On the Levi-flats in Complex Tori of Dimension Two

Abstract

Compact Levi flat real analytic hypersurfaces in complex tori of dimension two are completely classified. The method is a combination of Siu’s $\bar{\partial }$-regularity theory on weakly pseudoconvex domains, an updated variant of the $L^2$ estimate for the $\bar{\partial }$ operator originated from Berndtsson–Charpentier, and basic arguments in complex analytic geometry.