Suita Conjecture for a Complex Torus

TL;DR

This paper proves the Suita conjecture for complex tori, establishing a key inequality involving the Bergman kernel and logarithmic capacity, and discusses open problems for higher-genus Riemann surfaces.

Contribution

It extends the Suita conjecture to complex tori and provides insights into the case of higher-genus Riemann surfaces using elliptic function theory.

Findings

Suita conjecture holds for complex tori

Established inequality involving Bergman kernel and capacity

Discussed open problems for genus ≥2 surfaces

Abstract

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.