Question

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.

Answer 1

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.