/sci/ - Science & Math

>>11536167
>>11536175
>>11536185
Given a line bundle [math]E\rightarrow\Sigma_g[/math] you can count the number of non-constant holomorphic sections [math]\Gamma(\Sigma_g,E)[/math] via the holomorphic Euler characteristic [math]\chi(E)[/math]. If we write, WLOG, [math]E\cong L^k \otimes \mathcal{K}^s[/math] where [math]L[/math] is the Chern generator and [math]\mathcal{K}[/math] is the canonical line bundle with [math]k\in\mathbb{Z}[/math] and [math]s\in\frac{1}{2}\mathbb{Z}[/math] (existence of Spin structure), we see that by Riemann-Roch [math]\chi(E) = \operatorname{dim}H^1(\Sigma,E) - \operatorname{dim}H^0(\Sigma,E) = \operatorname{deg}(E) + 1-g = k + (2s-1)(g-1)[/math]. If applicable, you can then obtain [math]\chi(E)[/math] from [math]H^0(\Sigma_g,E)[/math] from Kodaira vanishing. Hence generally speaking, on the torus we have [math]\chi(E) \sim k[/math] where [math]k[/math] is the Chern number of [math]E[/math].

>>8976412
>practical implementations
>topological quantum computation
Not for a few decades, kid.