Semi-flat constant scalar curvature K\"ahler metric on elliptic surface
Abstract
We introduce and construct a novel type of canonical metric: the semi-flat constant scalar curvature K\"ahler (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 Ib or Ib*. These results contribute to a geometric uniformization program for elliptic surfaces.
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