Examples of Toric Scalar-flat Kähler Surfaces with Mixed-type Ends

Abstract: Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat K\"ahler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact K\"ahler surfaces. We also provide an alternative construction of toric scalar-flat K\"ahler metrics with conical singularity along the toric divisor, following the approach of Abreu and Sena-Dias.