the fact that in some sense the canonical bundle is the only bundle on the moduli space of curves is very non-trivial. – aginensky Jan 9 2011 at 17:01

The locus of holomorphic line bundles in $Pic_{g-1}(\Sigma)$ with a nontrivial holomorphic section is equivalently characterized as the image of $u_{g-1} : Sym^{g-1} \Sigma \to Pic_{g-1}(\Sigma)$ under the Abel-Jacobi map. In section 4 of chapter 1 of the book of Arbarello-Cornalba-Griffiths-Harris, the theta divisor is defined via the Riemann theta function. Then in the next section, they give a pretty low-tech proof of the fact that the cohomology classes of the theta divisor and of the image of $u_{g-1}$ agree (a special case of the Poincare formula), by reducing to the case where $\Sigma$ is a product of elliptic curves. Finally, they prove Riemann's theorem precisely in the way that you suggest.