Examples of Toric Scalar-flat K\"ahler Surfaces with Mixed-type Ends

TL;DR

This paper constructs explicit examples of complete and singular toric scalar-flat K"ahler metrics on non-compact 4-manifolds, expanding the known classes of such geometries with detailed constructions.

Contribution

It provides explicit constructions of scalar-flat K"ahler metrics on unbounded toric symplectic 4-manifolds, including metrics with conical singularities, using novel and existing methods.

Findings

Explicit metrics on non-compact toric 4-manifolds

Construction of metrics with conical singularities

Extension of known methods to new classes of surfaces

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.