Deformation for coupled K\"ahler-Einstein metrics

TL;DR

This paper studies how coupled K"ahler-Einstein metrics on Fano manifolds can be deformed, providing conditions for such deformations and exploring the effects of changing complex structures.

Contribution

It establishes a necessary and sufficient condition for deforming coupled K"ahler-Einstein metrics on Fano manifolds with holomorphic vector fields.

Findings

Derived a deformation criterion for coupled K"ahler-Einstein metrics.

Analyzed deformation behavior under complex structure variations.

Extended understanding of metric stability on Fano manifolds.

Abstract

The notion of coupled K\"ahler-Einstein metrics was introduced recently by Hultgren-WittNystr\"om. In this paper we discuss deformation of a coupled K\"ahler-Einstein metrics on a Fano manifold. In particular we obtain a necessary and sufficient condition for a coupled K\"ahler-Einstein metric to be deformed to another coupled K\"ahler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled K\"aher-Einstein metric on a Fano manifold when the complex structure varies.