Deformations of extremal toric manifolds

TL;DR

This paper establishes a combinatorial criterion for complex deformations of extremal toric Kähler manifolds to admit extremal metrics, leading to new constant scalar curvature metrics on specific blow-ups.

Contribution

It introduces a novel combinatorial criterion based on the fan structure to identify deformations with extremal metrics in toric manifolds.

Findings

Provides a criterion ensuring existence of extremal metrics on deformations

Finds new CSC metrics on 4-point blow-ups of ^1 imes ^1

Connects combinatorial fan data with geometric metric properties

Abstract

Let $X$ be a compact toric extremal K\"ahler manifold. Using the work of Sz\'ekelyhidi, we provide a combinatorial criterion on the fan describing $X$ to ensure the existence of complex deformations of $X$ that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of $\C\P^1\times\C\P^1$.