Scalar-flat K\"ahler orbifolds via quaternionic-complex reduction

TL;DR

This paper classifies certain scalar-flat K"ahler 4-orbifolds with torus symmetry as quaternionic quotients, extending the understanding of their geometric structure through quaternionic-complex reduction techniques.

Contribution

It establishes a classification of asymptotically locally Euclidean scalar-flat K"ahler 4-orbifolds with torus symmetry as quaternionic quotients, linking them to quaternionic-complex reduction.

Findings

Any such orbifold is isometric to a quaternionic-complex quotient.

Compact anti-self-dual orbifolds with positive Euler characteristic are conformally equivalent to quaternionic quotients.

Provides a new perspective on the structure of scalar-flat K"ahler orbifolds with symmetry.

Abstract

We prove that any asymptotically locally Euclidean scalar-flat K\"ahler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of $k$-dimensional quaternionic projective space by a $(k-1)$-torus.