  The author proves that the generalized Suita conjecture holds for any complex
torus, which means that $ \alpha\pi K \geq c^2(\alpha\in\mathbb R)$, $c$ being
the modified logarithmic capacity and $K$ being the Bergman kernel on the
diagonal. The open problems for general compact Riemann surfaces with genus
$\geq2$ is also elaborated. The proof relies in part on elliptic function
theories.

Complex Variables