Semi-flat constant scalar curvature Kähler metric on elliptic surface
Zhenqu Wang, Zhenlei Zhang
TL;DR
The paper constructs a canonical semi-flat constant scalar curvature Kähler (cscK) current on minimal elliptic surfaces with a section, proving existence of an $H(X,\Phi,s)$-invariant current of fiber volume $1$ and scalar curvature $-3$, and establishing uniqueness under the presence of a singular fiber beyond $\mathrm{I}_b$ or $\mathrm{I}_b^*$. It develops the metric on the nonsingular locus via a universal model, extends it to the whole surface as a closed positive current using Harvey-Polking, and analyzes its cohomology class, decomposing it into algebraic and essential pieces to define a distinguished class $D_X$ and a one-parameter family $\eta(t)$ in the class $K_X+tD_X$ with explicit fiber-volume and curvature formulas. A concrete relation $K_X=(2g-2+\chi)[F]$ connects the canonical class to the fibration, enabling a precise scaling that yields a distinguished canonical current at $t_0=\frac{2g-2+\chi}{\pi d/3+\chi}$ via $t_0^{-1}\eta(t_0)$. The results contribute to a geometric uniformization program for elliptic surfaces by showing the current is unique up to $H(X,\Phi)$-action and by tying the metric data to the Mordell-Weil and Neron-Severi structures of $X$, potentially linking to generalized Kähler-Einstein metrics in the fibration setting.
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 $I_b^*$. These results contribute to a geometric uniformization program for elliptic surfaces.