Toric separable geometries and extremal K\"ahler metrics

Abstract

This paper introduces the framework of (local) toric separable geometries, where toric separable K\"ahler geometries come in families, each uniquely determined by an underlying factorization structure. This unifying framework captures all known explicit Calabi-extremal toric K\"ahler metrics, previously constructed through diverse methods, as two distinct families corresponding to the simplest factorization structures: the product Segre and the Veronese factorization structure. Crucially, the moduli of typical factorization structures has a positive dimension, revealing an immensely rich and previously untapped landscape of toric separable geometries. The scalar curvature of toric separable geometries is computed explicitly, necessary conditions for the PDE governing extremality are derived, and new extremal metrics are obtained systematically. In particular, for a $2m$-dimensional toric separable geometry, solutions of the PDE are necessarily $m$-tuples of rational functions of one variable belonging to an at most $(m+2)$-dimensional real vector space and whose denominators are determined by the factorization structure. Toric separable geometries serve as a separation of variables technique and are well-suited for the analytic study of geometric PDEs.