An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons

TL;DR

This paper establishes effective criteria for weighted K-stability of polytopes, enabling the identification of new extremal Kähler metrics on semisimple principal toric fibrations through combinatorial conditions.

Contribution

It introduces verifiable sufficient conditions for weighted uniform K-stability, facilitating the discovery of new extremal Kähler metrics in low-dimensional cases.

Findings

Derived effective criteria for weighted K-stability

Provided new examples of extremal Kähler metrics

Enhanced understanding of stability conditions via polytopes

Abstract

The second author has shown that existence of extremal K\"ahler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal K\"ahler metrics it provides.