Semi-flat constant scalar curvature Kähler metric on elliptic surface

Abstract

We introduce and construct a novel type of canonical metric: the semi-flat constant scalar curvature Kähler (semi-flat cscK) current, which naturally arises in Calabi-Yau fibrations. For a given elliptic surface $X$ with a holomorphic section, We explicitly construct the desired semi-flat cscK current and analyze its behavior along singular parts. We establish its uniqueness under the condition that $X$ possesses at least one singular fiber other than of type $I_b$ or $I_b^{\ast }$. These results contribute to a geometric uniformization program for elliptic surfaces.

50 pages, All feedback and corrections are highly appreciated